AWM Research Symposium 2019

April 6-7, 2019 at Rice University, Houston, Texas

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ACxx: Women in Algebraic Combinatorics, I

Organizers: Elizabeth Niese, Elizabeth Drellich

Heather Russell

Comparing The Web And Specht Bases For Symmetric Group Representations

Abstract: We are interested in two different bases for the irreducible representation of the symmetric group corresponding to the partition [n, n, n]: the web basis and the Specht basis. The web basis arises in quantum representation theory and knot invariants. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representations. In this talk, we will describe these bases and discuss progress towards determining the entries of the transition matrix between them.

Anna Weigandt

Bumpless Pipe Dreams And Alternating Sign Matrices

Abstract: Lam, Lee, and Shimozono introduced bumpless pipe dreams to study back stable Schubert calculus. In particular, Schubert polynomials can be expressed as a weighted sum over bumpless pipe dreams in a square grid. Working from a different perspective, Lascoux gave a formula for Grothendieck polynomials as a sum over alternating sign matrices. We show that alternating sign matrices are in natural bijection with bumpless pipe dreams. Restricting to the lowest degree terms of Lascoux’s formula recovers the LLS formula for Schubert polynomials. We also discuss how to use the pipe dream perspective to compute keys of ASMs.

Bridget Eileen Tenner

Enumerations In Coxeter Groups

Abstract: Coxeter groups are of significant interest to communities in combinatorics, algebra, and geometry. Their structures and properties are both deeply beautiful and still not entirely understood. In this talk, we will explore some of the many enumerative aspects of these objects. We will count elements with desirable properties, we will give size orderings to certain features of all group elements, and we will relate some of these statistics in ways that give bounds and rankings − if not exact values − to their sizes. Our primary tools will come from leveraging different group presentations against each other, and interpreting element properties at the level of generator representations.

Sarah Bockting-Conrad

Tridiagonal Systems Of Racah Type

Abstract: In earlier papers, we have investigated the theory of tridiagonal systems by considering two linear transformations Δ, Ψ, which are associated with the tridiagonal system Φ and act on the underlying vector space in an attractive way. Using these transformations, we have been able to describe the situation in great detail for the q-Racah class of tridiagonal systems. In this talk, we describe the corresponding theory for the Racah class of tridiagonal systems and discuss the role of the universal enveloping algebra U(sl₂) in this theory. We do so by again considering the transformations Δ, Ψ, and describing their actions on various decompositions of the underlying space in detail. This leads to a number of interesting relations involving the transformations Δ, Ψ, and other maps associated with Φ.

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ACxx: Women in Algebraic Combinatorics, II

Organizers: Elizabeth Niese, Elizabeth Drellich

Maria Monks Gillespie

A Characterization Of Queer Supercrystals

Abstract: We provide a new combinatorial characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al.. This characterization is a combination of the local queer relations recently introduced by Assaf and Oguz, with further axioms and a new graph G characterizing the relations between the type A components of the queer crystal. We obtain a combinatorial description of the graph G on the type A components by providing explicit combinatorial rules for the odd queer operators, in terms of simple operations on certain highest weight words. This is joint work with Graham Hawkes, Wencin Poh, and Anne Schilling.

Kassie Archer

Pattern avoidance, cycle type, and characters of the symmetric group

Abstract:We say that a permutation π, written in its one-line notation, avoids a given pattern if there is no subsequence of π that appears in the same relative order as π. The combinatorial structure of pattern-avoiding permutations has been studied for several decades, but questions the cycle structure of these permutations remain largely open (with some notable exceptions). In particular, it remains open to enumerate the set of permutations comprised of a single cycle that, when written in its one-line notation, avoids a single pattern of length 3. Since pattern-avoiding permutations are inherently combinatorial objects, with no regard for the bijective or algebraic nature of a permutation, the study of their cycle structure is difficult. Investigating the cycle structure of these pattern-avoiding permutations is more than just an interesting exercise. The study of the cycle structure of permutations with certain one-line behavior has been shown to have applications to other fields of mathematics. We will talk about some results regarding pattern-avoidance and cycle type and some applications of these ideas to the study of characters of the symmetric group.

Martha Yip

A Minimaj-Preserving Crystal On Ordered Multiset Partitions

Abstract: One of the main objects of study in the Delta Conjecture is the polynomial Val_n,k(x; q, t). In this talk, we will give some background on the conjecture, and focus on two combinatorial aspects of the specialization of Val at q = 0. We will give a proof that the polynomial is Schur-positive via the use of crystal bases, and time allowing, we will show how the crystal structure leads to a bijective proof that the major index and the minimaj statistic on multiset partitions are equidistributed. This is joint work with Benkart, Colmenarejo, Harris, Orellana, Panova, and Schilling.

Margaret Readdy

Geometric Proofs Of Some Combinatorial Identities Of Morel

Abstract: Using the algebraic and geometric combinatorics of the permutahedron, we give proofs of combinatorial identities which arise in the technical heart of Morel’s computation of the intersection cohomology of Shimura varieties. No prior background will be assumed.
This is joint work with Richard Ehrenborg and Sophie Morel.

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Analysis and Numerical Methods for Kinetic Transport and Related Models, I

Organizer: Liu Liu

Kit Newton* and Q. Li

Two-Level Markov Chain Monte Carlo Methods For The Inverse Radiative Transfer Equation

Abstract: Optical tomography is the process of reconstructing properties of biological tissue using measurements of incoming and outgoing light intensity. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE). We study this problem in the Bayesian framework, meaning that we examine the posterior distribution of the scattering coefficient after the measurements have been taking. Sampling from this distribution is computationally expensive, so we employ a two-level MCMC technique, using the DE posterior distribution to make sampling from the RTE posterior distribution computationally feasible.

Liu Liu

Micro-Macro Decomposition Based Asymptotic-Preserving Schemes And Moments Conservation For Collisional Kinetic Equations

Abstract: In this talk, we first extend the micro-macro decomposition method for multiscale kinetic equations from the BGK model to general collisional kinetic equations, including the Boltzmann and the Fokker-Planck Landau equations. The main idea is to use a relation between the (numerically stiff) linearized collision operator with the nonlinear quadratic ones, the latter’s stiffness can be over- come using the BGK penalization method of Filbet and Jin for the Boltzmann, or the linear Fokker-Planck penalization method of Jin and Yan for the Fokker − Planck Landau equations. Such a scheme allows the computation of multiscale collisional kinetic equations efficiently in all regimes, including the fluid regime in which the fluid dynamic behavior can be correctly computed even without re- solving the small Knudsen number. A distinguished feature of these schemes is that although they contain implicit terms, they can be implemented explicitly. These schemes preserve the moments (mass, momentum and energy) exactly thanks to the use of the macroscopic system which is naturally in a conservative form. We further utilize this conservation property for more general kinetic systems, using the Vlasov-Ampére and Vlasov-Ampére-Boltzmann systems as examples. The main idea is to evolve both the kinetic equation for the probability density distribution and the moment system, the latter naturally induces a scheme that conserves exactly the moments numerically if they are physically conserved. This is a joint work with Irene Gamba and Shi Jin.

Anna Szczekutowicz

Velocity Dependent Coulomb Logarithm In The Landau Limit Of The Boltzmann Equation

Abstract: We propose to study numerical and analytically recent developments on a new spectral formulation of the Fokker-Planck-Landau (FPL) operator with velocity dependent Coulomb logarithm that fully revisits the derivation of the FPL limit operator as a grazing collision form starting from the classical Boltzmann operator form for binary elastic interactions where the transition probability rates depending on the scattering angle dominates the glancing limiting dynamics. The key problem is that using Rutherford cross section forms in the Boltzmann collision operator results in a logarithmic singularity. A revised derivation of the scattering cross section can be obtained noticing that the scattering angle depends on the impact parameter with is proportional to the total cross section. This observation yields an impact parameter depending on the angle and the relative speed resulting in a nonlinear relation between the total cross section and the relative particle speed, with a scattering angle with an angular cut-off cut off condition depending on Debye length λ_D given the range of charged particles being screened from one another. Then, the resulting cross section has a stronger physical meaning as the angular cut-off condition that depends on λ_D, and thus differs significantly from the one used in the classical formal derivation where the cut-off is given by a small constant that is completely unrelated to the Debye length between the two-body interaction.

Milana Pavic-Colic

Some Analytical Aspects For The Boltzmann System Of Monatomic Gas Mixtures: The Cauchy Problem, Generation And Propagation Of Polynomial And Exponential Moments

Abstract: We solve the Cauchy problem for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions in three dimensions. More precisely, we show existence and uniqueness of the vector value solution by means of an existence theorem for ODE systems in Banach spaces under the transition probability rates assumption corresponding to hard potentials rates γ ∈(0, 1], with a bounded angular section modeled by a bounded function of the scattering angle. The initial data for the vector valued solutions needs to be a vector of non-negative measures with finite total number density, momentum and strictly positive energy, as well as to have finite L¹_2+2γ(ℝ³)- integrability corresponding to a sum across each species of dimensionless 2+2γ-polynomial weighted norms depending in the corresponding mass fraction parameter for each species, referred as to the scalar polynomial moment of order 2 + 2γ. The scalar polynomial moments related to the whole mixture are independent of mass density units, a crucial assumption to obtain a new sharp form of Povzner estimate for the mixture. In addition, there is no assumption on the finiteness of the initial system associated scalar entropy. This set of initial data assumptions allows for some of the species to be initially a singular mass. The existence and uniqueness proof relies on a new angular averaging lemma adjusted to vector values solution that yield a Povzner estimate with explicit constants that decay with the order of the corresponding dimensionless scalar polynomial moment. In addition, such initial data yields global generation of such scalar polynomial moments at any order as well as their summability of moments to obtain estimates for corresponding scalar exponentially decaying high energy tails, referred as to scalar exponential moments associated to the system solution. Such scalar polynomial and exponential moments propagate as well.

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Analysis and Numerical Methods for Kinetic Transport and Related Models, II

Organizer: Liu Liu

Li Wang

Primal Dual Methods For Wasserstein Gradient Flows

Abstract: In this talk, I will introduce a variational method for nonlinear equations with a gradient fow structure, which arise widely in applications such as porous median fows, material science, animal swarms, and chemotaxis. Our method builds on the JKO framework and a reformulation of the Wasserstein distance into a convex optimization with a linear PDE constraint. As a result, we end up with one nested structure of optimization problem with two time scales, and we adopt a recent primal dual three operator splitting scheme. Thanks to the variational structure, our method has a built-in positivity preserving, entropy decreasing properties, and overcomes stability issue due to the strong nonlinearity and degeneracy. Upon discretization of the PDE constraint, we also show the Γ−convergence of the fully discrete optimization towards the semi-discrete JKO scheme. This is a joint work with Jose Carrillo, Katy Craig, and Chaozhen Wei.

Sona Akopian

On Global Weak L^p Solutions To A Class Of Boltzmann Equations With An Angle-Potential Concentrated Collision Kernel

Abstract: We study a class of Boltzmann equations, parameterized by small ε > 0, whose collision kernels are in the form
of a Dirac mass that is centered at small collision angles and relative velocities. This pseudo-Maxwellian kernel was
created in a paper of Bobylev and Potapenko for a Monte Carlo scheme of the Boltzmann and Landau equations.
Such a collision kernel is a big advantage because its angular average over the sphere is finite and has no singularities
that would otherwise be present in the traditional kernel. This allows us to split the collision operator into its gain
and loss parts, at which point we can perform analyses and L^p estimates, motivated by the works of Desvillettes,
Lions and Villani as well as Alonso, Gamba and Taskovic.

Jingwei Hu

A Discontinuous Galerkin Fast Spectral Method For The Multi-Species Boltzmann Equation

Abstract:We propose a fast Fourier spectral method for the multi-species Boltzmann collision operator and couple it with the discontinuous Galerkin method in the physical space to obtain a highly accurate deterministic solver for the full Boltzmann equation of mixtures. Various benchmark examples are simulated and compared with direct simulation Monte Carlo (DSMC) solutions. Joint work with S. Jaiswal and A. Alexeenko.

Yingda Cheng

An Adaptive High-Order Piecewise Polynomial Based Sparse Grid Collocation Method With Applications

Abstract: In this talk, we present the construction of adaptive sparse grid collocation methods based on interpolatory MRA onto general arbitrary order piecewise polynomial space. The motivation is to compute high-dimensional problems with reduced computational cost, while achieving high order accuracy with a natural framework for adaptivity. Theoretical justification will be provided, and applications in stochastic and partial differential equations are considered.

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Applied and Computational Harmonic Analysis, I

Organizers: Julia Dobrosotskaya, Xuemei Chen

Emily J King* with Rafael Reisenhofer and Johannes Kiefer, Georg Heygster, Zhen Li, & Wang-q Lim

Edge, Ridge, and Blob Detection With Symmetric Molecules

Abstract: In this talk a novel approach to the detection and characterization of edges, ridges, and blobs in two-dimensional images which exploits the symmetry properties of directionally sensitive analyzing functions in multiscale systems that are constructed in the framework of α-molecules−generalization of shearlets − will be presented. The proposed feature detectors are inspired by the notion of phase congruency, stable in the presence of noise, and by definition invariant to changes in contrast. It will also be shown that the behavior of coefficients corresponding to differently scaled and oriented analyzing functions can be used to obtain a comprehensive characterization of the geometry of features in terms of local tangent directions, widths, and heights. The accuracy and robustness of the proposed measures will be validated and compared to various state of the art algorithms in extensive numerical experiments in which sets of clean and distorted synthetic images that are associated with reliable ground truths will be considered. To further demonstrate the applicability, it will be shown that the proposed ridge and edge measures can be used to detect flame fronts, that the proposed ridge measure can be used to detect and characterize blood vessels in digital retinal images, and that the proposed blob measure can be applied to automatically count the number of cell colonies in a Petri dish.

Jing Qin

High-Resolution Fluorescence Microscopy Image Deconvolution

Abstract:

Yi Wang

ConceFT: Concentration Of Frequency And Time Via A Multitapered Synchrosqueezed Transform

Abstract: Time-frequency representations provide a powerful tool for the analysis of time series signals. Techniques that decompose the time-dependent signals into multiple oscillatory components, with time-varying amplitudes and instantaneous frequencies are very appealing and have been shown to be useful in a wide range of applications including geophysics, biology, medicine, finance and social dynamics. In this talk, I’ll give an introduction to time-frequency representations and review existing methods for the previously described decomposition. Then I’ll present a new method that applies the multitapering with synchrosqueezed transform. Numerical experiments as well as a theoretical analysis will be demonstrated to assess its effectiveness.

Rongrong Wang

A Simple Nonlinear Dimension Reduction Technique For High Dimension Data Visualization

Abstract: In this talk, we present the construction of adaptive sparse grid collocation methods based on interpolatory MRA onto general arbitrary order piecewise polynomial space. The motivation is to compute high-dimensional problems with reduced computational cost, while achieving high order accuracy with a natural framework for adaptivity. Theoretical justification will be provided, and applications in stochastic and partial differential equations are considered.

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Applied and Computational Harmonic Analysis, II

Organizers: Julia Dobrosotskaya, Xuemei Chen

Anna Ma

The Kaczmarz Algorithm For Multiple Measurement Vectors

Abstract: Nowadays, data is exploding at a faster rate than computer architectures can handle. For that reason, mathematical techniques to analyze large-scale data must be developed. Stochastic iterative algorithms in particular have gained recent interest due to their low memory footprint and adaptability for single measurement vector (SMV) models. While SMV models have been widely studied, there is a surging interest in addressing the multiple measurement vectors (MMV) problem. In the MMV setting, more than one measurement vector is available and the multiple signals to be recovered share some properties such as a common support. Applications in which MMV is a naturally occurring phenomenon include online streaming, medical imaging, and video recovery. In this talk, we will study the Randomized Kaczmarz algorithm for solving corrupt large-scale linear systems with multiple measurement vectors.

Wenjing Liao^*, Mauro Maggioni, Ha Dang and Yingjie Liu

Learning Low-Dimensional Manifolds, Functions On Manifolds And PDEs From Data

Abstract: Abstract: I will present methods to learn low-dimensional structures, such as low-dimensional manifolds, functions on manifolds, and parametric partial differential equations with low complexity, from data. When data are sampled on a low-dimensional manifold embedded in a high-dimensional space, we build efficient representations of these data for the purpose of compression and inference. We prove performance guarantees showing that our learning rate is cursed by the intrinsic dimension of the manifold, instead of the dimension of the ambient space. When data are sampled from a parametric PDE with sparse parameters, we develop numerical techniques to recover the underlying parameters from noisy data.

Longxiu Huang

Dynamic Sampling

Abstract: Dynamical sampling is a new area in sampling theory that deals with signals that evolve over time under the action of a linear operator. There are lots of studies on various aspects of the dynamical sampling problem. However, they all focus on uniform discrete time-sets T ⊂ {0, 1, 2, . . . , }. In our study, we concentrate on the case T = [0, L]. The goal of the present work is to study the frame property of the systems { A^t g : g ∈ G, t ∈ [0, L]}. To this end, we also characterize the completeness and Besselness properties of these systems.

