AWM Research Symposium 2017

Special Session Abstracts

WIN Special Session: Work from Women in Numbers

Organizers: Katherine Stange, Beth Malmskog

“Obstructions to the Hasse principle on Enriques surfaces”

Jennifer Berg, Rice University

Abstract: In 1970, Manin showed that the Brauer group of a variety can obstruct the existence of rational points, even when there exist points everywhere locally. Later, Skorobogatov defined a refinement of this Brauer-Manin obstruction, called the étale-Brauer obstruction. We show that this refined obstruction is necessary to understand failures of the Hasse principle on Enriques surfaces. This completes the case of Kodaira dimension 0 surfaces.

“A polynomial sieve in a geometric setting”

Alina Bucur, University of California, San Diego

Abstract: We consider an application of a polynomial sieve to counting points of bounded height on a cyclic cover of P_n over the rational function field. This is joint work with A.C. Cojocaru, M. Lalín and L. Pierce and it was started at WIN3.

“Shadow Lines in the Arithmetic of Elliptic Curves”

Mirela Ciperiani, Institute for Advanced Study/ Univ of Texas at Austin

Abstract: Let E be an elliptic curve of analytic rank 2 over ℚ, and p a prime of ordinary reduction such that the p-part of the Tate-Shafarevich group of E/ℚ is finite. This implies that E(ℚ) ⊗ ℚ_p’ ⊗ ℚ_2p . Consider imaginary quadratic fields K satisfying the Heegner hypothesis, such that the corresponding twisted elliptic curve has analytic rank 1 over ℚ. Each such field K gives rise to a copy of ℚ_p in E(K)⊗ℚp called the shadow line. We will describe work initiated in WIN3 which allows us to compute these shadow lines, and verify a conjecture which states that they in fact lie in E(ℚ) ⊗ ℚ_p. We will also discuss the use of this work in analyzing the distribution of shadow lines in E(Q) ⊗ ℚ_p as K varies. The WIN3 work described in this talk is joint with J. S. Balakrishnan, J. Lang, B. Mirza, and R. Newton.

“Galois action on homology of Fermat curves”

Rachel Davis, University of Wisconsin-Madison

Abstract: We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Pries, V. Stojanoska, and K. Wickelgren.

“Families of p-adic automorphic forms on unitary groups”

Jessica Fintzen, University of Michigan

Abstract: We will start with an introduction to p-adic automorphic forms and then discuss a variant of the q-expansion principle (called the Serre-Tate expansion principle) for p-adic automorphic forms on unitary groups of arbitrary signature. We outline how this can be used to produce p-adic families of automorphic forms on unitary groups, which has applications to the construction of p-adic L-functions. This is done via an explicit description of the action of certain differential operators on the Serre-Tate expansion.
The talk is based on joint work with Ana Caraiani, Ellen Eischen, Elena Mantovan and Ila Varma.

“Kneser-Hecke-Operators for Codes over Finite Chain Rings”

Jingbo Liu, The University of Hong Kong

Abstract: In this talk we will extend results on Kneser-Hecke-operators for codes over finite fields, to the setting of codes over finite chain rings. In particular, we consider chain rings of the form Z/p2Z for p prime.

This is a joint work with Amy Feaver, Anna Haensch and Gabriele Nebe.

“Zeta functions of curves with many automorphisms”

Padmavathi Srinivasan, Georgia Institute of Technology

Abstract: We will describe specific families of Artin-Schreier curves with many automorphisms in odd characteristic. These curves admit the action of a large extraspecial group, which helps us compute the zeta functions of these curves over the field of definition of these automorphisms. This generalizes results of Van der Geer and Van der Vlugt from characteristic two, and gives rise to new examples of maximal curves. This is joint work with Irene Bouw, Wei Ho, Beth Malmskog, Renate Scheidler and Christelle Vincent.

“Hypergeometric varieties and hypergeometric series”

Holly Swisher, Oregon State University

Abstract: We study generalized Legendre curves \(y_N = x_i(1-x_j)(1-\lambda x_k)\), using periods to determine for certain N a condition for when the endomorphism algebra of the primitive part of the associated Jacobian variety contains a quaternion algebra over Q. In most cases this involves computing Galois representations attached to the Jacobian varieties using Greene’s finite field hypergeometric functions. From here it is natural to explore higher dimensional hypergeometric algebraic varieties from this perspective, including relationships to classical hypergeometric functions and hypergeometric functions over finite fields. All of this work is joint with Alyson Deines, Jenny Fuselier, Ling Long, and Fang-Ting Tu. Part of this work is also joint with Ravi Ramakrishna.

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WinCompTop: Applications of Topology and Geometry

Organizers: Radmila Sazdanovic, Shirley Yap, Emilie Purvine

“Topological Complexity in Protein Structures”

Erica Flapan, Pomona College

Abstract: For DNA molecules, topological complexity occurs exclusively as the result of knotting or linking of the polynucleotide backbone. By contrast, while knots and links have been found within the polypeptide backbones of some protein structures, non-planarity can also result from the connectivity between a polypeptide chain and inter- and intra-chain linking via cofactors and disulfide bonds. In this talk, we survey the knots, links, and non-planar graphs that have been identified in protein structures and present models explaining how protein knots might occur and why certain non-planar con- figurations are more likely to occur than others.

“Data Science for Topologists”

Jenn Gamble, Noodle Analytics, Inc.

Abstract: In this talk, we will begin with the question “What is Data Science?” and outline some common statistical- and machine-learning-based approaches. We will next describe how a geometric/topological approach is also useful, and why this makes topologists/mathematicians uniquely situated to contribute to this growing field.
Concepts will be illustrated with a number of applications, including examples from network analysis (iterative simplicial collapse to identify core-periphery structure and community groups), and event-series analysis of the peri-operative period to identify surgical best practices.

“Identification of Copy Number Aberrations in Breast Cancer Subtypes Using Persistence Topology”

Georgina Gonzalez, University of California, Davis

Abstract: Chromosome aberrations are a hallmark of cancer initiation and progression. DNA copy number aberrations (CNAs), such as copy number gains and losses, are of particular interest because they may harbor oncogenes or tumor suppressor genes (driver aberrations). Genome wide experimental detection of copy number aberrations across the genome is achieved through microarray and DNA sequencing technologies. However, the identification of driver CNAs remains a challenge. Supervised methods address this problem by detecting CNAs that are common and specific to a given category (such as cancer subtype) or a cancer with specific clinical characteristics. We will present Topological Analysis of aCGH (TAaCGH), our complementary supervised method that identifies CNAs based on the topological properties of the CGH profile. TAaCGH focuses on the relationships between multiple genomic regions by mapping overlapping fragments of aCGH profiles into a 2D point cloud using a sliding window method. We then use the theory of computational algebraic homology to find patterns and associations within the data with β0, the number of connected components of a simplicial. As a result, β0 provides us with a measure of genomic instability that help us to identify aberrant regions.

“The Convergence of Mapper”

Elizabeth Munch, University at Albany – SUNY

Abstract: Mapper, a powerful tool for topological data analysis, gives a summary of the structure of data with respect to a filter function and a cover of the function range. Assuming that this data comes from a true, underlying (but possibly not accessible) topological space, a related construction, the Reeb graph, can be thought of as the ground truth and Mapper, its approximation. In particular, working with a better data sample and/or a more refined cover intuitively results in a Mapper graph which is more similar to the Reeb graph. In this talk, we will discuss a method for defining a distance on these objects via the interleaving distance idea from persistent homology. We can look at various ways to rigorously quantify the idea that Mapper converges to the Reeb graph, as well as ideas for approximation methods of the distance itself.