Karamatou Yacoubou-Djima

Diffusion Frames On Graphs

Abstract:Multiscale (or multiresolution) analysis is used to represent signals or functions at increasingly high resolution. In this talk, we construct frame multiresolution analyses (MRA) for \(L^2\)-functions of spaces of homogeneous type. In this instance, dilations are represented by operators that come from the discretization of a compact symmetric diffusion semigroup. The eigenvectors shared by elements of the compact symmetric diffusion semigroup can be used to define an orthonormal MRA for \(L^2\). We introduce several frame systems that generate an equivalent MRA, notably composite diffusion frames, which are built with the composition of two similar compact symmetric diffusion semigroups.

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Braid Groups and Quantum Computing

Organizers: Colleen Delaney, Jennifer Vasquez, Helen Wong

Jennifer Vasquez

Qubit Braid Group Representations

Abstract: We investigate solutions to the Yang Baxter which give rise to representations of the braid group. Certain solutions whose images are unitary and of the correct size may be useful in models of topological quantum computing. In this talk, we will present results about the images of some families of braid group representations

Colleen Delaney

Fusion Categories And Quantum Computing

Abstract: Unitary modular tensor categories (UMTCs) are algebraic models for anyons in (2+1)D topological phases of matter, which are the basis of the topological approach to quantum computation. Generalizations of UMTCs that either weaken the modularity condition or enrich the category with symmetry also provide meaningful models for topological phases and their phenomena. Moreover, the potential application of these systems for quantum information processing can be analyzed in terms of the fusion categories that model them.

Iris Cong

Universal Quantum Computation With Gapped Boundaries

Abstract: In this talk, I will discuss topological quantum computation with gapped boundaries of two-dimensional topological phases. I will first introduce the algebraic framework for topological quantum computation and gapped boundaries. Next, I will present systematic methods to encode quantum information topologically using gapped boundaries, and to perform topologically protected operations on this encoding. In particular, I will introduce a new computational primitive of topological charge measurement and present a symmetry-protected implementation of this primitive. Throughout the talk, a concrete physical example – the case of bilayer fractional quantum Hall 1/3 systems [mathematically, D(Z₃)], will be discussed. For this example, we have a qutrit encoding and an abstract universal gate set. If a practical implementation is found for the required topological charge measurement, these boundaries will give rise to a direct physical realization of a universal quantum computer based on a purely Abelian topological phase.

Julia Plavnik

Low-Dimensional Representations Of The Three Component Loop Braid Group

Abstract: The loop braid group LBn is the motion group of n oriented, unlinked circles in ℝ³. Recently, physical and topological applications have motivated the study of representations of LBn. Since the braid group Bn is a subgroup of LBn, one interesting question is when a representation of Bn❑ can be extended to a representation of LBn. In this talk, we will start by introducing the three component loop braid group LBn and then we will discuss some advances on the study of extensions of low-dimensional braid group representations to LBn. This talk is based on a joint work with P. Bruillard, L. Chang, S-M. Hong, E. Rowell and M. Sun (J. Math. Phys. 2015)

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Combinatorial Algebra

Organizers: Christine Berkesch, Laura Felicia Matusevich

Laura Escobar

Wall-Crossing Phenomena For Newton-Okounkov Bodies

Abstract: A Newton-Okounkov body is a convex set associated to a projective variety, equipped with a valuation. These bodies generalize the theory of Newton polytopes. Work of Kaveh-Manon gives an explicit link between tropical geometry and Newton-Okounkov bodies. We use this link to describe a wall-crossing phenomenon for Newton-Okounkov bodies. As an application we show how the wall-crossing formula for the tropicalization of Gr(2, n) is an instance of our phenomenon for Newton-Okounkov bodies. This is joint work with Megumi Harada.

Josephine Yu

Positivity Hyperbolic Varieties And Tropical Geometry

Abstract: A multivariate complex polynomial is called stable if any line in any positive direction meets its hypersurface only at real points. Stable polynomials have close relations to polymatroids and hyperbolic programming. We will discuss a generalization of this property to algebraic varieties of codimension larger than one. We will show that the tropicalization of these varieties, which we call positively hyperbolic varieties, have a nice structure which generalizes polymatroids (generalized permutohedra). In some cases such as curves, binomial ideals, and Bergman fans, we give a complete characterization of tropicalizations of positively hyperbolic varieties. This is based on joint work with Felipe Rincon and Cynthia Vinzant.

Jessica Sidman

Rigidity Theory And Algebraic Matroids

Abstract: Consider a framework consisting of fixed length bars attached at flexible joints. The central question in rigidity theory is to determine if the resulting framework is rigid or flexible. The combinatorics of a framework can be encoded by a graph, and a famous theorem of Asimow and Roth shows that if we fix a graph then either all generic realizations are locally rigid or all generic realizations are locally flexible. Hence, generically local rigidity is a combinatorial property. The minimally locally rigid graphs in dimension d are the bases of a matroid which can be realized as a linear matroid associated to the classical “rigidity matrix” or as the algebraic matroid associated to the Cayley-Menger variety. In this talk we will relate work on stresses on frameworks of White and Whitely derived from the rigidity matrix to the algebraic point of view. This is joint work with Zvi Rosen, Louis Theran, and Cynthia Vinzant.

Patricia Jacobs Klein

Edge ideals and liaison theory

Abstract: Let \(R = K[x_1, . . . , x_n]\) where K is a field, and let I and J be homogeneous ideals of R. Then I and J are said to be directly G-linked by the ideal L if L is a Gorenstein ideal with \((L : I) = J\) and \((L : J) = I\). The set of all ideals G-linked to I in finitely many steps is called the G-liason class of I. It is of particular interest which ideals are in the G-liason class of a complete intersection. We will discuss the special case of edge ideals, which are rich in algebraic and combinatorial data.

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Combinatorial Commutative Algebra, I

Organizers: Sara Faridi, Susan Morey

Jenna Rajchgot

Regularity Of Schubert Determinantal Ideals And Related Ideals

Abstract: Schubert determinantal ideals are polynomial ideals associated to partial permutation matrices. They are generated by certain minors of a generic matrix of variables. For example, each classical determinantal ideal is a Schubert determinantal ideal. I will explain how to obtain some combinatorial formulas for the Castelnuovo-Mumford regularity of Schubert determinantal ideals, and related ideals. This is joint work in progress with Yi Ren and Anna Weigandt

Kuei-Nuan Lin

Blow-Up Algebras Of Three-Dimensional Ferrers Diagrams

Abstract: In this talk, I will define the blow-up algebras relate to 3-dimensional Ferrers Diagram. In the joint work with Yi-Huang Shen, we are able to find the defining equations of these algebras. We then investigate the Castelnuovo-Mumford regularity and the multiplicity of the special fiber ring. We are able to provide direct formulas or bounds of these two important invariants in different situations.

Louiza Fouli

Initially Regular Sequences

Abstract: For an arbitrary ideal I in a polynomial ring R we define a new notion of initially regular sequences on R/I . These sequences share properties with regular sequences, and are relatively easy to construct using information obtained from the initial ideal of I. We will discuss situations where initially regular sequences are also regular sequences. These new sequences also give a lower bound on the depth of R/I and we will provide a concrete description of these sequences that often realize the bound. This is joint work with Tài Hà and Susan Morey.

Susan Cooper

Monomial Ideals & Symbolic Powers

Abstract: Relating regular and symbolic powers of homogeneous ideals has played a central theme in numerous problems. In this talk we will explore techniques developed to look at symbolic powers of monomial ideals. Topics surveyed will include linear programs and the symbolic polyhedron. We will highlight invariants such as the initial degree of a homogeneous ideal and the Waldschmidt constant which is an asymptotic invariant used to find failure of containments between symbolic and regular powers.

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Combinatorial Commutative Algebra, II

Organizers: Sara Faridi, Susan Morey

Selvi Beyarslan

Algebraic Invariants of Weighted Oriented Paths and Cycles

Abstract:
In this talk, we consider edge ideals of weighted oriented graphs. We first introduce a general lower bound in terms of the induced matching number of the underlying graph and then compute regularity and projective dimension of oriented paths and cycles with non-trivial weights (at least one vertex has a weight ≥2). This is a joint work with Jennifer Biermann, Kuei-Nua Lin, and Augustine O’Keefe.

Mavada Shahada

Sub-Additivity Property Of Syzygies Of Monomial Ideals Via Lattice Complements

Abstract: It is known that the sub-additivity property for the maximal degrees of the syzygies of a graded ideal I of a polynomial ring S = K[x₁, . . . , x_n] fails to hold in general, while, under restrictive conditions, many cases have been known to hold. On the other hand, the problem is still wide open for the class of monomial ideals and has been investigated by several researchers and using different approaches. We will reformulate the problem in terms of finding ‘Lattice Complements’. In this talk, we will consider special cases and build some examples to show the efficiency of this method in proving the sub-additivity conjecture for some monomial ideals.

Sonja Mapes* and Kuei-Nuan Lin

Computing Projective Dimension Of Monomial Ideals Via Associated Hypergraphs And Icm-Lattices

Abstract: Given a square-free monomial ideal I in a polynomial ring R over a field k, we would like to know the projective dimension of R/I. We recall the definition of the lcm-lattice of a monomial ideal introduced by Gasharov, Peeva and Welker, and the definition of the dual hypergraph of a square-free monomial ideal introduced by Kimura, Terai and Yoshida. Our work focuses on the relationship between the lcm-lattice and the dual hypergraph of a given square-free monomial ideal. We use the properties of lcm-lattice to find whether two different dual hypergraphs have the same projective dimension, and thus are able to extend some of the results by Lin and Mantero which compute the projective dimensions for ideals with certain hypergraphs.

Aihua Li

Zero Divisor Graphs of Matrices over Commutative Rings

Abstract:Let R be a commutative ring with identity 1 and T be the non-commutative ring of all n by n upper triangular matrices over R. In this talk, the zero divisor graph of T will be introduced. Some basic graph theory properties are given, including determination of the girth and diameter. The structure of such a graph is discussed and bounds for the number of edges are given. In the case that T is a 2 by 2 upper triangular matrix ring over a finite integral domain, the structure of the graph is fully determined. In particular, an explicit formula for the number of edges is given.

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Current Challenges in Mathematical Biology

Organizer: Renee Dale

Renee Dale

Identifying Critical Nodes From A Predicted Biological Network Using A Mathematical Model

Abstract: Computational analysis of gene expression in biological organisms produces a set of predictions of how components interest with one another. However, experimental testing and validation of these predictive relationship remains a challenge. Validation requires understanding of the function of both individual components as well as possible interactions between multiple components. We have found that modeling a small networks using a simplified system of ODEs allows us to predict critical needs and the dynamic behaviors of the system with space in silico data. These small network models greatly reduce the number and type of experiments necessary to predict whole-networks dynamics. How this method could be expended to work at a large scale is uncertain to us. Additional methods, such as sensitivity analysis or minimum-path techniques, are required to identity a minimum number of critical nodes as targets for experimental validation for very large networks.

Raffeal Bennett

Modelling The Conformational Behavior Of Protein Drug Templates In Industrial Chromatography

Abstract: Recent advancements in cancer treatment are owed in part to the development of novel protein pharmaceuticals. This success has garnered interest or the discovery of rapid characterization and purification methods amenable to these proteins. Chromatography remains a common industrial separation technique due to being highly repeatable and scalable, which stems from a reliance on modelling formulas coupled with design of experiment (DoE) methodologies including the factorial method. An understanding of separation behavior often comes is often modelled by well-proven semi-empirical derivations such as the Snyder’s linear solvent strength relationship, log k´ = a – S*x, and the vanDeemter relationship, H = A + B/u + C*u. When analytical methods are scaled up or large-scale purification and understanding of loading is guided by the modeling of nonlinear Langmuir adsorption isotherms, as described by C = q*K*m/(1+K*m). Though these models have historically yielded predictive intuition, they tend to be an oversimplification (if not entirely misleading) when applied to proteins. Proteins have a peculiar property in which their shape can greatly influence how they interact with their environment. We know that proteins often change shape when analysis or purification is performed, but standard models do not account or this effect. Studies have already been performed that track the change in protein shape (or conformation) as separations are performed. This existing information can be coupled with purpose-driven analytical data acquired during protein purification on the industrial scale. Therefore, the aim of this work is to develop a model that quantitatively incorporates this conformational property into our mathematical understanding of protein behavior during industrial processing.

Jessica L. Burnett

Advances in Ecological Regime Shift Detection

Abstract:

Shu-Xia Tang

Routing (Operators) in Traffice Flow Modelling With Semilinear PDE

Abstract: A model for a unidirectional road traffic network is presented as a coupled system of first-order semi-linear PDEs. At every junction of the network, a routing operator is imposed taking into account the status of the network at the instantaneous time and (probably also) the history of the network status for routing decisions to be made. The system also requires to incorporate multi-commodity dynamics, with each commodity representing a flow with specific group of drivers heading to designated destination in the network. To detail it more, the main focus of this contribution is on the appropriate choices of routing patterns at the junctions of different pathways subject to different commodities. Several classes of routing choices using different information (delayed, real time) are considered and thus define the boundary condition on every considered link which could also be treated as boundary control applied to the system. We show the well-posedness of the network system subject to different classes of the proposed routing operators with rigid proof. More importantly, the flow of the network system can be routed in a way that large flow on specific links is as far as possible avoided.

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Education Partnerships: University Mathematics Faculty and K-12 Mathematics Teachers

Organizer: Evan Rushton

Anne Papakonstantinou

The Rice University School Mathematics Project: Its Evolution And Current Efforts

Abstract: The Rice University School Mathematics Project (RUSMP), established in 1987 with NSF funding, has developed an extensive array of Science, Technology, Engineering, Arts, and Mathematics (STEAM) programs, courses, and interventions for K-12 teachers, teacher leaders, administrators, and students. As a regional hub for Code.org, RUSMP offers a variety of coding and computer science workshops for teachers throughout the state to expand pre-college students’ access to computer science. RUSMP also provides support for university faculty and departments at Rice and at other universities as well as for non-profit and for-profit organizations across the country. Through its research and evaluation efforts, RUSMP contributes to the growing body of research on learning, teaching, and professional development in K-12 STEAM education. Through its wide-ranging work, RUSMP values its role as a vital partner in meeting the educational needs of the greater Houston community and beyond. Learn how RUSMP has evolved into an important regional STEM center, how it is funded, how it builds relationships, and how it continues to grow and impact the K-16 educational community.

Adem Ekmekci

Being Research-Based And Research-Minded in Helping K-12 Mathematics Education

Abstract: This talk will provide the audience with insights into how to engage in research in public schools and school districts and other non-profit education centers. We will start with discussing the questions of why the research is needed and how it is relevant to improving mathematics education in a metropolitan area such as Houston, home to many high-poverty urban schools and school districts. Featured research studies conducted by the Rice University School Mathematics Projects (RUSMP) and their implications for and impact on the education community will be shared. The vital role of conducting research in future grant writing will be also discussed. In addition, basics of education research and evaluation of education programs will be shared with mathematics faculty around the nation who are involved in pure mathematics research but may not be as familiar with education research.

Cymra Haskell

Mathematicians In K-12 Mathematics Education

Abstract: K-12 education is a big business involving a complex collaboration between many stakeholders including children, parents, teachers, school and school-district administrators, schools of education, and professionals in universities and the workforce. It is something many people care deeply about but may feel powerless to influence. In this talk we will discuss the important contributions research mathematicians have to offer K-12 education, how individual research mathematicians can involve themselves in K-12 education, and some of the potential benefits research mathematicians can receive as a result of such involvement.

Evan Rushton

USC Math Initiative’s Teacher Trainer Of Trainers Pilot Study

Abstract: One goal of the Joan Herman and Richard Rasiej Mathematics Initiative at USC is to improve elementary mathematics instruction. Additionally, there is a broader goal to improve educational research around measuring teacher learning. A pilot study conducted during the 2017-2018 school year explored an inexpensive mentoring program designed to increase the confidence, reduce the anxiety, increase the content knowledge, and improve the instructional strategies of elementary teachers in their teaching of mathematics. The pilot study uncovered unexpected benefits of the program as well as flaws in its original design. Investigators will articulate key design features for a successful implementation of the program and how they might pitch it for larger-scale implementation.

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Graph Theory

Organizers: Carolyn Reinhart, Kate Lorenzo

Daphne Liu

Colouring Of Generalized Signed Triangle-Free Planar Graphs

Abstract: Assume G is a graph. We view G as a symmetric digraph, where each edge xy is replaced by two opposite arcs e = (x, y) and e^-1 = (y, x). Assume S is an inverse closed subset of permutations of the set of positive integers. We say G is S-k-colourable if for any mapping σ : E(G) → S, there is a mapping f : V (G) → [k] such that for each edge e = (x, y) of D, σ_e(f(x)) ≠ f(y). The concept of S-k-colourable is a common generalization of many colouring concepts. This paper studies the problem as to which sets S, every triangle free planar graph is S-3-colourable. We call a set S TFP-good if every triangle-free planar graph is S-3-colourable. Grötzsch’s theorem is equivalent to say that S = {id} is TFP- good. In this paper, we prove that for any inverse closed subset S of S₃ which is not isomorphic to {id, (12)}, S is TFP-good if and only if either S = {id} or there exists a ∈ [3] such that for each π ∈ S, π(a) ≠ a. It remains an open question whether or not S = {id, (12)} is TFP-good.
This is a joint work with Yiting Jiang, Yeong-Nan Yeh, and Xuding Zhu.