“Towards Spectral Sparsification of Simplicial Complexes based on Generalized Effective Resistance”

Bei Wang, University of Utah

Abstract: As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes have recently emerged as a useful tool for modeling higher-order interactions between three or more objects in complex systems. To apply spectral methods in learning to massive datasets modeled as simplicial complexes, we work towards the sparsification of simplicial complexes based on preserving the spectrum of the associated Laplacian operators. In particular, we introduce a generalized effective resistance for simplexes; provide an algorithm for sparsifying simplicial complexes at a fixed dimension; and verify a specific version of the generalized Cheeger inequalities for weighted simplicial complexes. This is a joint work with Braxton Osting and Sourabh Palande.

“Parameter-free topology inference and sparsification for data on manifolds”

Yusu Wang, The Ohio State University

Abstract: In recent years, a considerable progress has been made in analyzing data for inferring the topology of a space from which the data is sampled. Current popular approaches often face two major problems. One concerns with the size of the complex that needs to be built on top of the data points for topological analysis; the other involves selecting the correct parameter to build them. In this talk, I will describe some recent progress we made to address these two issues in the context of inferring homology from sample points of a smooth manifold sitting in an Euclidean space. I will describe how we sparsify the input point set and to build a complex for homology inference on top of the sparsified data, without requiring any user supplied parameter. Our sparsification algorithm guarantees that the data is sparsified at least to the level as specified by the so-called local feature size; and at the same time, the sparsified data is adaptive as well as locally uniform.

This is joint work with Tamal K. Dey and Dong Zhe.

“On minimum-area homotopies of curves in the plane”

Carola Wenk, Tulane University

Abstract: We study the problem of computing a homotopy from a planar curve C to a point that minimizes the area swept. The existence of such a minimum homotopy is a direct result of the solution of Plateau..s problem. We provide structural properties of minimum homotopies that lead to an algorithm. In particular, we prove that for any normal curve there exists a minimum homotopy that consists entirely of contractions of self-overlapping sub-curves (i.e., boundaries of immersed disks).

“Local Maxima of the Distance Function for Delaunay Triangulations on the Plane”

Shirley Yap, California State University East Bay

Abstract: Given a set of points P, its Voronoi diagram can be defined as the set of singularities (non-smooth points) of the distance function d. The minima of d are of the points of P themselves. The critical points of d coincide with points at which a face of the Voronoi diagram intersects its dual Delaunay triangulation. Such triangles are called anchored triangles are useful for reconstructing two dimensional surfaces. In this talk, I will discuss recent results about the density of the set of anchored triangles in the Delaunay triangulation of a set P of points in the plane under certain sampling, boundary, and packing conditions.

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WIMB Special Session: From cells to landscapes: modeling health and disease

Organizers: Carrie Manore, Erica Graham

“Models for Vector Transmitted Viral Disease of Crops with Different Replanting Strategies”

Vrushali Bokil, Oregon State University

Abstract: Vector-transmitted diseases of plants have had devastating effects on agricultural production worldwide, resulting in drastic reductions in yield for crops such as cotton, soybean, tomato and cassava. In this investigation, we formulate a new plant-vector-virus model with continuous replanting from density-dependent replanting of healthy and some infected plants. The new model is an extension of a model formulated by Holt et al., An epidemiological model incorporating vector population dynamics applied to African cassava mosaic virus disease, Journal of Applied Ecology, pages 793-806, 1997. Both models are analyzed and thresholds for disease elimination are defined in terms of the model parameters. Parameter values for cassava, whiteflies, and the virus, in African cassava mosaic virus serve as a case study. A numerical investigation illustrates how the equilibrium densities of healthy and infected plants for both models vary with changes in parameter values. Applications of insecticide and roguing to reduce plant disease and to increase the number of plants harvested are studied using optimal control theory.

“Mathematical modeling of hepatic insulin resistance in adolescent girls”

Cecilia Diniz Behn, Colorado School of Mines

Abstract: Insulin resistance (IR) is a crucial element of the pathology of the metabolic syndrome, which now affects more than a third of the population in the United States. Understanding the contribution to hyperglycemia of abnormal hepatic glucose release following a meal is crucial for the assessment of potential new medications. Using an oral glucose tolerance test (OGTT) protocol with two stable isotope tracers, both the rate of appearance of exogenous glucose coming from the drink and the suppression of endogenous glucose in response to the drink may be computed. We adapt a mathematical model of glucose-insulin dynamics during a labeled OGTT to describe hepatic IR in adolescent girls. We investigate the structural identifiability of the model, and this analysis informs the implementation of appropriate numerical approaches for subject-specific parameter estimation. Improved understanding of interactions between exogenous and endogenous hepatic glucose dynamics will facilitate the characterization of IR in individual patients and different disease conditions and may support the development of targeted therapeutic approaches.

“Identifiability and Parameter Estimation in Modeling Disease Dynamics”

Marisa Eisenberg, University of Michigan, Ann Arbor

Abstract: Connecting dynamic models with data to yield predictive results often requires a variety of parameter estimation, identifiability, and uncertainty quantification techniques. These approaches can help to determine what is possible to estimate from a given model and data set, and help guide new data collection. Here, we will discuss differential algebraic and simulation-based approaches to identifiability analysis, and examine how parameter estimation and disease forecasting are affected when examining disease transmission via multiple types or pathways of transmission. Using examples taken from cholera outbreaks in several settings, as well as the West Africa Ebola epidemic, we illustrate some of the potential difficulties in estimating the relative contributions of different transmission pathways, and show how alternative data collection may help resolve this unidentifiability. We also illustrate how even in the presence of large uncertainties in the data and model parameters, it may still be possible to successfully forecast disease dynamics.

“Toward a computational model of hemostasis”

Karin Leiderman, Colorado School of Mines

Abstract: Hemostasis is the process by which a blood clot forms to prevent bleeding at a site of injury. The formation time, size and structure of a clot depends on the local hemodynamics and the nature of the injury. Our group has previously developed computational models to study intravascular clot formation, a process confined to the interior of a single vessel. Here we present the first stage of an experimentally-validated, computational model of extravascular clot formation (hemostasis) in which blood through a single vessel initially escapes through a hole in the vessel wall and out a separate injury channel. This stage of the model consists of a system of partial differential equations that describe platelet aggregation and hemodynamics, solved via the finite element method. We also present results from the analogous, in vitro, microfluidic model. In both models, formation of a blood clot occludes the injury channel and stops flow from escaping while blood in the main vessel retains its fluidity. We discuss the different biochemical and hemodynamic effects on clot formation using distinct geometries representing intra- and extravascular injuries.