Daniela Ferrero

Generalizations Of Zero Forcing In Iterated Line Digraphs

Abstract: Zero forcing is a graph propagation process in which an initial set of vertices is iteratively augmented, through the application of certain rules, with the objective of finding the minimum cardinality of an initial set of vertices that can be augmented to the entire vertex set of a graph. Variants of zero forcing obtained by changes in the augmentation rules have proven useful in different applications. In this talk, I will present results about zero forcing and some of its variants in the case of digraphs. Some of the results were obtained jointly with T. Kalinowski and S. Stephen.

Mary Flagg

Rigid Linkage Forcing

Abstract: Given a simple graph, a linkage is a subgraph in which each connected component is a path. The pattern of a linkage is a set with elements the sets of endpoints of each component path. A linkage is called unique if it is the only linkage possible with its pattern, and spanning if it contains all the vertices of the graph. Robertson and Seymour defined a vital linkage as a spanning linkage which is unique. A rigid linkage is a special type of vital linkage defined by a rigid linkage forcing process. Rigid linkage forcing, as a variation of standard zero forcing, is a coloring game on a simple graph played according to a certain color change rule. The coloring process defines a rigid linkage for the graph. Zero forcing was introduced in the linear algebra community as an upper bound to the maximum multiplicity of an eigenvalue of a symmetric matrix whose off-diagonal pattern of zeros is determined by a graph. Rigid linkage forcing is a variation of zero forcing created to obtain bounds on the multiplicities of eigenvalues in this family of matrices. It this talk I will share the rules for rigid linkage forcing, and show how rigid linkages are connected to multiplicities of eigenvalues of symmetric matrices determined by the graph.

Shanise Walker

The Size Of N-Free Families

Abstract: In this talk, we are interested in estimating the maximum size of a family of subsets of the n -set avoiding a given subposet. The N poset consists of four distinct sets W, X, Y, Z such that W ⊂ X , Y ⊂ X , and Y ⊂ Z where W is not necessarily a subset of Z . We provide a lower bound for the size of a largest family avoiding the N poset, which makes use of error-correcting codes.

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Math on the EDGE

Organizers: Sarah Chehade

Keisha Cook

A Parallel Implementation of the Delay SSA

Abstract:We consider the spatial stochastic SIR model on a 2D elongated rectangular lattice. The SIR model consists of susceptible (S), infected (I), and resistant (R) cells regulated by 3 reactions: S+I -> I+I (k₁), I -> R (k₂) , R -> I(k₃), where k_i are reaction rates. The first reaction occurs only when susceptible and infected cells are in contact. Depending on parameters k_i the system may exhibit propagation of fronts of infected cells. Since the state in which all cells are susceptible is absorbing these fronts reach extinction, but the extinction time can be very long. We investigate the following questions: (1) What is the extinction time T of the infected cells front, (2) How many infected cells are needed to carry information (infection) over a specific distance, (3) What is the probability that the front start propagating backward. Keeping k₂=1 and k₃=0.01 we demonstrate that the T is a growing function of k₁ and height of the lattice H. Importantly, growth of T is abrupt when k₁ becomes larger than k₂=1. The T dependence to H is more gradual. As a consequence in the narrower the reactors but with large k₁, less cells are needed to send the signal over some distance (with respect to broader reactors with small k₁). For small k₃ susceptible cells behind the front are separated from the infected cells by the thick layer of resistant cells. The thickness of the separation layer decreases with increasing k₃. As a consequence the probability that the front starts propagating backward increases with increasing k₃.

Paula Egging

Rational Decay Of A Canonical Structural Acoustic PDE Dynamics

Abstract: A rate of rational decay is obtained for solutions of a PDE model which has been used in the literature to describe structural acoustic flows. This structural acoustics PDE consists partly of a wave equation which is invoked to model the interior acoustic flow within a given cavity. Moreover, a structurally elastic equation is invoked to describe time-evolving displacements along the flexible portion of the cavity walls. The coupling between these two distinct dynamics will occur across a boundary interface. We obtain this rational decay rate by establishing certain a priori inequalities for the PDE which will eventually allow the invocation of an abstract resolvent criterion for rational decay.

Victoria Day

Congruences Between Newforms And Modular Deformation Problems

Abstract: Given a newform f and a choice of a prime p, Deligne and Serre constructed a semisimple two dimensional residual (mod p) Galois representation associated to f . It is profitable to study lifts of this representation to Gl₂(A) for certain rings A, which can be done using the deformation theory of Galois representations. This theory has played a central role in the proofs of both the Taniyama-Shimura Conjecture and Serre’s Conjecture. In this talk, we will discuss the interplay between congruences among newforms and certain modular deformation problems.

Nida Kazi Obatake

Toric Ideals Of Neural Codes

Abstract: A rat has special neurons that encode its geographic location. These neurons are called place cells and each place cell points to a region in the space, called a place field. Neural codes are collections of the firing patterns of place cells. In this talk, we investigate how to algorithmically draw a place field diagram of a neural code, building on existing work studying neural codes, ideas developed in the field of information visualization, and the toric ideal of a neural code (joint work with Elizabeth Gross and Nora Youngs). We use the toric ideal of a neural code to show sufficient conditions for a code to be 1- or 2-inductively pierced (joint work with Molly Hoch and Samuel Muthiah).

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Multiphysics and Multiscale Problems

Organizers: Yue Yu, Xingjie Li

Xiaoxiao Zhang

Non-uniform Curvature And Anisotropy Confinement Control Wrinkling Patterns On Torus

Abstract: We investigate wrinkling patterns in a tri-layer torus, containing an expanding thin outer layer, an intermediate soft layer and an inner core with tunable elastic moduli. We show from large scale finite element simulations that hexagonal wrinkling patterns form for stiff cores and stripe wrinkling patterns develop for soft cores. Hexagons and stripes co-exist to form hybrid patterns for cores with intermediate stiffness. The governing mechanism for the pattern transition is that the stiffness of the inner core controls the global deformation of the whole torus, leading to an anisotropy confinement in the toroidal and poloidal directions. The anisotropy deformation alters stress states in the outer layer, which will be changed from biaxial (preferred hexagons) to uniaxial (preferred stripes) compression when reducing the core stiffness. As the outer layer continues to expand, stripe and hexagon patterns will evolve into Zigzag and Segment Labyrinth, respectively. In addition, we observe stripe wrinkles initiate from the inner surface of the torus while hexagon wrinkles start from the outer surface and find this is due to the curvature dependent stress state in the torus. We further show that the level of confinement can modify the dimpled patterns in the hexagonal wrinkling regime.

Emily Johnson

Penalty Coupling Of Non-matching Isogeometric Kirchhoff-Love Shell Patches With Application To Complex Structures

Abstract: Isogeometric analysis (IGA) has been a particularly impactful development in the realm of Kirchhoff–Love thin-shell analysis because the high-order basis functions employed naturally satisfy the requirement of C¹ continuity. Still, engineering models of appreciable complexity, such as wind turbine blades, are typically modeled using multiple surface patches and, often, neither rotational continuity nor conforming discretization can be practically obtained at patch interfaces. A penalty approach for coupling adjacent patches is therefore presented. The proposed method imposes both displacement and rotational continuity and is applicable to either smooth or nonsmooth interfaces and either matching or non-matching discretization. The penalty formulations require only a single, dimensionless penalty coefficient for both displacement and rotation coupling terms, alleviating the problem-dependent nature of the penalty parameters. Using this coupling methodology, numerous benchmark problems encapsulating a variety of analysis types, geometrical and material properties, and matching and non-matching interfaces are addressed. The coupling methodology produces consistently accurate results throughout all tests. Furthermore, the suggested penalty coefficient of α = 10³ is shown to be effective for the wide range of problem configurations addressed. Finally, a realistic wind turbine blade model, consisting of 27 patches and 51 coupling interfaces and having a chordwise- and spanwise-variant composite material definition, is subjected to buckling, vibration, and nonlinear deformation analyses using the proposed approach.

Weiqi Chu

Nonlinear Constitutive Models For Nano-scale Heat Conduction

Abstract: Heat conduction properties of nano mechanical systems have significant impacts on the performances of nano devices. There have been enormous experimental and numerical observations that indicate the breakdown of the conventional model of heat conduction, the Fourier’s Law. The indications include, but not limited to, the size dependence of the heat conductivity, propagation heat pulses behavior, and delay phenomena.
I will present a rigorous approach that leads, from a many-particle description, to a nonlinear, stochastic constitutive relation for the modeling of transient heat conduction processes at nanoscale. The nonlinearity gives rise to traveling waves, which also implies that temperature can propagate with finite speed. By enforcing statistical consistency, in that the statistics of the local energy is consistent with that from an all-atom description, we identify the driving force as well as the model parameters in these generalized constitutive models, such as heat propagation speed, relaxation time, thermal conductivity and its size dependence.

Xiaochuan Tian

Stability Of Nonlocal Dirichlet Integrals Using Nonlocal Gradient Operators

Abstract: Nonlocal gradient operators are basic elements of nonlocal vector calculus that play important roles in nonlocal modeling and analysis. We study a nonlocal Dirichlet integral that is given by a quadratic energy functional based on nonlocal gradients. Our main finding is that the coercivity and stability of this nonlocal continuum energy functional may hold for some properly chosen nonlocal interaction kernels but may fail for some other ones. This can be significant for applications in various nonlocal models, including the peridynamic correspondence material models. We will also discuss the implication of it to the well-posedness of a nonlocal Stokes equation.

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New Advances in Symplectic and Contact Topology, I

Organizers: Jo Nelson, Morgan Weiler

Ziva Myer

Product Structures For Legendrian Submanifolds With Generating Families

Abstract: In contact geometry, invariants of Legendrian submanifolds in 1-jet spaces have been defined through a variety of techniques. In my talk, I will discuss how I use the Morse-theoretical technique of generating families to enrich one such invariant. In particular, I have constructed a product on generating family cohomology groups and I will discuss current work extending this construction to an A-infinity category whose objects are generating families. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions whose critical points correspond to Reeb chords of the Legendrian submanifold.

Yu Pan

All The Augmentations Come From Immersed Lagrangian Fillings

Abstract: Augmentations are tightly connected to embedded exact Lagrangian fillings. However, not all the augmentations of a Lagrangian knot come from embedded exact Lagrangian fillings. In this talk, we show that all the augmentations come from possibly immersed exact Lagrangian fillings. In particular, for a 1-dimensional Lagrangian knot in J¹M, take an immersed exact Lagrangian filling that can be lifted to an embedded Lagrangian L in ℝ &times Symp(J¹M). For any augmentation of L, we associate an induced augmentation of the Lagrangian knot, whose homotopy class only depends on the compactly supported Lagrangian isotopy type of L and the homotopy class of its augmentation. This is a joint work with Dan Rutherford in progress.

Bahar Acu

Planarity In Higher-Dimensional Contact Manifolds

Abstract: Planar contact manifolds, those that correspond to an open book decomposition with genus zero pages, have been intensively studied to understand several aspects of 3- dimensional contact topology. In this talk, we present a higher-dimensional notion of planarity, iterated planarity, and provide several generalizations of results for planar contact 3-manifolds, to higher dimensions. This is joint work with A. Moreno.

Emmy Murphy

Weinstein Kirby Calculus

Abstract: Just as smooth 4-manifolds can be presented by links in S³, the symplectic topology of Stein 2n-manifolds can be presented by Legendrians links. We discuss Weinstein Kirby calculus, the complete set of moves which allow us to describe all Legendrians describing the same Stein manifold. We present a number of examples and applications.

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New Advances in Symplectic and Contact Topology, II

Organizers: Jo Nelson, Morgan Weiler

Susan Tolman

Beyond Semitoric

Abstract: A compact four dimensional completely integrable system f : M → ℝ² is semitoric if it has only non-degenerate singularities, without hyperbolic blocks, and one of the components of f generates a circle action. Semitoric systems have been well studied and have many nice properties; for example, the fibers f ^-1(x) are connected. Un- fortunately, although there are many interesting examples of semitoric systems, the class has some limitations. For example, there are blowups of S² × S² with Hamiltonian circle actions that cannot be extended to semitoric system. We show that, by allowing certain degenerate singularities, we can expand the class of semitonic systems but still prove that fibers f^-1(x) is connected. We hope that this class will be large enough to include not only all compact four manifolds with Hamiltonian circle actions, but more generally all complexity one spaces. Based on joint work with D. Sepe.

Saraswathi Venkatesh

Symplectic Cohomology Of Subdomains

Abstract:Mirror symmetry predicts the existence of Floer invariants that yield “local” information. Guided by this, we construct a Floer theory in the fashion of symplectic cohomology that detects whether a subdomain contains a Floer-essential Lagrangian. We illustrate the quantitative behavior of this theory by examining negative line bundles over toric symplectic manifolds.

Catherine Cannizzo

Homological Mirror Symmetry For The Genus 2 Curve In An Abelian Variety And Its Generalized Strominger-Yau-Zaslow Mirror

Abstract: Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs X and Y such that the complex geometry on X mirrors the symplectic geometry on Y . It allows one to deduce information about Y from known properties of X. Strominger-Yau-Zaslow (1996) described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich (1994) conjectured that a complex invariant on X (the bounded derived cate- gory of coherent sheaves) should be equivalent to a symplectic invariant of Y (the Fukaya category). This is known as homological mirror symmetry. In this project, we first use the construction of SYZ mirrors for hypersurfaces in abelian varieties following Abouzaid- Auroux-Katzarkov, in order to obtain X and Y as manifolds. The complex manifold comes from the genus 2 curve as a hypersurface in its Jacobian torus, and we equip the SYZ mirror manifold with a symplectic form. We then describe an embedding of the category on the complex side into a cohomological Fukaya-Seidel category of Y as a symplectic fibration.

Roberta Guadagni

New Advances In Symplectic And Contact Topology

Abstract: Darboux’s Theorem describes canonical symplectic coordinates close to a point P in a symplectic manifold (M, ω). Similarly, Weinstein’s neighborhood theorem states that the symplectic form is canonical in a neighborhood of any smooth submanifold S ⊂ M (and a model for it is given by the submanifold’s symplectic normal bundle). We will investigate what we are able to say about the symplectic form in a neighborhood of a symplectic submanifold S which we allow to be singular. The first formulation of this considers S = ∪_i S_i a transverse union of symplectic submanifolds. In this case, we can formulate a version of Weinstein’s theorem proving that the symplectic form in a neighborhood of S is uniquely determined by the restriction of the form along S. If the S_i intersect orthogonally, we can give an explicit model based on the union of normal bundles NS_i . I will discuss the result above and the possible extensions to a singular S admitting a Whitney stratification (for instance any S with algebraic singularities).

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New Developments in Algebraic Biology, I

Organizers: Anne Shiu, Brandilyn Stigler

Elizabeth Gross

Distinguishing And Inferring Phylogenetic Networks

Abstract: Phylogenetic networks are increasingly becoming popular in phylogenetics since they have the ability to describe a wider range of evolutionary events than their tree counterparts. In this talk, we discuss Markov models on phylogenetic networks, i.e. directed acyclic graphs, and their associated algebra and geometry. In particular, assuming the Jukes-Cantor model of evo- lution and restricting to one reticulation vertex, using tools from commutative algebra, we show that the semi-directed network topology of large-cycle networks is generically identifiable and discuss how these results can be used for inference.

Nora Youngs

Neural Ring Homomorphisms And Maps Between Neural Codes

Abstract: Neural codes are binary codes that are used for information processing and representation in the brain. An algebraic structure, called the neural ring, can be used to efficiently encode geometric and combinatorial properties of a neural code. In this talk, we consider maps between neural codes and the associated homomorphisms of their neural rings. In order to ensure that these maps are meaningful and preserve relevant structure, we find that we need additional constraints on the ring homomorphisms. This motivates us to define neural ring homomorphisms. We will characterize all code maps corresponding to neural ring homomorphisms as compositions of 5 elementary code maps. As an application, we find that neural ring homomorphisms behave nicely with respect to convexity. In particular, if C and D are convex codes, the existence of a surjective code map C → D with a corresponding neural ring homomorphism implies that the minimal convex embedding dimensions satisfy d(D) ≤ d(C).

Nida Kazi Obatake

The Capacity For Hopf Bifurcations In The Fully Distributive Dual-Site Phosphorylation Network

Abstract: Chemical Reaction Network theory is an area of mathematics that analyzes the behaviors of chemical processes. A major problem in this area is stability in these networks. This talk focuses on bifurcations in a particular network, the fully distributive dual-site phosphorylation network. Experimental results suggest that this network does not exhibit bifurcations, but as far as we know, there are no theoretical results to support this conjecture. In this work we examine the capacity for Hopf bifurcations, by analyzing the steady state locus of the ODE system obtained from the network. To reduce the number of variables, we compute a parameterization of the steady state locus. We use Maple to compute the corresponding Hurwitz matrix and its minors, so that we may apply a generalization of the Routh-Hurwitz criterion for Hopf bifurcations. Using SAGE, we examine the Newton polytope to understand the signs that the Hurwitz determinants take. We show for the first time that the relevant Hurwitz determinants change sign, and discuss the implications for bifurcations and oscillations in the network. Joint work with Anne Shiu and Xiaoxian Tang.

Xiaoxian Tang

Multistationarity In Structured Reaction Networks

Abstract: Many dynamical systems arising in applications exhibit multistationarity (two or more positive steady states), but it is often difficult to determine whether a given system is multistationary, and if so to identify a witness to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. In this talk we introduce a procedure to investigate multistationarity and to find a witness. In practice, the procedure is much less expensive than traditional quantifier elimination. Our method is based on two new sufficient conditions for multistationarity. First, when there are no boundary steady states and a positive steady-state parametrization exists, one can conclude multistationarity if a certain critical function changes sign. Particularly, if the steady states are defined by binomials, we have multistationarity if a certain critical function contains terms with different signs. Second, when the steady-state equations can be replaced by equivalent triangular-form equations, we have multistationarity if a positive degenerate steady state exists. We also investigate the mathematical structure of this critical function, and give conditions that guarantee that triangular-form equations exist by studying the specialization of Groebner bases.

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New Developments in Algebraic Biology, II

Organizers: Anne Shiu, Brandilyn Stigler

Elena Dimitrova

Network Control Through Multistate Canalization

Abstract: Boolean canalization, a type of hierarchical clustering of the inputs of a Boolean function, has been studied in the context of network modeling where each layer of canalization adds a degree of stability in the dynamics of the network. Multicellular populations give rise to emergent features such as patterns based upon the collective communication between neighboring and distant cells. This talk will present a recently introduced generalization of canalization to multistate functions and discuss the role of canalization in the study and control of multicellular populations.

Anyu Zhang

Partitioning Data Using Monomial Bases To Improve Network

Abstract: Network inference in systems biology is plagued by too few input data and too many candidate models which ft the data. When the data are discrete, models can be written as a linear combination of finitely many monomials. The problem of selecting a model can be reduced to selecting an appropriate monomial basis. Recently affine transformations were used to partition input data into equivalence classes with the same basis. We wrote a Python package to build the equivalence classes for small networks. We propose a “standard positions for data sets and developed a metric to measure how far a set is from being in standard position. We used this metric to define the representative of an equivalence class. The implication of this work is guidance for systems biologists in designing experiments to collect data that result in a unique model (set of predictions, thereby reducing ambiguity in modeling and improving predictions.

Kaitlyn Phillipson

Gröbner Bases Of Neural Ideals

Abstract: The neural ideal was introduced as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Gröbner basis with respect to that monomial order. How are these two types of generating sets – canonical forms and Gröbner bases &minus related? In this talk, we will demonstrate that when the canonical form of the neural ideal is a Gröbner basis, it is the union of all reduced Gröbner bases for the ideal (i.e. the universal Gröbner basis). A natural question to pursue, then, is under what conditions will the canonical form be a Gröbner basis? We will give some partial answers to this question. In addition, we will discuss what the Gröbner basis elements can tell us about the structure of the receptive field.

Ngoc M Tran

Automated Spike Sorting For Large-Scale, Long-Term Recordings

Abstract: Neural activity in a brain region forms a stochastic network. When a neuron ‘spikes’, it sends out a sharp waveform recorded by neighboring electrodes. Spike sorting is the problem of assigning spikes to individual neurons. Over a recording session, the neuron may move and its waveform may change, posing unique challenges to obtaining long-term recordings. In this talk, we review existing techniques, showcase recent advances and discuss open problems on stochastic networks based on spike sorting challenges. No prior knowledge on neuroscience is required.

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Origami, Belyi Maps, and Dessins D’Enfants

Organizers: Rachel Davis, Edray Goins

Edray Goins

Monodromy Groups of Compositions of Belyi Maps

Abstract: Given a Belyi map \(\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)\) of degree n, it is well known that its monodromy group \(\text{Mon}(\beta)\) is a subgroup of the symmetric group \(\text{Sym}(n)\). In fact, this group can be viewed as the “Galois closure” of the automorphism group \(\text{Aut}(\beta) \subseteq \text{Sym}(n)\). Given a composition \(\beta \circ \phi:\mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)\) of two Belyi maps \(\beta\) and \(\phi\), it is known that the monodromy group \(\text{Mon}(\beta \circ \phi) \subseteq \text{Mon}(\phi) \wr \text{Mon}(\beta)\) is contained in the wreath product of the monodromy groups of each of the maps. However, when do we have equality? And what exactly is the relationship between these three groups?

In the 2018 doctoral thesis of Jacob Bond, there was an explicit description of the relationship between these three groups. Explicitly, \(\text{Mon}(\beta \circ \gamma) \simeq \rho_\gamma(A) \wr \text{Mon}(\beta)\) where \(\rho_\gamma(A)\) is a subgroup of the collection of maps \(E_\beta \to \pi_1( X) \to \text{Mon}(\gamma)\) from the edges of the Dessin d’Enfant of β to the Fundamental Group of the thrice punctured sphere \(X = \mathbb P^1(\mathbb C) \backslash \{ 0, \, 1, \, \infty \}\) to the monodromy group of φ. In this talk, we explain the details, and provide some examples. This is joint work with Jacob Bond.

Rachel Davis

Square-Tiled Surfaces In Topology And Arithmetic

Abstract: Topologists have studied surfaces obtained by gluing unit squares together using “origami rules”. The resulting Riemann surfaces, called origami, are algebraic curves defined over number fields, and, therefore, admit an action by the absolute Galois group. In 2005, Möller showed that this action is faithful using the faithfulness of the Galois action on dessins. Noncongruence subgroups are more plentiful and less well understood than congruence subgroups. There is a conjectured connection (due to Chen) between origami with highly nonabelian monodromy groups and noncongruence subgroups.

Ozlem Ejder

Arboreal Galois Representations Of Dynamical Belyi Maps

Abstract: A dynamical Belyi map is a finite morphism f : ℙ¹_C → ℙ¹_C defined over C which is branched exactly at the three ordered points 0, 1, ∞ such that f({0, 1, ∞}) ⊆ {0, 1, ∞}. All iterates f ⁿ are also Belyi maps. Given a dynamical Belyi map defined over a field K and a non-preperiodic point α ∈ K, one can construct a tree of preimages of &alphs;. This construction leads to the phenomena: one has a tower of fields K = K₀ ⊆ K₁ ⊆ K₂ ⊆ ⋅⋅⋅ where K_n := K (φ^-n(α)). One also has a natural Galois representation on the tree of preimages, the so-called Arboreal Galois representation of the function f. In this talk, we describe the Arboreal Galois representations and the monodromy groups of iterations of a large class of dynamical Belyi maps. Studying these Galois groups has applications in the study of the density of prime divisors of elements of dynamical sequences. If time allows, I will mention some applications as well.

Bella Tobin

Dessins D’Enfants For Single-Cycle Belyi Maps

Abstract: Riemann’s Existence Theorem gives bijections between isomorphism classes of Belyi maps of degree d, equivalence classes of generating systems of degree d, and isomorphism classes of dessins d’enfants with d edges. Let f : X → ℙ¹ be a Belyi map. If f has a single ramification point above each branch point, we say that f is a single-cycle Belyi map. We describe dessins d’enfants for single-cycle Belyi maps and apply this to two infinite families of dynamical Belyi maps.

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Recent Developments in the Analysis of Obstacle Problems Associated to Nonlocal Operators

Organizers: Donatella Danielli, Camelia Pop

Ryan Hynd

Extremals Functions For Morrey’s Inequality

Abstract: Morrey’s inequality quantifies the continuity of functions whose derivatives have high enough integrability. We study the functions for which Morrey’s inequality is saturated. We will ex- plain how various qualitative properties of these extremal functions can be deduced from the partial differential equation they satisfy. Finally, we will discuss numerical approximations of these functions and other applications in optimization theory.

Ru-Yu Lai

Global Uniqueness For The Fractional Semilinear Schrodinger Equation

Abstract: We study global uniqueness in an inverse problem for the fractional semi- linear Schrödinger equation (-Δ)^su + q(x, u) = 0 with s ∈ (0, 1). In this talk, I will discuss how an unknown function q(x, u) can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to 2.

Stefania Patrizi

A Segregation Model With Local Vs Nonlocal Diffusion

Abstract: Segregation phenomena occurs in many areas of mathematics and science: from equipartition problems in geometry, to social and biological processes (cells, bacteria, ants, mammals) to finance (sellers and buyers). Segregation problems model a situation of high competition for resources and involve a combination of diffusion and annihilation between populations. We present a new model in which two competing species follow propagation equations, one of them involving a local diffusion while the other one involving a non-local diffusion. This is a joint paper with Luis Caffarelli.

Luca Spolaor

(Log)-Epiperimetric Inequality For The Thin Obstacle Problem

Abstract: In this talk I will introduce a new logarithmic epiperimetric inequality for the 2m- Weiss’ energy in any dimension and use it to improve the Garofalo-Petrosyan result [1] from C ¹ regularity of the singular set to C^1,log. Moreover I will give a very general proof of this inequality, which shows its applicability to many other variational problems.

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On Advances and New Techniques of Fluid Dynamics and Dispersive Equations

Organizer: Betul Orcan Ekmekci

Chengcheng Yang

Regularity Behavior Of Geodesics On A Complex Variety

Abstract: We study the shortest length curve in a space called algebraic variety that is defined as the set of zeros of finitely many polynomials. It is known that an algebraic variety can be partitioned into simpler components, which locally look like Euclidean space. One interesting question is whether the shortest length curve will cross each component only finitely many times. We will show some interesting results. The next question is whether the shortest length curve is finitely represented by explicit equations. We will also show something about this. In general, applying analytical viewpoints to algebraic objects is an exciting playground.

Maria Radosz

Blowup For Model Equations Of Fluid Mechanics

Abstract: Blowup for model equations of fluid mechanics. In this talk, I consider the 2D inviscid Boussinesq equations in vorticity form. It remains a challenge to decide if finite time blowup happens for smooth initial data or not. V. Hoang introduced a model problem for the Boussinesq equations associated to the hyperbolic flow scenario for which were able to show finite-time blowup (joint work with V. Hoang, B. Orcan-Ekmekci, H. Yang)

Betul Orcan-Ekmekci

Geometric Properties Of Euler Equation On The Torus

Abstract: Problems of fluid dynamics and dispersive equations as well as their applications play a critical role in nature, science, and engineering. For example, dynamics in the oceans and atmosphere is a prominent scientific topic. Despite the well-characterized modelings, some fundamental theoretical problems are still open such as finite-time singularity formation versus global regularity and growth rate estimates in different dimensions. To understand and characterize these behaviors, as a new approach, we can analyze the geometric properties of the trajectories. In this talk, we focus on incompressible Euler’s equation in 2-D and get some curvature estimates for the trajectories of the flow.

Kayla Bicol

A Deconvolution-Based Large Eddy Simulation Method For Incompressible Flows And Advection-Diffusion-Reaction Problems

Abstract: We consider a Leray model with a deconvolution-based indicator function for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of few thousand) with under-resolved meshes. For the implementation of the model, we adopt a three-step algorithm called evolve-filter-relax that requires (i) the solution of a Navier-Stokes problem, (ii) the solution of a Stokes-like problem to filter the Navier-Stokes velocity field, and (iii) a final relaxation step. To better understand the role of this method’s parameters, we then adapt the method to consider time-dependent Advection-Diffusion-Reaction problems, which are simpler than incompressible flow problems. We compare our deconvolution-based approach to classical stabilization methods and test its sensitivity to model parameters on 2D benchmark problems.

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Topology of 3- and 4-Manifolds, I

Organizers: Allison N. Miller, Arunima Ray

Rachel Roberts

CTFs In 3-Manifolds

Abstract: Co-oriented taut foliations (CTFs) are an important tool in the study of 3-manifolds; determining existence or nonexistence of CTFs in a given 3-manifold is therefore important. Heegaard Floer homology computations are currently the most effective method for establishing nonexistence, and, more recently, have been used to motivate the search for new constructions of CTFs. In this talk, I will give a brief overview of some constructions of CTFs. This will include work joint with Charles Delman.

Maggie Miller

Fibering 4-Manifolds Via Movies Of Singular Fibrations

Abstract: I will define movies of singular fibrations on 4-manifolds, and discuss what properties of such a movie imply that the underlying 4-manifold is fibered over S¹.

Christine Ruey Shan Lee

Quantum Invariants And Ribbon Links

Abstract: Recent research has shown that quantum invariants such as the Jones polynomial can provide criteria for ribbon links, though very little is known about what they detect. In this talk, I will discuss the criterion by Eisermann and the extension to colored Jones polynomial by Suzuki, and present open problems in this area. Specifically, I will discuss the possibility of making these criteria effective through finite-type expansions of quantum invariants.

Patricia Cahn

Invariants Of Fox-Colorable Knots From Branched Covers Of 4-Manifolds

Abstract: We study branched covering maps from a 4-manifold X to the 4-sphere, whose branching sets are embedded with one cone singularity modelled on a Fox p-colored knot K. Kjuchukova defined an invariant Ξ_p(K), whose value is the signature of the covering manifold X. We show that this invariant gives rise to a ribbon obstruction, as well as a bound on the 4-genus of a knot. We discuss two diagrammatic methods for computing the invariant.
The first is a computation involving linking numbers of curves in the dihedral branched cover of \(S^3\) along K. The second uses trisections of 4-manifolds; we use this technique to construct an infinite family of knots for which the above bound on 4-genus is sharp. This is joint work with R. Blair and A. Kjuchukova.

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Topology of 3- and 4-Manifolds, II

Organizers: Allison N. Miller, Arunima Ray

Emmy Murphy

Dissolving Large Index Covering Spaces Of 4-Manifolds

Abstract: Let X and Y be two homeomorphic smooth 4-manifolds with π₁ = ℤ, and let X_k and Y_k be their k-fold covering spaces. We show that there is a constant C not depending on k, so that X_k becomes diffeomorphic to Y_k after connect summing C copies of S ² × S ².

Melissa Zhang

Localization, Smith-Type Inequalities, And Khovanov Homology

Abstract: Whenever a topological object exhibits symmetry, one may consider how its algebraic invariants interact with the group action. Around 80 years ago, P. A. Smith studied topological spaces with finite cyclic action; his work lead Borel and others to formulate a ”localization theorem” relating the singular cohomology of the symmetric space with that of the fixed-point space. Nowadays, the prominence of categorified, homology-type invariants in low-dimensional topology has encouraged the study of symmetric objects via this localization framework. In this talk, I’ll give an overview of recent localization results in low-dimensional topology and introduce relevant machinery; as an example, I’ll discuss the application of this framework to the Khovanov homology of periodic links.

Caitlin Leverson

DGA Representations, Ruling Polynomials, And The Colored HOMFLY-PT Polynomial

Abstract: Given a pattern braid \(\beta \in J^1(S^1) \) braid, to any Legendrian knot Λ in braid \(\mathbb{R}^3 \) braid with the standard contact structure, we can associate the Legendrian satellite knot \(S(\Lambda,\beta) \). We will discuss the relationship between counts of augmentations of the Chekanov-Eliashberg differential graded algebra of \(S(\Lambda,\beta) \) and counts of certain representations of the algebra of Λ.
We will then define an m-graded n-colored ruling polynomial from the m-graded ruling polynomial, analogously to how the n-colored HOMFLY-PT polynomial is defined from the HOMFLY-PT polynomial, and extend results of the second author, to show that the 2-graded n-colored ruling polynomial appears as a specialization of the n-colored HOMFLY-PT polynomial.

Gordana Matic

Spectral Order Contact Invariant From Heegaard Floer Homology

Abstract: In 2005 Ozsváth and Szabó introduced an invariant of contact structures that lives in Heegaard Floer homology that was very successfully used by many people to study tightness and fillability. We provide a refinement that takes values in \(Z_{\geq 0} \cup \infty \) in the case when the contact invariant is 0. This order invariant is zero for overtwisted contact structures, ∞ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. It gives a criterion for tightness of a contact structure stronger than that given by the Heegaard Floer contact invariant, and an obstruction to existence of Stein cobordisms between contact 3-manifolds. We prove these claims by exhibiting an infinite family of examples with vanishing Heegaard Floer contact invariant on which our invariant assumes an unbounded sequence of finite and non-zero values. This is joint work with Çağatay Kutluhan, Jeremy Van Horn-Morris and Andy Wand.

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WIC: Women in Control, I

Organizers: Lorena Bociu, Irena Lasiecka

Suzanne Lenhart

Optimal Control Applied To Management Of Fishery Models

Abstract: Optimal control techniques are used to investigate management strategies in fishery models, while incorporating economic impacts. Harvesting of fishery stock can lead to habitat damage. We present a system of parabolic partial differential equations, which model the spatiotemporal dynamics of a fish stock and its habitat. Techniques of optimal control of PDEs are used to investigate the harvest rates that maximize the discounted profit value while minimizing the negative effects on the habitat.