“Simulating Within-Vector Generation of the Malaria Parasite Diversity”

Olivia Prosper, University of Kentucky

Abstract: Plasmodium falciparum, the malaria parasite causing the most severe disease in humans, undergoes an asexual stage within the human host, and a sexual stage within the vector host, Anopheles mosquitoes. Because mosquitoes may be superinfected with parasites of different genotypes, this sexual stage of the parasite life-cycle presents the opportunity to create genetically novel parasites. To investigate the role that mosquitoes’ biology plays on the generation of parasite diversity, which introduces bottlenecks in the parasites’ development, we first constructed a stochastic model of parasite development within-mosquito, generating a distribution of parasite densities at five parasite life-cycle stages: gamete, zygote, ookinete, oocyst, and sporozoite, over the lifespan of a mosquito. We then coupled a model of sequence diversity generation via recombination between genotypes to the stochastic parasite population model. Our model framework shows that bottlenecks entering the oocyst stage decrease diversity from the initial gametocyte population in a mosquito’s blood meal, but diversity increases with the possibility for recombination and proliferation in the formation of sporozoites. Furthermore, when we begin with only two distinct parasite genotypes in the initial gametocyte population, the probability of transmitting more than two unique genotypes from mosquito to human is over 50% for a wide range of initial gametocyte densities.

“Viruses like it hot: modeling the effects of temperature variation on dengue transmission”

Helen Wearing, The University of New Mexico

Abstract: The recent emergence of Zika and chikungunya viruses has shed a spotlight on the potential of mosquito-borne viruses to cause disease outbreaks beyond tropical climes. Dengue virus, which is transmitted by the same principal mosquito, has also been causing minor outbreaks in more temperate zones during the past decade, following introductions from tropical regions where it is endemic. In this talk, we discuss mathematical models of dengue transmission that explicitly account for temperature effects on mosquito life history traits and on viral dissemination within the mosquito. We use these models to examine the impact of seasonal and diurnal temperature fluctuations on the potential for dengue outbreaks in six U.S. cities with differing temperature profiles. We demonstrate that the timing of viral introduction and the temperature profile of the city interact to determine the potential for, and magnitude of, a subsequent outbreak. In addition, we highlight how different assumptions about the relationship between temperature, mosquito mortality, and viral dissemination within the mosquito affect our results. We discuss our findings in the context of a current collaboration to integrate mathematical model development and experimental data collection, which aims to improve our understanding of how temperature impacts the transmission dynamics of other mosquito-borne viruses.

“Mathematical Modeling of Cardiovascular Dynamics during Head-up Tilt”

Nakeya Williams, United States Military Academy

Abstract: This study considers pulsatile and non-pulsatile models for the prediction of short-term cardiovascular responses during head-up tilt (HUT). HUT refers to tilting a patient from supine position to an upright position. To explore potential deficits within the autonomic control system, which maintains the cardiovascular system at homeostasis, many people suffering from chronic fainting or light-headedness are often exposed to the head-up tilt test. This system is complex and difficult to study in vivo. As a result, we show how mathematical modeling can be used to extract features of the cardiovascular system that cannot be measured experimentally. More specifically, we show that it is possible to develop a mathematical model that can predict changes in cardiac contractility and vascular resistance, quantities that cannot be measured directly, but which are useful to assess the state of the system.
The cardiovascular system is pulsatile, yet predicting the control in response to head-up tilt for the complete system is computationally challenging, and limits the applicability of the model. In this work we show how to develop a simpler non-pulsatile model that can be interchanged with the pulsatile model, which is significantly easier to compute, yet it is still able to predict internal variables. The models are validated using head-up tilt data from healthy young adults.

Determining Near-Optimal Treatment Protocols via Nonlinear Cancer Models”

Shelby Wilson, Morehouse College

Abstract: This work aims to develop evidence-based treatment protocols designed to optimize the effectiveness of combined cancer therapies. Here, we study chemotherapy in a context where it is combined with anti-angiogenic drugs (drugs that prevent blood vessel growth). Model parameters corresponding to tumor growth and monotherapy are estimated in a mixed-effect manner using Monolix (Lixoft) while parameters corresponding to drug synergism are estimated in a fixed-effect manner using a Nelder-Mead Simplex Method. We then evaluate the hypothesis that our two drugs interact synergistically when administered together. A direct consequence of this interaction is the creation of a therapeutic window in which the relative timing between drug administrations is most effective. Finally, we use our model, in combination with heuristic algorithms, to propose drug treatment schedules that are designed to maximize treatment outcomes.

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ACxx Special Session: Algebraic Combinatorics

Organizers: Gizem Karaali, Hélène Barcelo

“Combinatorial models for Schubert polynomials”

Sami Hayes Assaf, University of Southern California

Abstract: Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this talk, I will introduce several new combinatorial models for Schubert polynomials that relate them to other known bases including key polynomials and fundamental slide polynomials. I will present a generalization of the insertion algorithm of Edelman and Greene to give a bijection between reduced expressions and pairs of tableaux of the same key diagram shape and use this to give a simple formula, directly in terms of reduced expressions, for the key polynomial expansion of a Schubert polynomial. Finally, I will show how this model can be used to give a simple proof of Kohnert’s algorithm for computing Schubert polynomials.

“Uniqueness of Berline-Vergne’s valuation”

Fu Liu, University of California, Davis

Abstract: Berline-Vergne constructs a valuation that assigns values to faces of polytope, and it satisfies what we call “the McMullen’s formula”. There are different solutions to the McMullen’s formula. Any solution provides a way to write the coefficients of the Ehrhart polynomial of a polytope as positive sums of these values.

We study the Berline-Vergne’s valuation on generalized permutohedra, and show that their construction is the unique solution to the McMullen’s formula that is symmetric about the coordinates. This is joint work with Federico Castillo.

“The Remarkable Ubiquity of Standard Young Tableaux of Bounded Height”

Marni Mishna, Simon Fraser University

Abstract: Standard Young tableaux are a classic family of combinatorial objects that appeared in algebra early in the previous century. Their utility is widely appreciated. The subfamily of tableaux of bounded height also appears in many guises in bijective and enumerative combinatorics. The generating functions are particularly lovely for their algebraic and analytic properties. We will explore these combinatorial classes, focusing on recent bijections that illustrate new, non-trivial connections between some very classic objects. We conclude by tracing the shadows of these results in representation theory.
Work in collaboration with Julien Courtiel, Eric Fusy and Mathias Lepoutre.

“Discrete affairs with Macdonald and Gromov-Witten”

Jennifer Morse, University of Virginia

Abstract: After discussing the nature of problems in Schubert calculus, we will see how our lasting affair with Macdonald symmetric functions has revealed that the Lascoux-Sch\”utzenberger charge on tableaux can be used as a tool in quantum, affine and equivariant Schubert calculus. We will also give a new formula for the monomial expansion of Macdonald polynomials using the charge statistic.

“Face numbers and the fundamental group”

Isabella Novik, University of Washington

Abstract: We will discuss a proof of Kalai’s conjecture positing a lower bound on the number of edges of a (d − 1),-dimensional triangulated manifold Δ in terms of d, the minimum number of generators of the fundamental group of Δ, and the number of vertices of Δ. Our proofs rely on the μ-numbers introduced by Bagchi and Datta and on their algebraic and topological interpretations.

“The partition algebra, symmetric functions and Kronecker coefficients”

Rosa Orellana, Dartmouth College

Abstract: The Schur-Weyl duality between the symmetric group and the general linear group allows us to connect the representation theory of these two groups. A consequence of this duality is the Frobenius formula which connects the irreducible characters of the general linear group and the symmetric group via symmetric functions.