Weiwei Hu

Second Order Optimality Conditions For Boundary Control Of Optimal Mixing Via Fluid Flows

Abstract: We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via an active control of the flow velocity, governed by Stokes or Navier-Stokes equations, in an open bounded and connected domain. The problem is motivated by mixing the fluids within a cavity or vessel by moving the walls or stirring at the boundaries. It is natural to consider the velocity field that is induced by a control input tangentially acting on the boundary of the domain through the Navier slip boundary conditions. Our main objective is to design an optimal Navier slip boundary control that optimizes mixing at a given final time. This essentially leads to a finite time nonlinear control problem. First-order necessary optimality conditions and second-order necessary and sufficient optimality conditions will be presented.

Luz De Teresa

Some Hierarchic Control Problems For The Heat Equation

Abstract: In this conference we present some results concerning a hierarchic Stackelberg strategy to control the heat equation. That means that we act on the equation with two controls with different objectives: the leader and the follower. The follower acts once the leader control has been established. In the present situation the follower objective is a null control one and the leader objective is an optimal control objective.

Valeria Neves Domingos Cavalcanti

Uniform Stability For The Wave Equation With Localized Memory

Abstract: We discuss the asymptotic stability of a damped wave equation subject to a locally distributed viscoelastic effect with supercritical sources and damping.

Yulia Gorb

Singular behavior of the gradient of the solution to high contrast PDEs

Abstract: This talk is about a particular blow up phenomena in high contrast two-phase dispersed composites whose inhomogeneities are closely spaced. These composites are described by PDEs with high contrast and rapidly varying coefficients. The gradients of solutions to such problems exhibit singular behavior – blow up. This blow up is captured in terms of the distance between inhomogeneities and is fully characterized and justified. Both linear and non-linear formulations will be explored in the talk.

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WIC: Women in Control, II

Organizers: Lorena Bociu, Irena Lasiecka

Bozenna Pasik-Duncan

Advances in Noise Modeling in Stochastic Systems and Control

Abstract: Many continuous time stochastic systems that are modeled by SDE and SPDE have been limited to noise processes being Brownian motions. Brownian motion models have a well developed stochastic calculus and limiting behaviors that reflect the martingale, Markov and Gaussian properties of Brownian motion. However for many physical systems the empirical data do not justify the use of Brownian motion as the model for random disturbances. In fact Brownian motions provide models that are often far from the physical data. Thus it is necessary to find more general noise models and tractable methods to solve the associated problems of control or adaptive control. These other noise models include more general Gaussian processes and non-Gaussian processes. The talk focuses on new developments and challenges in noise models for stochastic control and adaptive control problems.

Giusy Mazzone

On The Motion Of Rigid Bodies With A Fluid-Filled Gap

Abstract: Consider the system S constituted by a rigid body B having a hollow cavity which (strictly) contains a rigid ball BR. The gap between these rigid bodies is completely filled by a viscous incompressible fluid, whose motion is governed by the Navier-Stokes equations. I will present results concerning existence, regularity and stability properties of solutions to the equations governing the motion of the whole system S.

Marta Lewicka

Random Tug Of War Games For The p-Laplacian: 1<p<∞

Abstract: We propose a new finite difference approximation to the Dirichlet problem for the homogeneous p-Laplace equation posed on an N-dimensional domain, in connection with the Tug of War games with noise. Our game and the related mean-value expansion that we develop, superposes the “deterministic averages” “1/2 (inf+sup)” taken over balls, with the “stochastic averages” “ƒ”, taken over N-dimensional ellipsoids whose aspect ratio depends on N, p and whose orientations span all directions while determining inf / sup. We show that the unique solutions u_ε of the related dynamic programming principle are automatically continuous for continuous boundary data, and coincide with the well-defined game values. Our game has thus the min-max property: the order of supremizing the outcomes over strategies of one player and infimizing over strategies of their opponent, is immaterial. We further show that domains satisfying the exterior corkscrew condition are game regular in this context, i.e. the family {u_ε} converges, as ε → 0, uniformly to the unique viscosity solution of the Dirichlet problem.

Ivonne Rivas^*, Phillipe Martin, Ivonne Rivas, Lionel Rosier,Pierre Rouchin

Exact Controllability Of A Linear Korteweg-de Vries Equation By The Flatness Approach

Abstract: We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a system was proved in the last decade by using some Carleman estimate. Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.

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WICA: Commutative Algebra, I

Organizers: Sandra Spiroff, Adela Vraciu

Janet Vassilev

Tight Interiors And Related Ideals In Stanley-Reisner Rings

Abstract: Recently Epstein and Schwede defined the tight interior of an R-module. In Stanley- Reisner rings, it is fairly straightforward to compute the tight interior of an ideal. We will show that *-core I ⊆ I_*. Similarly, we will discuss *-extensions and *-hulls of ideals and show that \(I^* \subseteq \mbox{*-hull }I\). In both cases we will discuss when equality is achieved.

Haydee Lindo

Trace modules, rigidity and ring classifications

Abstract: I will speak on some recent developments in the theory of trace modules over commutative Noetherian rings. This will include applications of trace modules in classifying rings and a discussion of ongoing work examining the relationship between trace modules and modules having no self- extensions.

Alexandra Seceleanu

Lefschetz Properties For Connected Sums And Fibered Products

Abstract: The Lefschetz properties are desirable algebraic properties of graded artinian algebras inspired by the Hard Lefschetz Theorem for cohomology rings of complex projective varieties. A standard way to create new varieties from old is by forming connected sums. This corresponds at the level of their cohomology rings to an algebraic operation also termed a connected sum, which has recently started to be investigated in commutative algebra by Ananthnarayan-Avramov-Moore. It is natural to ask whether algebraic connected sums of graded Gorenstein artinian algebras enjoy the Lefschetz properties in the absence of any underlying topological information. We investigate this question, as well as the analogous question concerning a closely related construction, the fibered product. This is joint work with Chris McDaniel and Tony Iarrobino.

Rebecca R.G.

Characteristic-Free Test Ideals

Abstract: We define the test ideal of a general closure operation cl, and give some of its properties. We highlight connections to the trace ideal and interior operations, and the applications of these viewpoints to the study of singularities of commutative rings. In all characteristics, test ideals coming from big Cohen-Macaulay modules or algebras can take on the role of the tight closure test ideal used in characteristic p > 0 to study singularities.

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WICA: Commutative Algebra, II

Organizers: Sandra Spiroff, Adela Vraciu

Oana Veliche

Linkage And Classification Of Grade Three Perfect Ideals

Abstract: The possible graded-commutative algebra structures on the Tor algebra \(Tor_Q^* (Q/I, k )\), where I is a grade three perfect ideal in a regular local ring Q with residue field k , were identified by Weyman (1989) and by Avramov, Kustin, and Miller (1988). Based on these algebra structures one can classify, in term of numerical parameters, all grade 3 perfect ideals of a regular local ring. Employing linkage theory methods we are able to give a detailed structure of this classification. The talk is based on a work done in collaboration with L.W. Christensen and J. Weyman.

Liana Sega

The Structure Of Quasi-Complete Intersection Ideals

Abstract:We prove that every quasi-complete intersection (q.c.i.) ideal is obtained from a pair of nested complete intersection ideals by way of a flat base change. As a by- product we establish a rigidity statement for the minimal two-step Tate complex associated to an ideal I in a local ring R. Furthermore, we define a minimal two- step complete Tate complex T for each ideal I in a local ring R and prove a rigidity result for it. The complex T is exact if and only if I is a q.c.i. ideal, and in this case, T is the minimal complete resolution of R/I by free R-modules. This is joint work with Andrew Kustin.

Hema Srinivasan

Structure Of Some Semigroup Rings And Their Resolutions

Abstract: Let G = < C > be a numerical semi-group minimally generated by a subset C = (c₁, . . . , c_n) of relatively prime integers in ℕ. If k is a field, then S = k[t^a | a ∈ G] is called the semigroup ring associated to G. This semi-group ring S is isomorphic to k[x₁, . . . , x_n]/IC = R/IC where IC is the kernel of the map φ: k[x₁, . . . , x_n] → S given by φ(x_i) = t^c_i. The minimal R-free resolution of S are known only when either n is small or when C is special in some way. In this note, we will illustrate the known results via a series of examples and explicitly demonstrate their resolutions and other numerical invariants such as Betti numbers, Hilbert Series, Regularity and Frobenius numbers. We will report on some of the results in higher dimension, namely semi-group rings associated to a subset C = (c₁, . . . , c_n)ℕ^d . This is joint work with Philipee Gimenez.

Susan Cooper

The Generalized Minimum Distance Function

Abstract: Relating regular and symbolic powers of homogeneous ideals has played a central theme in numerous problems. In this talk we will explore techniques developed to look at symbolic powers of monomial ideals. Topics surveyed will include linear programs and the symbolic polyhedron. We will highlight invariants such as the initial degree of a homogeneous ideal and the Waldschmidt constant which is an asymptotic invariant used to find failure of containments between symbolic and regular powers.

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WIG: Women in Geometry, I

Organizers: Liz Stanhope, Chikako Mese

Pamela Sargent

Index Bounds For Free Boundary Minimal Surfaces Of Convex Bodies

Abstract: In this talk, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in ℝ₃ tends to infinity as its genus or the number of boundary components tends to infinity.

Xuan Nguyen

Finding Shrinking Doughnuts, An Alternate Proof

Abstract: In this talk, we give an alternate proof of the existence of self-shrinking doughnuts for the mean curvature flow. We use variational methods, a modified curve shortening flow, and a special family of initial rectangles.

Priyanka Rajan

Exotic Spheres Of Cohomogeneity Two

Abstract: Eldar Straume classified non-linear isometric group actions of cohomogeneity two on homo- topy spheres. In a joint work Searle, we provide some evidence for the following: Let Σⁿ be an n−dimensional homotopy sphere of strictly positive sectional curvature. Suppose a compact Lie group G acts on Σⁿ isometrically and effectively by cohomogeneity two. Then Σⁿ is G−equivariantly diffeomorphic to Sⁿ (1) with a linear G−action.

Allie Ray

Eigenvalue And Multiplicity Bounds For The Steklov Spectrum On Orbifolds

Abstract: Previous results on surfaces have found upper bounds for both eigenvalues of the Steklov spectrum and the multiplicity of these eigenvalues based on geometric informaton of the surfaie. Afer reviewing these, we will introduce the notion of an orbifold and see which of these results can be generalized to the orbifold setting.

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WIG: Women in Geometry, II

Organizers: Liz Stanhope, Chikako Mese

Colleen Robles

Title: TBA

Abstract:

Sema Salur

Calibrations On Manifolds With Special Holonomy

Abstract: Examples of n-dimensional Ricci fat manifolds are Riemannian manifolds whose holonomy groups Hol(g) are subgroups of SU(n), for n=2m, and subgroups of the exceptional ie group G_2, for n=7. We call them Calabi-Yau and G_2 manifolds, respectively. They are also examples of manifolds with special holonomy.
Calibrated submanifolds of Calabi-Yau and G_2 manifolds are volume minimizing in their homology classes and their moduli spaces have many important applications in geometry, topology and physics.

Chikako Mese

Harmonic maps into CAT(1) spaces

Abstract: We discuss the Sacks-Uhlenbeck bubbling phenomenon in the context of harmonic maps in singular geometry. The main result is that, given a continuous map from a compact surface into a metric space of curvature bounded from above by 1, either there exists a harmonic map homotopic to this map or there exists a bubble, i.e. nontrivial conformal harmonic map from the standard sphere. The proof uses the harmonic map replacement technique which can be interpreted as a discrete heat flow.

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WIMB: Women in Math Biology, I

Organizers: Angela Peace, Wenjing Zhang

Christina Edholm^*, Blessing O. Emerenini, Anarina L. Murillo, Omar Saucedo, Nika Shakiba, Xueying Wang, Linda J. S. Allen, Angela Peace

Searching For Superspreaders: Identifying Epidemic Patterns Associated With Superspreading Events In Stochastic Models

Abstract:In an infectious disease outbreak, individuals who transmit the disease to a large number of susceptible individuals are know as superspreaders. To investigate the role superspreaders play in an outbreak, we construct both deterministic and stochastic models with two classes of individuals, superspreaders and nonsuperspreaders. We analyze these models and then run numerical simulations for the cases of Middle East respiratory syndrome (MERS) and Ebola. From the analysis and simulations, we gain insight into superspreaders role in the outbreak timeline and severity of the outbreak.

Marissa Renardy

Modeling Tumor Immune Dynamics In Multiple Myeloma

Abstract: We propose a mathematical model that describes the dynamics of multiple myeloma and three distinct populations of the innate and adaptive immune system: cytotoxic T cells, natural killer cells, and regulatory T cells. The model includes significant biologically- and therapeutically- relevant pathways for inhibitory and stimulatory interactions between these populations. We focus on five main aspects: 1) obtaining and justifying parameter ranges and point estimates; 2) determining which parameters the model is most sensitive to; 3) determining which of the sensitive parameters could be uniquely estimated given various types of data; 4) exploring the model and updated parameter estimates numerically; and 5) analytically exploring the equilibria and stability of a reduced model. Using multiple sensitivity analysis techniques, we found that the model is generally most sensitive to parameters directly associated with M protein levels. This analysis provides the foundation for a future ultimate application of the model: prediction of optimal combination regimens in patients with multiple myeloma.

Rebecca Everett

Fronts Of Locusts: Modeling Foraging Behavior In The Australian Plague Locust

Abstract: Locusts gather in large numbers to feed on crops, destroying agricultural fields. Wingless juveniles marching together through a field demonstrate collective behavior that forms a coherent front of advancing insects. We examine this front through two models: an agent-based model and a set of partial differential equations. We construct the agent-based model using observations of individual behavior from the biological literature. The PDE model yields insight into collective behavior of the front. We find selection principles that determine the speed of the front and the amount of food resources left behind as a function of the initial resources and the number of locusts in the group.

Miranda Teboh-Ewungkem^*, Carrie A. Manore, Olivia Prosper, Angela L. Peace, Katharine Gurski, Zhilan Feng

Intermittent Preventive Treatment (IPT) And Antimalarial Drug Resistance Spread

Abstract: We develop an age-structured ODE model to investigate the role of Intermittent Preventive Treatment (IPT) in averting malaria-induced mortality in children, and its related cost in promoting the spread of anti-malarial drug resistance. IPT, a malaria control strategy in which a full curative dose of an antimalarial medication is administered to vulnerable asymptomatic individuals at specified intervals, has been shown to reduce malaria transmission and deaths in children and pregnant women. However, it can also promote drug resistance spread. Our mathematical model is used to explore IPT effects on drug resistance and deaths averted in holoendemic malaria regions. The model includes drug-sensitive and drug-resistant strains as well as human hosts and mosquitoes. The basic reproduction and invasion reproduction numbers for both strains are derived. Numerical simulations show the individual and combined effects of IPT and treatment of symptomatic infections on the prevalence of both strains and the number of lives saved. Our results suggest that while IPT can indeed save lives, particularly in high transmission regions, certain combinations of drugs used for IPT and to treat symptomatic infection may result in more deaths when resistant parasite strains are circulating. Moreover, the half-lives of the treatment and IPT drugs used play an important role in the extent to which IPT may influence spread of the resistant strain. A sensitivity analysis indicates the model outcomes are most sensitive to the reduction factor of transmission for the resistant strain, rate of immunity loss, and the natural clearance rate of sensitive infections.

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WIMB: Women in Math Biology, II

Organizers: Angela Peace, Wenjing Zhang

Katharine Gurski

A Sexually Transmitted Disease Model With Longterm Partnerships In Homogeneous And Heterogeneous Populations

Abstract:Sexually transmitted diseases are spread through short term casual relationships and longterm partnerships. However, longterm relationships, much less the issues of concurrent relationships, are not simple to model in population based ODE systems. So many modelers have abandoned population, pair formation, and pair approximation models in favor of network models and simulations. Despite the continued growing power of computers to run more complex numerical simulations, there is still a need for analytic models where it is easy to understand the effect of heterogeneity on parameter estimation, to develop and validate approximation schemes for epidemics, to strengthen the link between modeling and epidemiologically relevant data, and to design intervention strategies. In this talk I present a population model that can account for the possibilities of an infection from either a casual sexual partner or a longtime partner who was uninfected at the start of the partnership. The model allows for multiple longterm partnerships, which adds the advantage that network models have, the means to include serially monogamous and concurrent relationships, within the traditional strengths of a population. The model can be further diversified by including populations divided by sexual behavior, age, and/or race/ethnicity.

Zhilan Feng^*, John W. Glasser

Implications For Infectious Disease Models Of Heterogeneous Mixing On Control Thresholds

Abstract: Mixing among sub-populations, as well as heterogeneity in characteristics affecting their reproduction numbers, must be considered when evaluating public health interventions to prevent or control infectious disease outbreaks. n this talk, we model preferential within- and proportional among-group contacts in compartmental models of disease transmission and derive results for the overall effective reproduction number (Rv) assuming different levels of vaccination in the sub-populations. Specifically, we unpack the dependency of Rv on the fractions of contacts reserved for individuals within one’s own subgroup and show that Rv increases as this fraction increases in a given sub-population. These considerations lead to our proposing the gradient of Rv with respect to subgroup vaccination fractions as a measure by which to evaluate interventions. Another significant result is that for general mixing schemes, both R0 and Rv are bounded below and above by their corresponding expressions when mixing is proportionate and isolated, respectively. This work is based on (1) Glasser et al., Lancet infectous Diseases (001)) htp://dx.doi.org/10.101)/S173—09999(1))00007-99, (0) Feng et al., J. Theor. Biol. -8) (0015) 133–183, and (-) Poghotanyan et al. J. Math. Biol. (0018) htps://doi.org/10.1003/s00085-018-101)-1.

Heather Brooks^*, Maryann E. Hoh, Candice R. Price, Ami Radunskaya, Suzanne S. Sindi, Nakeya D. Williams, Shelby Wilson, Nina H. Fefferman

Parasites And The Evolution Of Sociality: How Social Complexity And Grooming Efficiency Affect The Selective Pressures On Group Organization

Abstract: Individuals who live in close, collaborative social groups are susceptible to infectious diseases such as pathogens and parasites. Ectoparasites are a particularly interesting case because social grooming (allogrooming) reduces the parasite load of one individual while potentially exposing both the groomer and groomee to additional transmission. Using an agent-based model that shows parasite spread based on individual behavior in a dynamic network, we model the interactions between social organization and allogrooming efficiency to consider whether or not certain physiological or energetic expenditures may have been required to allow the evolution/existence of increasingly complex social systems. Conversely, we explore whether social complexity may have been an adaptation to alleviate burdens of allogrooming under parasitic threat. We also consider the role of social position for individuals (where status is often correlated with frequency of allogrooming) and con- textualize the fitness consequences for individuals in both high and low ranking positions as they feed back to determine the fitness of the whole population. The collaborative research group for this project was formed in April 2017 at the Women Advancing Mathematical Biology conference, which was hosted at the Mathematical Biosciences Institute (The Ohio State University).

Wenjing Zhang

Global Stability And Re-emergence In A Cholera Model Considering Stochastic Fluctuations In Pathogen-Host Encounter

Abstract: Endemic and re-emergent cholera outbreaks are still a great threat to both developing and developed countries. In this project, complicated cholera transmission dynamics are captured by a simple compartmental model with a novel encounter rate, which considers stochastic fluctuations in pathogen-host encounters. The transmission rates for human-to-human and environment-to- human routes are investigated for the existence of two endemic equilibriums and backward bi- furcation when the basic reproduction number is less than one. The condition for a complete disease elimination is obtained from a globally stable disease-free equilibrium. The condition for a persistent endemic cholera is obtained from a globally stable endemic equilibrium. A Lyapunov function and compound metrics are employed for global analyses. Hopf bifurcation occurs and provides an oscillating source for semi-annual to multi-annual cholera outbreak patterns. Two-parameter bifurcation analysis and diagrams (with respect to two transmission rates for human-to-human and environment-to-human routes) illustrate the parameter regions for the array of disease dynamics. Numerical simulations are shown to demonstrate the corresponding dynamical behaviors.

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WIMM: Women in Math Materials, I

Organizers: Malena Espanol, Hala Ah Shehadeh

Lidia Mrad

Chromonic Liquid Crystals And Applications To Modeling DNA In Free Solution

Abstract: Liquid crystals are a distinct phase of matter existing between the chaos of isotropic liquids and the order of crystalline solids. In addition to being partially ordered, some liquid crystals manifest sensitivity to changes in concentration when added to a solution. Chromonic liquid crystals, which fall under this category, have important biological applications. They are composed of disc-like molecules that form rings when reaching a certain concentration, and these in turn aggregate into interesting geometrical shapes. An important question in this setup is how the dominant mechanism – shape formation in this case – is affected by specific system parameters. We formulate the model as an energy minimization problem allowing us to use several variational tools. Our results address existence and computation of solutions to the ensuing partial differential equations, taking into account key parameters. As confined DNA forms chromonic liquid crystal phases, these results in turn shed light on the packing mechanism of a viral DNA in a capsid. This is joint work with M. C. Calderer, M. Espanol, E. Panagiotou, R. Selinger, L. Xu, and L. Zhao.

Silvia Jimenez Bolanos*, Anna Zemlyanova, and Marta Lewicka

Materials Science And Differential Geometry

Abstract: In this talk, I will present the results obtained by our group as a consequence of the collaboration started at the Women in Mathematics of Materials Workshop. Our project deals with analytical and geometrical questions coming from the study of elastic material that exhibit residual stress at free equilibria; for example, plastically strained sheets, growing tissues, atomically thin graphene layers, etc. These and other phenomena can be studied through a variational model, pertaining to the non-Euclidean version of nonlinear elasticity, which postulates formation of a target Riemannian metric, resulting in the shape formation of the tissue that attains an orientation-preserving configuration closest to being the metric’s isometric immersion.

Xingjie Helen Li^*, Derek Olson

Bending Admissible Blended Force-Based Coupling Method For Single-Layered 2D Crystal

Abstract: Out-of-plane effects due to defects are observed experimentally in single-layered two- dimensional structures such as graphene and hexagonal boron-nitride. In this talk, we propose a new continuum approximation which allows large out-of-plane deformations observed in two-dimensional materials. Based on this new formulation of continuum energy, we apply the force-based quasi-continuum method to such 2D materials for which point defects and out-of-plane behavior including bending are incorporated into the model. The rigorous analyses confirm the consistency and stability of the coupling model and provide precise guidelines for practical finite element implementations.

Ling Xu

On The Viscous Lamb Dipole

Abstract: Self-propelled Lamb dipoles are fundamental units of large-scale vortical flows, for example, the ocean currents. Here we present a numerical study of the viscous Lamb dipole for Reynolds numbers in the range [125, 1000]. The focus is on the effects of convective and diffusion. The influence of these two terms are isolated and discussed by comparing solutions of the Navier-Stokes equation (NSE) with solutions of the diffusion equation (DE).

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WIMM: Women in Math Materials, II

Organizers: Malena Espanol, Hala AH Shehadeh

Yue Yu^*, Huaiqian You, Xin Yang Lu, Nathaniel Trask

A Neumann-Type Boundary Condition For Nonlocal Problems

Abstract: In this talk we consider 2D nonlocal models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, we present a new generalization of classical local Neumann conditions that recovers the local case as O(δ²) in the L^∞(Ω) norm. This convergence rate is optimal considering the O(δ²) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for domains with corners. To verify the analysis, we discretize the nonlocal boundary condition using an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ²) convergence.

Amanda Howard^*, Lei Cao, Hala AH Shehadeh, Yekaterina Epshteyn, Pania Newll

A Mathematical Model For Fluid Flow In A Fractured Media

Abstract: The modeling of fluid flow is a key challenge in many fields, including petroleum engineering and material sciences. Accurate modeling is essential for understanding and predicting the fluid flow and the behaviors of the fractured bulk, as well as the coupling between the two. In spite of very active recent research on phase-field modeling of fracture initiation and growth, there are still many questions remain open. Existing models often assume that properties such as the porosity are constant across the bulk, however, in these applications the domains can contain anisotropic inclusions and discontinuous material properties. This talk will focus on the development and implementation of a more accurate mathematical model for cou- pling fluid flow and fracture in porous media to improve understanding of the interaction between the fluid and the fractured system to study the effect of porosity on fracture evolution.

Xiaochuan Tian

Numerical Mathematics For Peridynamics And Nonlocal Models

Abstract: Nonlocal continuum models are in general integral-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in numerical analysis with nonlocality involved. We present in this talk numerical analysis for nonlocal models characterized by a horizon parameter which measures the range of nonlocal interactions. Considering their close connections to classical local PDE models in the limit when the horizon parameter shrinks to zero and to global fractional PDEs in the limit when the horizon parameter tends to infinity, we present numerical schemes that are robust under the changes of the horizon parameter. Those schemes are effective to deal with multiscale models where different scales of nonlocality are presented.

Malena Espanol

Discrete-To-Continuum Modeling Of Weakly Interacting Incommensurate Lattices

Abstract: In this talk, we present a formal discrete-to-continuum procedure to derive a continuum variational model for two chains of atoms with slightly incommensurate lattices. The chains represent a cross-section of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. We show that numerical solutions for both models demonstrate the presence of large commensurate regions separated by localized incommensurate domain walls. We show how this approach can be extended to modelling incommensurate square lattices and hexagonal honeycomb lattices.

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WIN: Women in Numbers, I

(organized by Michelle Manes and Ila Varma)

Min-Joo Jang

Quantum Modular Forms And Singular Combinatorial Series

Abstract: Since Dyson defined the rank of a partition, a number of studies have been done on this statistic. For example, a celebrated result of Bringmann and Ono showed that the rank generating function is essentially a mock modular form. Andrews introduced k-marked Durfee symbols and more generally defined the ranks for them. In particular, when k = 1 one recovers Dyson’s rank. In this talk, we establish the quantum modular properties of this combinatorial series, the rank generating function for k-marked Durfee symbols. This is joint work with Amanda Folsom, Sam Kimport, and Holly Swisher.

Christelle Vincent

Computing Hyperelliptic Modular Invariants From Period Matrices

Abstract: We define the modular invariants of a hyperelliptic curve to be the value of certain Siegel
modular functions that correspond to classical invariants of hyperelliptic curves evaluated at a
period matrix of the Jacobian of the curve. In this talk we discuss this correspondence between
modular functions and invariants of curves as well as certain computational considerations that
arise when recognizing the invariants as algebraic numbers from their floating point
approximation. This is joint work with Ionica, Kilicer, Lauter, Lorenzo Garcia, Massierer and
Manzateanu.

Jiuya Wang

Inductive Method in Counting Number Field

Abstract: We propose general frameworks to inductively count number fields building on previously known counting results and good uniformity estimates in different flavors. By using this method, we prove new results in counting number fields with Galois groups ranging from direct product to wreath product. We will also mention interesting applications eu route.

Alina Bucur

Statistics For Points On Curves Over Finite Fields

Abstract:A curve is a one-dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families. This is joint work with Chantal David, Brooke Feigon, Kiran S. Kedlaya, and Matilde Lalin.

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WIN: Women in Numbers, II

Organizers: Michelle Manes and Ila Varma

Yuan Liu

On The Level Of Modular Curves That Give Rise To Sporadic J-Invariants

Abstract: In this talk, we discuss points on X₁(n) of unusually low degree, the so-called sporadic points, and we focus particularly on sporadic points corresponding to non-CM elliptic curves. We show that non-CM non-cuspidal sporadic points on X₁(n) map to sporadic points on X₁(d), for d some bounded divisor. This is joint work with Abbey Bourdon, Özlem Ejder, Frances Odumodu and Bianca Viray.

Catalina Camacho Navarro

Modular Curves Of Low Composite Level And Genus Zero Subgroups

Abstract: Let E be an elliptic curve defined over ℚ without complex multiplication. For every positive integer N, the Galois group Gal\((\bar{\mathbb{Q}}/\mathbb{Q})\) induces an action on the N-torsion points of E and so there is a representation \(\rho_{E,n} : \mbox{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mbox{GL}_2(\mathbb{Z}/n\mathbb{Z})\). In recent years, Zureick– Brown, Rouse, Sutherland and Zywina have made significant progress towards classifying the subgroups of GL₂(\(\widehat{\mathbb{Z}}\)) which contain subgroups that are conjugate to images of Galois for some elliptic curve. Based of work by Sutherland–Zywina, Morrow began the study of the composite-(m₁ , m₂ ) image of Galois in the case where m₁ is a power of 2 and m₂ is a prime ≤ 13. We continue the study of composite-(m1, m2) image of Galois, using Galois representations and modular curves. For each subgroup of G ⊂ GL₂(Z/m₁m₂Z) where the modular curve has low genus, we construct a hyperelliptic model and use different methods, along with the software Magma to find almost all rational points. Finally we determine which ones correspond to sporadic points. This is joint work with Catalina Camacho-Navarro, Wanlin Li, Jack Petok, Jackson S. Morrow and David Zureick-Brown.

McKenzie Rachel West

Brauer-Manin Computations For A Family Of K3 Surfaces

Abstract: To study rational points on surfaces, we generally begin by looking at equivalence classes of curves, the Picard Group, on that surface. Elements of the Picard group can then be used to describe algebras in the Brauer group of the surface. In turn, one can use these Brauer algebras along with an observation of Manin to disprove the existence of global points despite the existence of local points. We examine a particular family of K3 surfaces, finding the Picard group and studying the possibility of an obstruction.

Jennifer Berg

Odd Ordered Transcendental Obstructions To The Hasse Principle On K3 Surfaces

Abstract: K3 surfaces are 2-dimensional analogues of elliptic curves, but lack a group structure. Moreover, they need not have rational points. However, in 2009 Skorobogatov conjectured that the Brauer group (a torsion abelian group which encodes reciprocity laws) should account for all failures of the local-to-global principle for rational points on K3 surfaces. In this talk I will briefly describe the geometric origin of certain 3-torsion classes in the Brauer group of a K3 surface. We utilize this geometric description to show that these classes can in fact obstruct the existence of rational points. This is joint work with Tony Várilly-Alvarado.

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WINART: Women in Noncommutative Algebra and Representation Theory, I

Organizers: Van C. Nguyen, Julia Plavnik, Sarah Witherspoon

Khrystyna Serhiyenko

Mutation Of Type D Friezes

Abstract: Frieze is a lattice of positive integers satisfying certain rules. Friezes of type A were first studied by Conway and Coxeter in 1970’s, but they gained fresh interest in the last decade in relation to cluster algebras. Moreover, the categorification of cluster algebras developed in 2006 yields a new realization of friezes in terms of representation theory of Jacobian algebras. Thus, a frieze is an array of positive integers on the Auslander-Reiten quiver of a finite Jacobian algebra such that entires on a mesh satisfy a certain rule. In this talk, we will discuss friezes of type D and their mutations. This is joint work with A. Garcia Elsener.

Lauren Grimley

Deformations Of Quantum Complete Intersections

Abstract: We consider deformations of quantum complete intersections ex- tended by finite groups. As deformations of an associative algebra are dic- tated by the Hochschild cohomology of that algebra, we will characterize the cohomological structure and compare to conditions on totally ordered monomial bases given by the Poincare-Birkoff-Witt Theorem. Among the deformations of our choice algebra are a class of algebras which we call trun- cated quantum Drinfeld Hecke algebras in view of their relation to classical Drinfeld Hecke algebras. This is joint work with Christine Uhl.

Elizabeth Wicks

Frobenius-Perron Theory Of Modified ADE Bound Quiver Algebras

Abstract: The Frobenius-Perron dimension for an abelian category was recently introduced. We apply this theory to the category of representations of the finite-dimensional radical square zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius-Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius-Perron dimension of a radical square zero bound quiver algebra. We use this result to introduce a family of abelian categories which produce arbitrarily large irrational Frobenius-Perron dimensions.

Gordana Todorov

Cyclic Posets And Triangulation Clusters

Abstract: Triangulated categories coming from cyclic posets were originally introduced in [IT15] as a generalization of the constructions of various triangulated categories with cluster structures. We will give an overview, and analyze triangulation clusters which are those corresponding to topological triangulations of the 2- disk. Locally finite non-triangulation clusters give topological triangulations of the cactus space associated to the cactus cyclic poset.
[IT15] Continuous Cluster Categories, Algebras and Representation Theory, Vol.18, (2015), pp.65-101 (arXiv:1209.1879).

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WINART: Women in Noncommutative Algebra and Representation Theory, II

Organizers: Van C. Nguyen, Julia Plavnik, Sarah Witherspoon

Ellen Kirkman

Bounds On The Degrees Of Minimal Generators Of Invariants

Abstract: Let k be a field of characteristic zero. In 1916 E. Noether proved that if G is a finite group acting naturally on a polynomial ring k[x₁, . . . , x_n], then the degrees of a minimal set of generators of the subring of invariants are bounded above by the order of the group, and in 2011 P. Symonds proved that for general k, an upper bound is n(|G| − 1) when n ≥ 2 and |G| > 1. Replacing k[x₁, . . . , x_n] by an Artin- Schelter regular algebra A, we obtain examples (in characteristic zero) where the Noether bound on the degrees of a minimal set of generators of A^G does not hold, and we prove upper bounds for some particular algebras.

Qing Zhang

Classification Of Super-Modular Categories By Rank

Abstract: A super-modular category is a unitary pre-modular category with Müger center equivalent to the symmetric unitary category of super-vector spaces. Super-modular categories are important alternatives to modular categories. For example, any unitary pre-modular category is the equivariantization of a modular or super-modular category. Physically, super-modular categories are related to the study of fermionic topological phases of matter. In this talk, we will talk about classifying these categories up to rank 8.