The symmetric group is also in Schur Weyl duality with the partition algebra. This duality allows us to introduce a new Frobenius type formula that connects the characters of the symmetric group and those of the partition algebra. Due to this connection we have introduced a new basis of the ring of symmetric functions which specialize to the characters of the symmetric group when evaluated at roots of unity. Furthermore, the structure coefficients for this new basis of symmetric functions are the stable (or reduced) Kronecker coefficients. In this talk we will discuss how this new basis allows us to use symmetric functions to study the representation theory of the partition algebra and the Kronecker coefficients.

This is joint work with Mike Zabrocki.

“Schur expansion of parabolic Hall–Littlewood polynomials”

Anne Schilling, University of California at Davis

Abstract: In 2000, Shimozono and Weyman carried out a combinatorial study of the Poincaré polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials, H_λ^•[X; t], defined for any sequence of partitions λ^• = (λ⁽¹⁾,… λ^(d)), generalize the Kostka–Foulkes polynomials, are q-analogues of Littlewood–Richardson coefficients, and relate to Macdonald–Schur transition matrices in special cases. Shimozono and Weyman conjectured that the Schur expansion of parabolic Hall–Littlewood polynomials can be elegantly described as the generating functions of katabolizable semistandard tableaux with the charge statistic of Lascoux and Schützenberger. We outline how to attack this conjecture.

“Chromatic bases for symmetric functions”

Stephanie van Willigenburg, University of British Columbia

Abstract: The chromatic polynomial was generalized to the chromatic symmetric function by Stanley in 1995. This function has recently experienced a renaissance, such as Shareshian and Wachs introducing a quasisymmetric refinement to study the positivity of chromatic symmetric functions into the basis of elementary symmetric functions, that is e-positivity.

In this talk we approach the question of e-positivity from a different angle through resolving which of the classical symmetric functions can be realized as the chromatic symmetric function of some graph, and identifying families of graphs whose chromatic symmetric functions give rise to new e-positive bases of the algebra of symmetric functions. This is joint work with Soojin Cho, Samantha Dahlberg and Angele Hamel.

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WINASC Special Session: Recent Research Development on Numerical Partial Differential Equations and Scientific Computing

Organizers: Chiu-Yen Kao, Yekaterina Epshteyn

“Minimization of the Principal Eigenvalue of a Mixed Dispersal Model”

Baasansuren Jadamba, Rochester Institute of Technology

Abstract: In this work, we study a mixed dispersal model of population dynamics and its corresponding linear eigenvalue problem. The model describes the evolution of a population which disperses both locally and nonlocally. We investigate how long-term dynamics depend on parameter values. We also study the minimization of the positive principal eigenvalue; the problem that is motivated by the determination of optimal spatial arrangement of favorable and unfavorable regions for the species to die out more slowly or survive more easily. Numerical results are presented to show various scenarios for the case of Dirichlet and Neumann boundary conditions. This work is in collaboration with Marina Chugunova, Chiu-Yen Kao, Christine Klymko, Evelyn Thomas and Bingyu Zhao.

“Decoupling algorithms for a fluid-poroelastic structure interaction problem”

Aycil Cesmelioglu, Oakland University

Abstract: In this work, we propose decoupling algorithms for the finite element solution of the interaction of a free fluid and a poroelastic structure described as a coupled Stokes-Biot system. The decoupling of the problem is done by casting it as a constrained optimization problem which enforces the continuity of the normal stress on the interface through a Neumann type control. The objective functional is designed to minimize the violation of the other interface conditions. Numerical algorithms based on a residual updating technique will be presented together with some numerical results. This is joint work with H. Lee, A. Quaini, K. Wang, S.-Y. Yi.

” An efficient and high order accurate solution technique for partial differential equations”

Adrianna Gillman, Rice University

Abstract: Solving linear boundary value problems often limits what practitioners (scientist and engineers) simulate numerically. Having efficient and high accuracy solution techniques for these boundary values problems will increase the range of problems that can be modeled numerically. This talk presents a high order discretization technique that comes with a fast direct solver. The high order discretization technique is robust even for problems with highly oscillatory solutions. The computational cost of the direct solver scales linearly (or nearly linearly) with respect to the number of unknowns. The tiny constant prefactor makes this technique ideal for problems involving many solves. Numerical results will illustrate the performance of the method and its use in applications.

“The effect of the sensitivity parameter in weighted essentially non-oscillatory methods”

Yan Jiang, Michigan State University

Abstract: Weighted essentially non-oscillatory methods (WENO) were developed to capture shocks in the solution of hyperbolic conservation laws while maintaining stability and without smearing the shock profile. WENO methods accomplish this by assigning weights to a number of candidate stencils, according to the smoothness of the solution on the stencil. These weights favor smoother stencils when there is a significant difference while combining all the stencils to attain higher order when the stencils are all smooth. When WENO methods were initially introduced, a small parameter . was defined to avoid division by zero. Over time, it has become apparent that . plays the role of the sensitivity parameter in stencil selection. WENO methods allow some oscillations, and it is well-known that these oscillations depend on the size of .. In this work we show that the value of . must be below a certain critical threshold .c, and that this threshold depends on the function used and on the size of the jump discontinuity captured. Next, we analytically and numerically show the size of the oscillations for one time-step and over long time integration when . < .c and their dependence on the size of ., the function used, and the size of the jump discontinuity.

“Computational model of biofilm evolution with a variational inequality”

Malgorzata Peszynska, Oregon State University

Abstract: In the talk we present analysis and computational results for a recently developed numerical model of flow and transport in which the geometry of the (pore-scale) domain is changing in time. This research is inspired by available experimental micro-imaging data showing biofilm growing at pore-scale. Biofilm is a collection of microbial cells which adhere to each other and to fluid and fluid-solid interfaces. We propose a new model for this process using a parabolic variational inequality. In our model, a system of advection-diffusion-reactions for biomass and nutrient evolution is coupled to viscous (Navier-Stokes) fluid model, and the fluid-biofilm interface is described similarly to that in one-phase Stefan problem. The most interesting and challenging part is how to account for the constraint on the maximum density of biofilm that can be present, and for the associated growth through interfaces. This is joint work with Anna Trykozko, Interdisciplinary Centre for Modeling, University of Warsaw, and Azhar Alhammali from Oregon State.

“A C0 finite element method for elliptic distributed optimal control problems with pointwise state constraints”

Sara Pollock, Wright State University

Abstract: We consider a class of nonconforming finite element methods for elliptic distributed optimal control problems with pointwise state constraints, in three-dimensional convex polyhedral domains. The optimal control problem can be reformulated as a fourth-order variational inequality, to which a quadratic C0 interior penalty method may be applied. Taking this approach, we obtain numerical results for the three-dimensional problem, demonstrating predicted convergence rates.

“Numerical methods for solving linear poroelasticity equations”

Beatrice Riviere, Rice University

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.

“On Metrics for Computation of Strength of Coupling in Multiphysics Simulations”

Anastasia Wilson, Augusta University

Abstract: Many multiphysics applications arise in the world of mathematical modeling and simulation. Much of the time in scientific computation these multiphysics applications are solved by decoupling the physics, giving no heed to how this affects the numerical results. However, a fully coupled approach is often not computationally cost effective. Consequently, having a metric for determining the strength of coupling could give insight into whether a simulation should be decoupled in the computation. If the fully coupled approach is not available, then a metric that measures the strength of coupling dynamically in time could help determine when smaller time steps are required to better incorporate coupling into the split solution. In this paper, we report on an Institute for Mathematics and Its Applications student project where we explored metrics for dynamically measuring the strength of coupling between two physical components in a model multiphysics simulation. Four metrics were considered: two based on measured components of the Jacobian matrix, one on error estimates, and the last on time scales of the system components. The metrics are all developed based on previous work found in the literature and tested on a diffusion-reaction problem.

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WINART Special Session: Representatives of Algebras

Organizers: Susan Montgomery, Maria Vega

“Mutation of friezes”

Karin Baur, University of Graz

Abstract: A frieze pattern is a lattice of shifted rows positive integers satisfying the diamond rule: for any four entries b a d c we have ad-bc=1. These patterns were first studied in the 70s by Coxeter and Conway who proved that frieze patterns with finitely many rows are in correspondence with triangulations of polygons. Over the last years, they gained fresh interest because of connections between triangulations of surfaces and cluster algebras. Mutation is a key notion in cluster theory. We introduce mutation as an operation on frieze patterns and show how it is compatible with mutation in the cluster theory set-up. This is joint work with E. Faber, S. Gratz, K. Serhiyenko and G. Todorov.

“Noncommutative Resolutions of Toric Rings”

Eleonore Faber, University of Michigan

Abstract: We consider endomorphism rings A = End_R(M) of Cohen–Macaulay modules M over commutative rings R. If A has finite global dimension, then it is called a noncommutative resolution of singularities (NCR) of R (or Spec(R)). When R is a domain of characteristic p > 0, one possible M to consider is the module of p e -th roots.

In this talk, we consider toric rings R, where the module of p^e -th roots gives a non-commutative resolution, and show how the precise module structure of the endomorphism ring can be described combinatorially. In particular, we are interested in the case when the global dimension of A is equal to the Krull-dimension of R. This is joint work with Greg Muller and Karen E. Smith.

“Separating Ore sets for Prime Ideals of Quantum Algebras”

Sian Fryer, University of California Santa Barbara

Abstract: The prime ideals of various families of quantized coordinate rings can be studied via a finite set of primes known as the H-primes, which stratify the prime spectrum. This allows us to phrase questions about the Zariski topology of the prime spectrum in terms of quotients and localizations of the algebra with respect to the H-primes. Of course, localization in noncommutative algebras isn’t necessarily easy: we can only invert sets of elements which satisfy the Ore conditions. I will talk about what these Ore sets should look like in general, and then relate them to a combinatorial construction called the Grassmann necklace which allows us to easily compute examples. (Joint work with M Yakimov and K Casteels.)

“Representations of signed affine Brauer algebras”

Mee Seong Im, United States Military Academy

Abstract: I will explain the construction of unital associative algebras called signed affine Brauer algebras, which are an extension of Brauer algebras constructed by D. Moon. Our algebras could also be realized as the periplectic version s ⩔ of the affine Nazarov-Wenzl algebras. I will introduce s ⩔ algebraically and diagrammatically, and I will discuss the representation theory of these algebras. This is joint with M. Balagovic, Z. Daugherty, I. Entova-Aizenbud, I. Halacheva, J. Hennig, G. Letzter, E. Norton, V. Serganova, and C. Stroppel.

“Auslander’s Theorem for Permutation Actions on \(A = \mathbb{k}_{-1}[x_1, \dots, x_n]\)”

Ellen Kirkman,Wake Forest University

Abstract: Let \(\mathbb{k}\) be an algebraically closed field of characteristic zero. Maurice Auslander proved that when a finite subgroup G of \(\mathrm{GL}_n(\mathbb{k})\), containing no reflections, acts on \(A = k[x_1, . . . , x_n]\) naturally, with fixed subring \(A^G\), then the skew group algebra A#G is isomorphic to \(\mathrm{End}_{A^G} (A)\) as algebras. We prove that Auslander’s Theorem holds for \(A = \mathbb{k}_{-1}[x_1, \ldots , x_n]\) under the action of any group of permutations of {\(x_1, . . . , x_n\)}. In some cases \(A^G\) is a graded isolated singularity in the sense of Mori-Ueyama (work wtih J. Gaddis, W. F. Moore, and R. Won).

“Cluster-tilted and quasi-tilted algebras”

Khrystyna Serhiyenko, University of California, Berkeley

Abstract: It is known that relation-extensions of tilted algebras are cluster-tilted algebras, and various resulting relationships between them have been studied with great interest. In this work, we investigate a wider class of algebras of global dimension at most two, namely the quasi-tilted algebras. We show that relation-extensions of quasi-tilted algebras are 2-CalabiYau tilted. To study the module category of cluster-tilted algebras of euclidean type, we generalize the notion of reflections of local slices and develop an algorithm for constructing the tubes. Finally, we characterize all quasi-tilted algebras whose relation-extensions are cluster-tilted of euclidean type. This is joint work with Ibrahim Assem and Ralf Schiffler.

“Dominant Dimension and Tilting Modules”

Gordana Todorov, Northeastern University

Abstract: We study which algebras have a tilting module which is both generated and cogenerated by projective-injective modules. Auslander algebras have such a tilting module and for algebras of global dimension 2, Auslander algebras are classified by the existence of such a tilting module.

In this paper we show that, independently of global dimension, the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2. Furthermore, as special cases, we show that algebras obtained from Auslander algebras by extensions on certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

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WiSh Special Session: Shape Modeling and Applications

Organizers: Kathryn Leonard, Asli Genctav

“3D Shape Representations for Interactive Images and Videos”

Duygu Ceylan, Adobe Research

Abstract: Images and videos are becoming one of the most popular medium for capturing the 3D world due to the increasing availability of mobile and low-cost-high-resolution devices. This popularity is creating a desire for interacting with this content similar to real-world interactions, e.g. rotating an object in an image, viewing a video from a different viewpoint. Such interactions pave the road to a much more immersive user experience. One way to accomplish such interactions is to compute an intermediate 3D shape representation from the available image and video data and use this to guide an interactive editing process. Even though accurate 3D shape modeling from images and videos is still a challenging task, often an intermediate 3D proxy representation is sufficient for various editing applications.

In this talk, I will present our recent work that converts monocular 360-videos to sterescopic videos that can be viewed on a headset with 6 degree-of-freedom (DOF). We first compute an intermediate 3D representation of the scene from the monocular video. We then playback the video in a VR headset where we track the 6-DOF motion of the headset. We synthesize novel views for each eye in real time by a warping method that is guided by the intermediate 3D representation. This results in a full 3D VR experience that responds both to translational and rotational motion of the headset significantly eliminating motion sickness and increasing immersiveness.

“Discriminating Placentas of Increased Risk for Autism with Chorionic Surface Vascular Network Features”

Jen-Mei Chang, Colorado State University

Abstract: It was hypothesized that variations in the placental chorionic surface vascular network (PCSVN) structure may reflect both the overall effects of genetic and environmentally regulated variations in branching morphogenesis within the conceptus and the fetus’ vital organs. Significant differences in certain PCSVN features in children at increased risk for autism have been identified; however, a comprehensive understanding of how these, and possibly a lot more, features work together to provide discriminating power is lacking. 28 vessel-based and 8 shape-based PCSVN attributes from a high-risk ASD cohort of 89 placentas and a population-based cohort of 201 placentas were examined for ranked relevance using the Boruta algorithm. Principal component analysis (PCA) was applied to isolate principal effects of vessel growth on the fetal surface. Linear discriminant analysis with a 10-fold cross validation was performed to establish classification statistics. Boruta algorithm selected 15 vessel-only attributes as relevant, implying the difference in high and low ASD risk is better explained by the vascular features alone. The five principal features, which accounted for about 88% of the data variability in PCA, indicated that PCSVNs associated with placentas of high-risk ASD pregnancies generally had fewer branch points, thicker and less tortuous vessels, better extension to the surface boundary, and smaller branch angles than their population-based counterparts.