Jieru Zhu

Two Boundary Centralizer Algebras For q(n)

Abstract: The Sergeev duality states that the action of the Type Q Lie superalgebra q(n) and the Sergeev algebra fully centralize each other on the tensor space. Hill-Kujawa-Sussan (2011) generalized this work to the one boundary setting. We further study the two boundary generalization and define the degenerate two boundary affine Hecke-Clifford algebra Bd using generators and relations. It admits a q(n)-linear action on M ⊗ N ⊗ V^⊗d for the natural representation V and arbitrary q(n)-modules M and N. When M and N are polynomial modules parametrized by a staircase and a single row partition, respectively, the action of B_d factors through a quotient algebra H_d. Using combinatorial tools such as the Bratteli diagram and shifted Young tableaux, we construct simple modules for H_d. These modules occur as irreducible H_d-summands of M ⊗ N ⊗ V^⊗d.

Frauke Bleher

Top Exterior Quotients Of Iwasawa Modules

Abstract: This talk is about joint work with Ted Chinburg, Ralph Greenberg, Mahesh Kakde, Romyar Sharifi and Martin Taylor about higher codimension Iwasawa theory. Iwasawa theory produces from Galois theory certain finitely generated torsion modules M, called Iwasawa modules, over certain Noetherian integral domains R, called Iwasawa algebras. In this talk, I will concentrate on a result in which we obtain an isomorphism between a quotient module of a top exterior power of an Iwasawa module and a quotient of an Iwasawa algebra modulo a sequence of L-functions.

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WINASC: Women in Numerical Analysis and Scientific Computing, I

Organizers: Bo Dong, Adrianna Gillman

Annalisa Quaini

A Computational Study Of Lateral Phase Separation In Biological Membranes

Abstract: Conservative and non-conservative phase-field models are considered for the numerical simulation of lateral phase separation and coarsening in biological membranes. An unfitted finite element method is devised for these models to allow for a flexible treatment of complex shapes in the absence of an explicit surface parametrization. For a set of biologically relevant shapes and parameter values, we compare the dynamic coarsening produced by conservative and non-conservative numerical models, its dependence on certain geometric characteristics and convergence to the final equilibrium. This is joint work with V. Yushutin and M. Olshanskii (Mathematics, UH), and S. Majd (Biomedical Engineering, UH).

Beatrice Riviere

Numerical Methods For Solving Linear Poroelasticity Equations

Abstract: The modeling of poroelastic deformation arises in many fields including biomechanics, energy and environmental engineering. We propose and analyze discontinuous Galerkin methods for solving the linear poroelasticity equations. In a first approach, the flow and mechanics equations are solved fully implicitely. In a second approach, the equations are decoupled and solved sequentially at each time step. Theoretical error estimates are derived. Applications to reservoir engineering and bio-medicine are shown.

Lise-Marie Imbert-Gerard

Integral Equation Methods For Acoustics In Smoothly Varying, Anisotropic Media

Abstract: Our goal is to develop integral equation based numerical methods for the solution of acoustic scattering problems involving anisotropic, inhomogeneous media. We will present a collection of well-conditioned integral equation formulations and will illustrate their performance using iterative solution methods coupled with an FFT-based technique to discretize and apply the relevant integral operators.

Karin Leiderman

A Density-Dependent FEM-FCT Algorithm With Application To Modeling Platelet Aggregation

Abstract: Upon injury to a blood vessel, flowing platelets will aggregate at the injury site, forming an plug to prevent blood loss. Through a complex system of biochemical reactions, a stabilizing mesh forms around the platelet aggregate forming a stable blood clot that further seals the injury. Computational models of clot formation have been developed to a study thrombosis, where a vessel injury does not cause blood leakage outside the blood vessel but blocks blood flow. To model scenarios in which blood leaks from a main vessel out into the extravascular space, new computational tools need to be developed to handle the complex geometries that represent the injury. We have previously modeled intravascular clot formation under flow using a continuum approximation wherein the transport of platelet densities into some spatial location is limited by the volume fraction of platelets that already reside in that location, i.e., the densities satisfy a maximum packing constraint through the use of a hindered transport coefficient. To extend this notion to extravascular injury geometries, we have modified a finite element method flux corrected transport (FEM- FCT) scheme by prelimiting anti-diffusive nodal fluxes. We show that our modified scheme, under a variety of test problems, including mesh refinement, structured vs unstructured meshes and for a range of reaction rates, produces numerical results that satisfy a maximum platelet-density packing constraint.

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WINASC: Women in Numerical Analysis and Scientific Computing, II

Organizers: Bo Dong, Adrianna Gillman

Jingwei Hu

A Second-Order Asymptotic-Preserving And Porosity-Preserving Exponential Runge-Kutta Method For A Class Of Stiff Kinetic Equations

Abstract: We introduce a second-order time discretization method for stiff kinetic equations. The method is asymptotic-preserving (AP) — can capture the Euler limit without numerically resolving the small Knudsen number; and positivity-preserving — can preserve the non-negativity of the solution which is a probability density function for arbitrary Knudsen numbers. The method is based on a new formulation of the exponential Runge-Kutta method and can be applied to a large class of stiff kinetic equations including the BGK equation (relaxation type), the Fokker-Planck equation (diffusion type), and even the full Boltzmann equation (nonlinear integral type). Furthermore, we show that when coupled with suitable spatial discretizations the fully discrete scheme satisfies an entropy-decay property. Various numerical results are provided to demonstrate the theoretical properties of the method. This is joint work with Ruiwen Shu (University of Maryland).

Chiu-Yen Kao

Maximal Convex Combinations Of Sequential Steklov Eigenvalues

Abstract: In this work, we study a shape optimization problem in two dimensions where the objective function is the convex combination of two sequential Steklov eigenvalues of a domain with a fixed area constraint. We show the existence of the optimal domain and the nondecreasing, Lipschitz continuity, and convexity of the optimal objective function with respect to the convex combination constant. On one-parameter family of rectangular domains, asymptotic behaviors of lower eigenvalues are found. For general shapes, numerical approaches are used to find optimal shapes. The range of the first two Steklov eigenvalues are discussed for several one-parameter families of shapes including Cassini oval shapes and Hippopede shapes. (This is joint work with Weaam Alhejaili.)

Wei Wang

A High Order Well-Balanced Particle-In-Cell Method For Shallow Water Equations

Abstract: A hybrid Eulerian–Lagrangian particle–in–cell (PIC) type numerical method is developed for the solution of advection dominated flow problems. For smooth flows the method presented is of formal high–order accuracy in space. The method is applied to solve the non–linear shallow water equations resulting in a new, and novel, shock capturing shallow water solver. The approach is able to simulate complex shallow water flows which can contain an arbitrary number of discontinuities. Both trivial and non–trivial bottom topography is considered and it is shown that the new scheme is inherently well–balanced exactly satisfying the C–property. The scheme is verified against several 1D benchmark shallow water problems.

Yingda Cheng

Numerical Methods For Nonlinear Maxwell’s Equations In Optics

Abstract: In this talk, we present several high order numerical scheme for Maxwell’s equation in nonlinear optics. The considered methods include finite difference/discontinuous Galerkin methods in space, and energy stable time integrators. We study the property of the scheme by investigating the dispersion relations in linear Lorentz media.

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WinCompTop: Women in Computational Topology, I

Organizers: Erin Chambers, Brittany Terese Fasy, Elizabeth Munch

Sara Kalisnik Verovsek*, Bernd Sturmfels, Paul Breiding and Madeline Weinstein

Learning Algebraic Varieties From Samples

Abstract:I will discuss how to determine a real algebraic variety from a fixed finite sample of points and what to do with that information. For example, from the equations defining a variety one can learn the degree and the dimension of the variety. One can also construct ellipsoid complexes which, based on the experiments, strengthen the topological signal for persistent homology. All the algorithms needed are made available in a Julia package.

Rachel Neville

Topological Techniques For Characterization Of Pattern Forming Systems

Abstract:Complex spatial-temporal patterns can be difficult to characterize quantitatively. In particular, distinguishing between visually similar patterns formed under different conditions is challenging. These small differences are detectable by persistent homology. We describe how persistent homology can be used as a low-dimensional quantitative summary of topological structure of dynamic data. These summaries retain a remarkable amount of information that allows for the investigation of the influence of nonlinear parameters, classification of data by parameters, and study of defect evolution.

Michelle Feng

Title: TBA

Abstract:

Violeta Kovacev-Nikolic

Visual And Statistical Comparison Of Simplicial Complexes

Abstract: Starting from the graphical representation of a simplicial complex as it undergoes changes with increasing filtration, we will use this visual tool to qualitatively compare two different complexes. For quantitative comparisons we will implement a statistical approach. Both methods of comparison will be demonstrated on an example of a simplicial complex built from point-cloud data.

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WinCompTop: Women in Computational Topology, II

Organizers: Erin Chambers, Brittany Terese Fasy, Elizabeth Munch

Radmila Sazdanovic

Machine Learning Revelations From the Color Jones Polynomial

Abstract: A multitude of knot invariants have been found and calculated in the last few decades. In thin talk we discuss several lessons machine learning has gathered from analyzing this menagerie of information. Critically, we find that the color Jones polynomial encodes a significant level of information about the signature of a knot and other invariants. This has important ramifications for a conjecture if Garoufalidis and the calculation of the Khovanov.

Moira Chas

Computer Driven Questions And Theorems And In Geometry

Abstract: On an orientable surface S with negative Euler characteristic, there exists a minimal set of generators of the fundamental group of S, and a hyperbolic metric. Associated to each homotopy class C of closed oriented curves on S are three numbers: the word length, the geometric self-intersection number, and the geometric length. These three numbers can be explicitly computed. These computations lead to counterexamples to existing conjectures, new conjectures and sometimes to new theorems.

Mao Li

Applications Of Topological Data Analysis In Plant Science

Abstract: Plants are the backbone of all life on Earth as they provide food, oxygen; they purify the water; they provide the starting materials for clothing, fuel, and medicine. However, efforts to understand the genetic and environmental conditioning of plant morphology is hindered by the lack of flexible and effective tools for quantifying morphology. Here, we show a few examples to apply topological data analysis methods to quantify diverse plant morphologies that can significantly increase the ability to capture phenotypic variation. We describe the applications on different crops such as sorghum, tomato, and maize; different type of data such as point cloud, 2D image, and 3D X-ray image; different scale of image from microscope to satellite images.

Sarah Tymochko

Using Persistent Homology To Quantify A Diurnal Cycle In Hurricane Felix

Abstract: The tropical cyclone (TC) diurnal cycle is a regular, daily cycle in hurricanes that may have implications for the structure and intensity of hurricanes. This pattern can be seen in a cooling ring forming in the inner core of the storm near sunset and propagating away from the storm center overnight, followed by warmer clouds on its inside edge. The current state of the art for diurnal cycle measurement has a limited ability to analyze the behavior beyond qualitative observations. Our method creates a more advanced mathematical method for quantifying the TC diurnal cycle using tools from Topological Data Analysis, specifically one dimensional persistent homology. Using geostationary operational environmental satellite (GOES) IR imagery data from Hurricane Felix in 2007, our method is able to detect an approximately daily cycle in the hurricane.

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WISDM: Women in the Science of Data and Mathematics, I

Organizers: Linda Ness, Carlotta Domeniconi

Carolyn Mayer

Erasure Coding Techniques For Content Download

Abstract: Large volumes of data must be stored reliably and efficiently. Furthermore, there is demand for fast and efficient retrieval and processing of large data files. Recent work has proposed using erasure codes to address both goals. We will discuss progress on emerging problems that involve coding. In this talk, we will focus on service rates in coded storage systems. A combination of coding and replication can be used to shape service rate regions of distributed storage systems.

Anna Ma^* and Jamie Haddock

A Dynamic Sampling Approach To SKM Method

Abstract: Stochastic iterative algorithms have gained recent interest in machine learning and signal processing for solving large-scale systems of equations, Ax=y. One such example is the Randomized Kaczmarz (RK) algorithm, which acts only on single row of the matrix A at a time. While RK randomly selects a row of A to work with, Motzkin’s algorithm employs a greedy row selection. Connections between the two algorithms resulted in the Sampling Kaczmarz-Motzkin (SKM) algorithm which samples a random subset of rows of A and then greedily selects the best row of the subset. In this work, previous analysis of RK and its greedy counterpart, Motzkin’s algorithm, motivates the question “How greedy is too greedy?”

Karamatou Yacoubou-Djima

Heuristic Framework For Multi-Scale Testing Of The Multi-Manifold Hypothesis

Abstract: Global linear models often overestimate the number of parameters required to analyze or efficiently represent datasets, for example when a data set in sampled from a manifold of lower dimension than the ambient space. The manifold hypothesis consists in asking whether data lies on or near a d-dimensional manifold or is sampled from a distribution supported on a manifold. In this talk, we outline a heuristic framework for a hypothesis test suitable for computation and empirical data analysis. We consider two datasets made of multiple manifolds and test our manifold hypothesis on a set of spline-interpolated manifolds constructed based variance-based intrinsic dimensions computed from the data.

Yang Chen^*, Xiao-Li Meng, Xufei Wang, David A. van Dyk ,Herman L. Marshall, Vinay L. Kashyapk

Calibration Concordance For Astronomical Instruments Via Multiplicative Shrinkage

Abstract: Calibration data are often obtained by observing several well-understood objects simultaneously with multiple instruments, such as satellites for measuring astronomical sources. Analyzing such data and obtaining proper concordance among the instruments is challenging when the physical source models are not well understood, when there are uncertainties in “known” physical quantities, or when data quality varies in ways that cannot be fully quantified. Furthermore, the number of model parameters increases with both the number of instruments and the number of sources. Thus, concordance of the instruments requires careful modeling of the mean signals, the intrinsic source differences, and measurement errors. In this paper, we propose a log-Normal model and a more general log-t model that respect the multiplicative nature of the mean signals via a half-variance adjustment, yet permit imperfections in the mean modeling to be absorbed by residual variances. We present analytical solutions in the form of power shrinkage in special cases and develop reliable Markov chain Monte Carlo (MCMC) algorithms for general cases, both of which are available in the Python module CalConcordance. We apply our method to several datasets including a combination of observations of active galactic nuclei (AGN) and spectral line emission from the supernova remnant E0102, obtained with a variety of X-ray telescopes such as Chandra, XMM-Newton, Suzaku, and Swift. The data are compiled by the International Astronomical Consortium for High Energy Calibration (IACHEC). We demonstrate that our method provides helpful and practical guidance for astrophysicists when adjusting for disagreements among instruments.
Keywords: Adjusting attributes; shrinkage estimator; Bayesian hierarchical model; log-Normal model; half-variance adjustment; log-t model.

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WISDM: Women in the Science of Data and Mathematics, II

Organizers: Linda Ness, Carlotta Domeniconi

Giseon Heo

Analysis Of Facial Morphology Of Pediatric Obstructive Sleep Apnea Patients

Abstract: Among children and adolescents, the prevalence of obstructive sleep apnea (OSA) has been
reported to range from 1% to 5%. The gold standard for diagnosis of pediatric OSA is
polysomnography. However, access to polysomnography is severely limited and many children
do not have an appropriate diagnosis before treatment. In this study, we explore the possibility
of classifying patients at risk of OSA from their facial morphology. We will compare the
supervised learning results obtained by 2D persistence and convolutional neural networks. This
is joint work with Milad Kiaee.

Patricia Medina

Deep Learning In Crowd Flow Exit Data

Abstract: We consider black box simulation data modeling crowds exiting different configurations of a one story building. We explore machine learning techniques to estimate the exit times of N agents, using a feed forward neural network for supervised multioutput regression. Based upon a number of experiments we have settled on a “sliding window” approach for feature generation, whereby a particular agent is represented by features derived across a number of time slices, rather than restricting ourselves to a more naive agent representation involving information at only a single time step. Such a representation promised to provide a richer representation for each agent and thereby provide for more robust predictions. In particular, our methodology follows a three stage process. First, we perform feature engineering on the original data by using a “sliding window technique”. Second, we take the input features and preform various types of dimensionality reduction. Herein we study the differences between linear dimensionality reduction using Principal Component Analysis (PCA) and using an auto-encoder to do non-linear dimensionality reduction. In both cases, after the new lower-dimensional features are produced, a feed-forward neural network is used to make predictions of the agent exit times. Our proposed methodology promises a number of advantages. First, since the dimensionality reduction is performed without access to the measured exit times of the agents, at least that part of our procedure is safe from over-fitting. Second, as the problem of interest is quite complex, there is likely a non-linear relationship between our measured features (e.g., the initial position of the agents) and their final exit time.

Jessica Metcalf-Burton

Hubness: What Is It, And What’s It Good For?

Abstract: In this talk we introduce the concept of hubness, consider its potental applicatons, and discuss some open problems. Hubness is like a backwards version of nearest neighbors, where instead of asking “what are my nearest neighbors” a point asks “how many other points have me as their nearest neighbor?” More formally, given a collecton of items, the (k-)hubness score of an item j is the number of other items that consider j a (k-)nearest neighbor. Even in high dimensions where distances between items are virtually indistnguishable, points of high hubness emerge. There are indicatons that hubness scores may be useful for clustering, outlier detecton, and computaton of intrinsic dimensionality. Since the propertes of hubness are not fully understood, there is plenty of room for mathematcal exploraton. This talk is based on work begun at the 2017 Women in Data Science and Mathematcs esearch Collaboraton Workshop.