“Medial Fragments for Segmentation of Articulating Objects in Images”

Ellen Gasparovic, Union College

Abstract: We propose a method for extracting objects from natural images by combining fragments of the Blum medial axis, generated from the Voronoi diagram of an edge map of a natural image, into a coherent whole. Using techniques from persistent homology and graph theory, we combine image cues with geometric cues from the medial fragments in order to aggregate parts of the same object. We demonstrate our method on images containing articulating objects, with an eye to future work applying articulationinvariant measures on the medial axis for shape matching between images. This is joint work with Erin Chambers and Kathryn Leonard.

“A Joint Segmentation and Nonlinear Elasticity Registration algorithm using FFT”

Weihong Guo, Case Western Reserve University

Abstract: We present a Fourier transform based solution of joint image registration and segmentation. The combined registration and segmentation framework is optimized in a way that the displacement (solution to registration) and the segmenting curve of the deforming template converge at the same time. The images are modeled as hyperelastic (specifically St. Venant Kirchhoff) materials allowing for nonlinear strain-displacement relationship and consequently larger deformation. We iteratively solve the segmentation and registration subproblems by first solving for the displacement using the fast fourier transform. The second task, which is the segmentation of the template image is based on the dual formulation of the piecewise constant Mumford-Shah in the framework of active contour model with a weighted TV regularity. Numerical experiments show the advantages of the proposed method.

“Shapes and Other Things”

Terry Knight, Massachusetts Institute of Technology

Abstract: Shape grammars have offered a unique computational theory of design over the past forty or so years. Shape grammars are comprised of visual, shape rules that specify seeing and doing actions (see this ® do that). Shape rules apply in computations to generate, or compute, designs made of shapes. Underpinning shape grammar computations are formal definitions of shapes based on their visual properties.
Recently, shape grammars have been adapted to define making grammars comprised of rules that apply to compute material, real-world objects or things, as opposed to abstract shapes. Underpinning making grammars and their computations are formal definitions of things based on their physical, sensory properties.
In this talk, I will overview (1) shape computing with shape grammars, (2) different ways that shapes in shape grammars have been augmented with material properties to describe physical things, and (3) new work with making grammars for computing physical things and their properties directly. I will highlight some merits, drawbacks, and peculiarities of computing with shapes, computing with augmented shapes, and computing with things in the realm of design.

“Measures on the Blum medial axis: Toward automated shape understanding”

Kathryn Leonard, California State University Channel Islands

Abstract: The Blum medial axis offers a skeletal shape representation whose desirability has long been hampered by the perceived instability of its branching structure. Instead of the usual approach of trying to prune spurious medial points, we define measures on the medial axis that capture salient qualities of the associated shape regions. These measures provide a framework for tasks beyond pruning, including decomposing a shape into parts, determining the similarity of parts, and evaluating the shape’s inherent complexity. We present an overview of these measures and apply the resulting framework to a wide range of shapes.

“Conveying and Analyzing Shapes: From Art to Science”

Alla Sheffer, University of British Columbia

Abstract: Humans have developed multiple ways to communicate about both tangible and abstract shape properties. Artists and designers can quickly and effectively convey complex shapes to a broad audience using traditional mediums such as paper, while both experts and the general public can analyze and agree on intangible shape properties such as style or aesthetics. While perception research provides some clues as to the mental processes involved, concrete and quantifiable explanations of this process are still lacking. Our recent line of research aims to quantify the geometric properties and tools involved in shape communication and analysis, and to develop algorithms that successfully replicate human abilities in this domain. In my talk I will survey our efforts in this domain – describing methods for creation of 3D looking shaded production drawings from concept sketches; sketch based modeling algorithms that automatically create complex 3D shapes from artist-generated line drawings in a range of domains, including industrial design, character modeling, and garment design; and methods for style analysis and transfer for a range of man-made shapes. The common thread in these approaches is the use of insights derived from perception and design literature combined with subsequent perceptual validation via a range of user studies.

“The Intelligent Search and Mapping of Shipwrecks in the Coastal Waters of Malta”

Zoe Wood, Cal Poly – San Luis Obispo

Abstract: With its rich maritime history, the coastal waters of Malta contain numerous shipwrecks of archeological importance. For marine archeologists searching for undiscovered wrecks, the sheer magnitude of the search space is a major challenge. Where should the archeologist begin? Towed side scan sonar has been used to detect potential wreck sites, and more recently Autonomous Underwater Vehicles (AUVs) equipped with side scan sonar have been deployed in the search for wrecks. However, the approach to determining potential wreck site areas remains largely based on labor-intensive human research into historical archives. How can robotics based software and hardware be advanced to enable efficient search and discovery of underwater archeological sites? And how might robot obtained sensor information be processed to produce 3D visualizations that effectively convey the required information to the archeological community? This interdisciplinary project focuses on developing novel AUV planning, control, and visualization techniques that can be applied to a general class of autonomous robot exploration tasks. These techniques are being applied in actual AUV shipwreck search and mapping in coastal areas of Malta and Sicily. We present our work on probabilistic algorithms for AUV motion planning that maximize information gain when mapping a wreck and visualization techniques that construct 3D models of wrecks and the ocean environment. This talk presents the background and current state of this ongoing joint research project between Cal Poly, Harvey Mudd College and the University of Malta.

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WIT Special Session: Topics in Homotopy Theory

Organizers: Julie Bergner, Angelica Osorno

“Topological coHochschild homology: Tools for computations”

Anna Marie Bohmann, Vanderbilt University

Abstract: Hochschild homology is a classical invariant of algebras. A “topological” version, called THH, has important connections to algebraic K-theory, Waldhausen’s A-theory, and free loop spaces. For coalgebras, there is a dual invariant called “coHochschild homology” and Hess and Shipley have recently defined a topological version called “coTHH,” which also has connections to K-theory, A-theory and free loops spaces. In this talk, I’ll talk about coTHH (and THH) are defined and then discuss work with Gerhardt, Hogenhaven, Shipley and Ziegenhagen in which we develop some computational tools for approaching coTHH.

“Derived A-infinity algebras and their homotopies”

Daniela Egas Santander, Freie Universität Berlin

Abstract: The notion of a derived A-infinity algebra, introduced by Sagave, is a generalization of the classical A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. Special cases of such algebras are A-infinity algebras and twisted complexes (also known as multicomplexes). We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between their morphisms. In this talk I will define these objects and describe two different interpretations of them as A-infinity algebras in twisted complexes and as A-infinity algebras in split filtered cochain complexes. We use this reinterpretation to show that this hierarchy of homotopies is an extension of the special case of twisted complexes. This is joint work with Joana Cirici, Muriel Livernet and Sarah Whitehouse

“Galois extensions in motivic homotopy theory”

Magdalena Kedziorek, EPFL

Abstract: Galois extensions of ring spectra in classical homotopy theory were introduced by Rognes. In this talk I will discuss a general formal framework to study homotopical Galois extensions and concentrate on the applications to motivic homotopy theory. I will discuss several examples of homotopical Galois extensions in motivic setting comparing them to the ones known from classical homotopy theory.

This is a joint project with Agn`es Beaudry, Kathryn Hess, Mona Merling and Vesna Stojanoska.

“A Higher Order Chain Rule for Abelian Functor Calculus”

Christina Osborne, University of Virginia

Abstract: One of the most fundamental tools in calculus is the chain rule for functions. Huang, Marcantognini, and Young developed the notion of taking higher order directional derivatives of functions, which has a corresponding higher order iterated directional derivative chain rule. When Johnson and McCarthy established abelian functor calculus, they constructed the chain rule for functors which is analogous to the directional derivative chain rule when n = 1. In joint work with Bauer, Johnson, Riehl, and Tebbe, we defined an analogue of the iterated directional derivative and provided an inductive proof of the analogue to the HMY chain rule. Our initial investigation of this result involved a concrete computation of the case when n = 2, which will be presented in this talk. If time permits, we will discuss the cartesian differential category structure, which is used for the more general proof.

“2-Segal spaces and the Waldhausen construction”

Angelica Osorno, Reed College

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.

“Mapping Spaces for Orbispaces”

Laura Scull, Fort Lewis College

Abstract: Orbifolds, and more generally orbispaces, are a class of spaces which have wellbehaved singularities. These are often modelled using topological groupoids. Using this approach, the category of orbispaces can be described as a bicateogory of fractions of groupoids, where a certain class of maps, the Morita equivalences, have been inverted.

Using this approach, we can define a mapping object which is another groupoid. However, because we are dealing with a bicategory, the mapping object definition is not completely straightforward. I will discuss the work of the WIT team (Coufal, Pronk, Rovi, Scull, Thatcher) done at the first WIT meeting, in exploring the structure of this mapping object. I will also present newer results, done after the WIT meeting but following up on the foundational work done there, that allow us to give this groupoid a topology so that it becomes another orbispace, and (with certain compactness conditions) becomes an exponential object in the category of orbispaces.

“A Homotopical Generalisation of the Bestvina-Brady Construction”

Elizabeth Vidaurre, University of Rochester

Abstract: Using polyhedral products (X, A) K, we recognise the BestvinaBrady construction as the fundamental group of the fibre of
(S 1 , ∗) L → (S 1 , ∗) K = S 1 , where L is a flag complex and K is a one vertex complex. We generalise their construction by studying the homotopy fibre F of (S 1 , ∗) L → (S 1 , ∗) K for an arbitrary simplicial complex L and K an (m − 1)-dimensional simplex. We describe the homology of F, its fixed points, and maximal invariant quotients for coordinate subgroups of Z_m. This generalises the work of Leary and Saadeto˘glu who studied the case when m = 1.

“Inverting Operations in Operads”

Sarah Yeakel, University of Maryland

Abstract: The Dwyer-Kan hammock localization provides a simplicially enriched model for a homotopy category in which maps in a subcategory are inverted. In this talk, I will define a variant of this construction which gives a localization for an operad with respect to a submonoid of one-ary operations and discuss some of its various winsome properties. This is joint work with Maria Basterra, Irina Bobkova, Kate Ponto, and Ulrike Tillmann.

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Special Session: Women in Sage Math

Organizers: Alyson Deines, Anna Haensch

“Minimal Integral Weierstrass equations for genus 2 curves”

Lubjna Beshaj, The University of Texas at Austin

Abstract: We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers O_F . This is done via reduction theory and the Julia quadratic of binary sextics. We show that when the binary sextics has extra automorphisms this is usually easier to compute. Moreover, we build a database of genus 2 curves defined over Q which contains all curves with minimal absolute height ≤ 5 and all curves with extra automorphisms in standard form y 2 = f(x 2 ) defined over Q with height ≤ 101.

“Belyi maps and effective bounds”

Lily Khadjavi, Loyola Marymount University

Abstract: Belyi’s theorem, mapping algebraic curves to the projective line with ramification over at most three points, is a linchpin to deep work in algebra and number theory. These include Grothendieck’s program to understand the structure of the absolute Galois group and Mochizuki’s purported proof of the ABC Conjecture. (Indeed, regarding Belyi’s theorem, Grothendieck noted, “Never was such a profound and disconcerting result proved in so few lines!’’) The fact that Belyi’s proof is constructive has useful implications; we will use Sage to illustrate examples of interest.

“A Census Of Quadratic Post-Critically Finite Rational Functions Defined Over Q”

Michelle Manes, University of Hawaii at Manoa

Abstract: A result of Benedetto, Ingram, Jones, and Levy provides a specific height bound on quadratic post-critically finite (PCF) rational functions defined over Q, guaranteeing a finite number of such maps. A natural question is to find all such rational functions. We describe an algorithm, prototyped in Sage and implemented in both Sage and C, to search for possibly PCF maps. Using the algorithm, we eliminate all but twelve quadratic functions, all of which are verifiably PCF. We also give a complete description of possible rational preperiodic structures for quadratic PCF maps defined over Q

“On the Field of Definition of a Cubic Rational”

Bianca Thompson, Harvey Mudd College

Abstract: Using essentially only algebra, we give a proof that a cubic rational function over C with real critical points is equivalent to a real rational function. We also show that the natural generalization to Qp and number fields fails.

“Constructing hyperelliptic curves of genus 3 whose Jacobians have CM”

Christelle Vincent, University of Vermont

Abstract: For cryptographic applications, it is convenient to be able to, given a CM field, be able to construct an abelian variety with complex multiplication by an order in the ring of integers of that field.

It is currently well-understood how to do this in dimension 1, and a lot of progress has been done in dimension 2. We discuss here the challenges of constructing an abelian threefold with complex multiplication by the ring of integers of a sextic CM field and the work that has been done recently in this direction, both by members of Women in Sage and Women in Numbers projects as well as other mathematicians.

“Solving the S-unit equation in Sage”

Mackenzie West, Reed College

Abstract: Inspired by work of Tzanakis–de Weger, Baker–W¨ustholz and Smart, we use the LLL methods available in Sage to implement an algorithm that returns all S-units τ0, τ1 ∈ O× S such that τ0 +τ1 = 1. Portions of this code were developed during the Women in Sage 5 workshop and at ICERM as part of a Collaborate@ICERM project.

“Arithmetic Mirror Symmetry and Isogenies”

Ursula Whitcher, Mathematical Reviews

Abstract: Arithmetic mirror symmetry is a relationship between the number of points on appropriately chosen mirror pairs of Calabi-Yau varieties over finite fields. We investigate whether arithmetic mirror relationships observed for pencils in weighted projective spaces can be extended to mirror families obtained via the Batyrev-Borisov construction. Our results show that arithmetic mirror symmetry is controlled by an isogeny structure. This talk describes joint work with Christopher Magyar.

“Parameter space analysis for algebraic Python programs in SageMath”

Yuan Zhou, University of California, Davis

Abstract: A metaprogramming trick transforms algebraic programs for testing a property for a given input parameter into programs that compute simplified semialgebraic descriptions of the input parameters for which the property holds. Our implementation of this trick is for Python programs (within the Python-based computer algebra system SageMath and using Mathematica for semialgebraic computations). We illustrate it with an application in the theory of integer linear optimization, the automatic discovery and proof of certain cutting plane theorems in integer programming.