Anna Little^*, James Murphy, Mauro Maggioni

Path-Based Spectral Clustering: Guarantees, Robustness To Outliers And Fast Algorithms

Abstract:This talk will discuss new performance guarantees for robust path- based spectral clustering and an efficient approximation algorithm for the longest leg path distance (LLPD) metric, which is based on a sequence of multiscale adjacency graphs. LLPD-based clustering is informative for highly elongated and irregularly shaped clusters, and we prove finite- sample guarantees on its performance when random samples are drawn from multiple intrinsically low-dimensional clusters in high-dimensional space, in the presence of a large number of high-dimensional outliers. More specifically, we derive a condition under which the Laplacian eigengap statistic correctly determines the number of clusters for a large class of data sets, and prove guarantees on the number of points mislabeled by our method. Our methods are quite general and provide performance guarantees for spectral clustering with any ultrametric. We also propose a fast algorithm for implementing path-based spectral clustering which has complexity quasilinear in the number of data points.

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WiSh: Women in Shape Modeling, I

Organizers: Kathryn Leonard, Terry Knight

Athina Panotopoulou

Scaffolding A Skeleton

Abstract: A skeleton allows one to quickly describe a shape, while its boundary surface allows one to visualize the shape. We present an algorithm to construct a quadrilateral mesh around a one-dimensional skeleton that is as coarse as possible, the ”scaffold”. This scaffold is then a potential support for the surface representation: it provides a topology, a domain for parametric representation, and together with the skeleton, a grid to project on an implicit surface.

Theodora Vardouli

Shapes Beyond Structures

Abstract: This paper focuses on uses of graph theory for the generation of architectural floor plans circa 1960. It discusses methods of translating structures representing connectivity and adjacency of rooms into geometric compositions, and vice versa, and the automation of these methods in early computer aided design programs. The paper contextualizes these methods within shifts in atti- tude toward geometric shape and the visual realm that have been associated with mathematical formalism and structuralism, and which reached broader audiences through 1960s reforms in mathematical education. Specifically, it examines how the severance of geometry from its empir- ical and perceptual aspects, put forward through these movements, was perceived by certain ar- chitects as enabling new relationships between architecture and mathematics: ones that would purify architecture from aesthetic preferences and stylistic conventions. Making structural ab- straction visible and workable, graphs were widely adopted by architects as heralding these new possibilities. In floor plan generation, specifically, graphs were viewed as underlying and delim- iting possibilities of the plan’s geometric shape, implicitly adopting the view that shapes are gov- erned by and appear on top of abstract structures. Aside from historicizing this view, the paper discusses its limitations for conceptualizing active, creative processes such as drawing and de- signing; limitations that still vex computer aided design software. Drawing from the design for- malism of shape grammars, the paper puts forward the idea that shapes precede and give rise to structures; that they are their precondition as opposed to their epiphenomenon.

Ilke Demir

On The Importance Of Shape Representations For Deep Learning

Abstract: Deep learning approaches have started simplifying several traditional computer vision tasks to be conducted accurately and efficiently. Although recognition, detection, and segmentation tasks are heavily explored by deep learning approaches, deep learning met shape and geometry understanding with a delay. One reason for this delay is the variety in shape representations to feed DNNs. In this talk, I will introduce different shape representations for several deep learning tasks in 2D and 3D domains for geometric shape understanding. Then I will share initial explorations to derive shape abstractions corresponding to these representations. I will end with some potential use cases and applications benefiting these learned shape abstractions.

Geraldine Morin

Representation And Distribution Of 3D Shapes And Environments

Abstract: Whereas screens are the 2D interface we use for communicating with the virtual objects and virtual environments, visualizing, sensing and interacting with 3D content has a special and attractive flavour — maybe simply because 3D is our native context in the real world.   Representations are needed for modeling 3D objects and we alway seek representations adapted to a targeted application use, but also for easing access to such content: we seek intuitive, expressive and compact 3D models.
This talk will introduce the different geometric models their advantages and drawbacks, and consider their capacity to efficiently code a shape, and to provide a multi-level, progressive representation for streaming such 3D content. We will illustrate the choice of models for different purposes: skeleton based models for modeling plants or providing a reconstruction; image based models for saving rendering cycles, or adapted scene coding for scalable distribution.

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WiSh: Women in Shape Modeling, II

Organizers: Kathryn Leonard, Terry Knight

Mine Ozkar

The Matter Of Shape: Purpose And Techniques That Bring About The Visible

Abstract: Shapes of design, as often as they are considered visual and spatial, emerge as manifests of functional specifications, and more importantly of means of production. A door handle will in some way address how a hand will grab and apply pressure, and how the material will feel to the skin. The production of the handle, out of that particular material, and how it is extruded, molded or cut, will influence the contour. It is imperative to talk of design shapes with these factors taken into consideration, as how to interpret, represent or transform them rely on the contexts thus established.

Shape studies in design rely on visual parts and their relations such as overlap and embedding. These terms already provide a rich field of study that challenges discrete parts that the physical and digital worlds impose in shape representation. Just as relevant to design are the material properties that complement the visual relations. Recent studies in computational making, earlier studies such as those in weights, color grammars and hands-on play with Kindergarten building blocks have already recognized these properties in shapes. We have particularly looked at how shapes in design physically emerge upon application of an idea to the material, and are dependent on the operations that bring them about. Additional to the operations that are strictly related to parameters in visual and spatial relations, we have studied parameters in applications of designs on actual materials. With examples from traditional stone relief carving, this talk will emphasize how the tool tip and its movement on the material surface factor into and determine the final shape, and what constitutes it. The discussion will extend to the techniques of capturing and modeling the physical information from built heritage, and how consequent representations of shapes in these designs may incorporate material aspects.

Cindy Grimm

Metrics For Modeling Robotic Grasping

Abstract: Fundamentally, the goal of a robotic grasping metric or representation is to reduce the complex physics of robotic hand-object interaction down to a small set of values that can be reasoned about. Desirable properties include: Stability with respect to noise (joint angles, pose uncertainty, object shape variation), concise, useful for prediction, efficient to calculate, hand morphology agnostic, and suitable for machine learning/ control strategies.
In this talk I will describe a novel representation based on distance and orientation of contact surfaces, and discuss several experiments we’ve run to evaluate its effectiveness.

Emily Whiting

Mechanics-Based Design For Computational Fabrication

Abstract: Advancements in rapid prototyping technology are closing the gap between what we can simulate with computers and what we can build, as it is now possible to create shapes of astounding complexity. Despite innovations in hardware, however, costly bottlenecks still exist in the design phase. Today’s computational tools for design are largely unaware of the fundamental laws that govern how geometric models will behave in the real world. In this talk I will present recent work combining digital geometry processing, engineering mechanics, and rapid prototyping. The aim is to infuse principles of mechanics into design processes for fabrication. I will highlight specific applications including balance, buoyancy, acoustics, and architectural construction.

Caitlin Mueller

Shaping For Structural Performance In Architecture: Computational Design And Optimization

Abstract: In the realm of buildings and large-scale structures, shape and geometry often have a direct and important influence on measures of performance, such as material efficiency. Since Galileo’s Two New Sciences in 1638, scholars, engineers, and architects have explored how to shape structural elements and carve material away to achieve strength and stiffness targets while using as little matter as possible. Today, many computational methods are available to optimally distribute material in 3D space for maximum performance across a number of disciplines, such as shaping a building’s mass for maximum solar performance or its structure for maximum stiffness. However, despite the existence of such methods in academia since the 1970s, their use in practice at the architectural scale has been extremely limited. This presentation will address two key reasons for the limited uptake of performance-driven shaping. The first is the difficulty in reconciling the singular outputs of optimization methods with the multi-faceted and qualitative nature of architectural design. Designers need to be active agents in the process of designing shape to incorporate aesthetics, context, culture, etc., so geometries generated solely by the computer without human input are of minimal value. New methods to integrate designers into shape optimization processes, such as interactive optimization and design space exploration methods, have the potential to overcome this challenge. The second problem with conventional optimization in architecture is that existing methods typically disregard materialization and construction. This presentation will discuss several new directions in the field of digital fabrication that directly link the performance-driven design generation of shape with processes for material production, fabrication, and assembly. Together, these two research directions offer new potential for shape in architecture to be responsive to performance, materialization, and designer intention simultaneously.

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WIT: Women in Topology, I

Organizers: Sarah Yeakel, Martina Rovelli

Agnes Beaudry

Picard Groups And Orientability

Abstract: Let G be a finite group and E be an equivariant ring spectrum. In this talk, I will discuss techniques to detect the E-orientability of some representations and how this is related to the computation of certain Picard groups.

Eva Belmont

Mahowald Invariants And The R-Motivic Adams Spectral Sequence

Abstract: The Mahowald invariant is a highly nontrivial operator from the stable homotopy of spheres to itself (with indeterminacy). I will discuss a new technique to compute the Mahowald invariant of certain low-dimensional classes, which leverages calculations of the R-motivic Adams spectral sequence. This is joint work with Dan Isaksen.

Safia Chettih

Configurations With Sinks And On Graphs

Abstract: Given a graph Γ, we cam construct discretized models for its n-point configuration space that are cubical complexes. The model constructed by A. Abrams in his 2000 PhD thesis is the most well-known, but in 2001 Światkowski constructed a lesser-known model who’s dimensions stabilizes as the number of points increases. In this recent work with Daniel Lütgehetmann, we have considered a Światkowski-style discretized model for configurations with sinks, where the multiple points are allowed to occupy certain vertices of the graph. In my talk I will discuss these various constructions and sketch the techniques used to prove torsion-freeness and representation stability for the homology of configurations on trees with loops.

Brittany Terese Fasy*, Robin Lynne Belton, Robyn Brooks, Stefania Ebli, Liseth Fajstrup, Catherine Ray, Nikki Sanderson, and Elizabeth Vidaurre

Directed Homotopy Collapses

Abstract: Defining homotopy or collapsibility in directed topological spaces is surprisingly difficult. In the directed setting, we use the (nonextendible) path space to describe the space of all (nonextendible) paths in the directed space. The simplest case is when the path space is a single path; we call this the trivial path space. Surprisingly, spaces that are trivial topologically may have a non-trivial path space, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of weak directed contractibility and directed collapsibility, using the path space of a directed topological space. In addition, we give a sufficient condition for a directed topological space to be weakly directed contractible.

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WIT: Women in Topology, II

Organizers: Sarah Yeakel, Martina Rovelli

Kathryn Lesh

Connectivity Of Complexes Related To Homological Stability

Abstract: I will describe joint work with Bridget Schreiner ad Nathalie Wahl, in which we investigate the connectivity of certain simplicial complexes related to homological stability. As a tool, we introduce a new doubling” construction for simplicial complexes and show that, given at weak Cohen-Macauley bound for a simplicial complex, it is possible to obtain at weak Cohen-Macauley bound for its double.

This work began at the WIT workshop at MSRI in December 2017.

Angelica Osorno

2-Segal Spaces And The Waldhausen Construction

Abstract: The notion of 2-Segal spaces was introduced by Dyckerhoff and Kapranov as a higher dimensional version of Rezk’s Segal spaces. In this talk we will explore the motivation for this notion, give examples, and show that it is related to a certain class of double categories via a version of Waldhausen’s construction. This is joint work with J. Bergner, V. Ozornova, M. Rovelli, and C. Scheimbauer.

Kate Ponto

Refining Fixed Point Invariants

Abstract: We often think of the Euler characteristic as a number. This is a useful perspective when first working with the Euler characteristic ,but it becomes very limiting when trying to extend and refine this invariant. Alternatively, the Euler characteristic is an element of the stable homotopy groups of the sphere. While the relevant homotopy group is isomorphic to the integers, this shift in perspective enables similar and even more essential shifts for the Lefschetz number and the Reidemeister trace.

I’ll describe the approach to fixed point theory that enables these richer invariants and indicate some of the generalization of the Euler characteristic that become available.

Carmen Rovi

Abstract:

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Women in Data Science, I

Organizers: Jing Qin, Yifei Lou

Sung-Ha Kang

Identifying Differential Equations Using Numerical Techniques

Abstract:We propose a new method to identify differential equation from a given time dependent data. Identifying unknown differential equation is challenging, since there are a wide range of possible combinations of terms which can contribute to the current data realization, a small amount of noise can quickly make the recovery unstable. Nonlinearity and varying coefficients of differential equation add complexity to the problem. We will propose and discuss a new method on this direction.

Yifei Lou

Nonconvex Approaches In Data Science

Abstract: Although “big data” is ubiquitous in data science, one often faces challenges of “small data,” as the amount of data that can be taken or transmitted is limited by technical or economic constraints. To retrieve useful information from the insufficient amount of data, additional assumptions on the signal of interest are required, e.g. sparsity (having only a few non-zero elements). Conventional methods favor incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent. I will present a nonconvex approach that works particularly well in the coherent regime. I will address computational aspects in the nonconvex optimization. Various numerical experiments have demonstrated advantages of the proposed method over the state-of-the-art. Applications, ranging from super-resolution to low-rank approximation, will be discussed.

Jing Qin

Graph Regularizations In EEG Source Localization

Abstract: Electroencephalogram (EEG) serves as an essential tool for brain source localization due to its high temporal resolution. However, the inference of brain activities from the EEG data is a challenging ill-posed inverse problem. In this talk, we investigate several EEG source localization methods based on various graph regularizations, including graph total generalized variation, graph fractional-order total variation, and temporal graph regularization. Numerical results have shown that the proposed methods localize source extents more effectively than the benchmark methods.

Li Wang

Probabilistic Dimensionality Reduction Via Structure Learning

Abstract: We propose an alternative probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data. Based on this framework, we present a new model, which is able to learn a set of embedding points in a low- dimensional space by retaining the inherent structure from high-dimensional data. The objective function of this new model can be equivalently interpreted as two coupled learning problems, i.e., structure learning and the learning of projection matrix. Inspired by this interesting interpretation, we propose another model, which finds a set of embedding points that can directly form an explicit graph structure. We proved that the model by learning explicit graphs generalizes the reversed graph embedding method, but leads to a natural interpretation from Bayesian perspective. This can greatly facilitate data visualization and scientific discovery in downstream analysis.

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Women in Data Science, II

Organizers: Jing Qin, Yifei Lou

Mingchang Ding

Efficient And Highly Accurate Semi-Lagrangian Discontinuous Galerkin Method For Convection-Diffusion Problems

Abstract: We propose to organically combine Semi-Lagrangian Discontinuous Galerkin (SLDG) method with local DG (LDG) approximations to diffusion terms for convection-diffusion problems. In particular, we apply a weak formulation of the SLDG method for the convection term (Cai, Guo and Qiu, JSC, 2017), and along characteristics using high order implicit Runge-Kutta method for a LDG discretization of the diffusion term. The proposed scheme is shown to be mass conservative, high order accurate in both space and in time, and highly efficient due to large time stepping sizes allowed from the semi- Lagrangian and implicit nature of time discretization. The scheme can be straightforwardly extended to 2D problems in the truly multi-D SLDG framework previously proposed. The performance of the scheme will be showcased by several classical linear test problems such as rigid body rotation, swirling deformation, as well as the nonlinear incompressible Navier-Stokes and guiding center Vlasov models.

Ngoc Tran

Title: Neuron Spike Sorting

Abstract:Neural activity in a brain region forms a stochastic network. When a neuron `spikes’, it sends out a sharp waveform recorded by neighboring electrodes. Spike sorting is the problem of assigning spikes to individual neurons. Over a recording session, the neuron may move and its waveform may change, posing unique challenges to obtaining long-term recordings. In this talk, we review existing techniques, showcase recent advances and discuss open problems on stochastic networks based on spike sorting challenges. No prior knowledge on neuroscience is required.

Miju Ahn

Difference-Of-Convex Programming For Sparse Learning

Abstract: For sparse learning problems, the method of sample average approximation involving non-convex sparsity functions is widely solved in practice. We introduce a unified difference-of-convex formulation for the learning problem, and study properties of the directional stationary solutions. The solution kind is compared to a vector which is possibly the global optimum of an underlying expec- tation minimization problem. We provide a bound for the distance between the two solutions, a bound on the difference between their model outcomes, and a result showing inclusion relationships among their support sets.

Andrea Arnold

Time-Varying Parameter Estimation Via Nonlinear Filtering

Abstract:Many applications in modern day science involve unknown system parameters that must be estimated using little to no prior information A subset of these problems includes parameters that are known to vary with time but have no known evolution model. We show how nonlinear sequential Monte Carlo filtering techniques can be employed to estimate time- varying parameters, while naturally providing a measure of uncertainty in the estimation. In particular, we show how structural characteristics of the time-varying parameters can be exploited in the estimation. Results are demonstrated on several applications from the life

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