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Women in Government Labs

Organizers: Cindy Phillips, Carol Woodward

“Next-Generation Adaptive Mesh Refinement”

Ann Almgren, LBNL

Abstract: Block-structured adaptive mesh refinement (AMR) is a powerful tool for improving the computational efficiency and reducing the memory footprint of structured-grid numerical simulations. AMR techniques have been used for over 25 years to solve increasingly complex problems. I will talk about the challenges for designing AMR algorithms and software for solving large multiscale, multiphysics problems on next-generation multicore architectures.

“Optimization and Concrete Problems in Design of Complex Adaptive Systems”

Natalia Alexandrov, NASA Langley Research Center

Abstract: Complex adaptive systems, such as air transportation, have been managed via strict complexity bounding, to enable control by humans. Arguably, the traditional transportation system has reached saturation. Growing density of traffic and diversity of aircraft, including unmanned aerial systems (UAS), require increasing reliance on hybrid human-machine control, autonomous control, and, potentially, a complete clean-slate re-design of the transportation system. These developments give rise to a number of difficult unsolved problems in design, system control, and artificial intelligence. While it is easy to develop trust in an autonomous vacuum cleaner, developing trustworthy and trusted safetycritical systems is a much harder problem. In this talk, we discuss an approach to resolving some of these problems via optimization.

“Unique Challenges of Multiphysics High Performance Computing for DOE Labs”

Anshu Dubey, Argonne National Laboratory

Abstract: The Department of Energy (DOE) laboratories develop and deploy scientific software for a great deal of mission critical work. The useof such software ranges from aiding in scientific insight and discovery to development of new devices and other research prototypes. Many of these software packages model multiple physical phenomena in form of numerical components that need to interoperate with one another. These applications pose several unique challenges to their developers and users. There is often more than one kind of discretization in the same application, and several different numerical algorithms are used. The developers come from a wide range of expertise and it is critical to have some with breadth of knowledge spanning several domains. The lifecycle of the software far exceeds the lifecycles of machines or specific problems. Many aspects of the software, including discretization methods, numerical algorithms, and optimization techniques, are typically subject of ongoing research themselves. In this presentation I will discuss these and other challenges, and how they are met in the DOE laboratories with a focus on applications from high energy physics and climate modeling.

“Development of novel sparse matrix algorithms and software for large scale simulations and data analyses”

Sherry Li, Lawrence Berkeley National Laboratory

Abstract: Efficient solution of large-scale indefinite algebraic equations often relies on high quality preconditioners together with iterative solvers. Because of their robustness, factorization-based algorithms play a significant role in developing scalable solvers. We discuss the recent advances in high-performance sparse factorization techniques which are used to build sparse direct solvers, domain-decomposition type direct/iterative hybrid solvers, and approximate factorization preconditioners. In addition to algorithmic principles, we also address the key parallelism issues and practical aspects in order to fully utilize the highly heterogeneous architectures of the current and future HPC systems.

“Using Supercomputing to Solve Large Energy Grid Planning Problems”

Carol Meyers, Lawrence Livermore National Laboratory

Abstract: We discuss the use of supercomputing to solve energy grid planning problems, based on work with energy stakeholders in the state of California. With the increased introduction of renewable resources (such as wind and solar) into the electric grid, planning models must account for increased intermittency of generation, which leads to larger and more complex optimization problems. The underlying model in many of these instances is a mixed-integer linear unit commitment problem, which can solve very slowly when the number of variables and constraints are very large. In the first part of the talk we describe our experiences in speeding execution of a commercial Windows-based energy grid software package via the use of improved formulations (to speed each individual instance) and supercomputing (to enable many instances to solve at once). In the second part of the talk we describe different parallelization strategies that we developed to solve the even larger (millions of variables and constraints) stochastic version of the problem.

“Parallel solution algorithms and modeling tools for dynamic optimization”

Bethany Nicholson, Sandia National Laboratories

Abstract: Dynamic optimization problems directly incorporate detailed dynamic models as constraints within an optimization framework. Applications of dynamic optimization can lead to significant improvements in process efficiency, reliability, safety, and profitability. A well-established method to solve dynamic optimization problems is direct transcription where the differential equations are replaced with algebraic approximations using some numerical method such as a finite-difference or Runge-Kutta scheme. However, for problems with thousands of state variables and discretization points, direct transcription may result in nonlinear optimization problems that exceed memory and speed limits of most serial computers. In particular, when applying interior point optimization methods, the computational bottleneck and dominant computational cost lies in solving the linear systems resulting from the Newton steps that solve the discretized optimality conditions. To overcome these limits, we exploit the parallelizable structure of the linear system to accelerate the overall interior point algorithm. We investigate two algorithms which take advantage of this property, cyclic reduction and Schur complement decomposition and study the performance of these algorithms when applied to dynamic optimization problems. We also briefly discuss pyomo.dae, an open-source modeling framework that enables high-level abstract representations of dynamic optimization problems.

“A Frequentist Approach to Multi-Source Classification”

Katherine Simonson, Sandia National Laboratories

Abstract: The classification of unknown entities based on measured data is a fundamental challenge across applications as diverse as medical diagnostics, treaty monitoring, and electronic fraud detection. In many cases, the data available to support classification decisions arise from multiple sources, each with its own unique signal and noise characteristics. The method to be discussed here, known as Probabilistic Feature Fusion (PFF), provides a means to combine multi-source classification information in a manner that is statistically rigorous and accounts for the uncertainties associated with the constituent sources. PFF provides final class consistency scores that are readily interpretable within in a Frequentist framework, and allows complete traceability back to the contributing sources. It is particularly appropriate in applications related to high consequence decision support, where training data may be limited, and “black box” classifiers struggle to gain trust and cultural acceptance. The method will be illustrated with a practical application related to the segmentation of human skin in color imagery.

“The Impact of Computer Architectures on the Design of Algebraic Multigrid Methods”

Ulrike Meier Yang, Lawrence Livermore National Laboratory

Abstract: Algebraic multigrid (AMG) is a popular iterative solver and preconditioner for large sparse linear systems. When designed well, it is algorithmically scalable, enabling it to solve increasingly larger systems efficiently. While it consists of various highly parallel building blocks, the original method also consisted of various highly sequential components. A large amount of research has been performed over several decades to design new components that perform well on high performance computers. As a matter of fact, AMG has shown to scale well to more than a million processes. However, it is facing several major challenges with future architectures: non-increasing clock speeds are being offset with added concurrency (more cores) and limited power resources are leading to reduced memory per core, and highly complex heterogeneous architectures. To meet these challenges and yield fast and efficient performance, solvers need to exhibit extreme levels of parallelism, and minimize data movement.

In this talk, we will give an overview on how AMG has been impacted by the various architectures of high performance computers to date and discuss our current efforts to continue to achieve good performance on emerging computer architectures.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. LLNL-ABS-723278.

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Special Session: EDGE-y Mathematics: A Tribute to Dr. Sylvia Bozeman and Dr. Rhonda Hughes

Organizers: Alejandra Avlarado, Candice Price

“Maximum nullity, zero forcing, and power domination”

Chassidy Bozeman, Iowa State University

Abstract: Zero forcing on a simple graph is an iterative coloring procedure that starts by initially coloring vertices white and blue and then repeatedly applies the following color change rule: if any vertex colored blue has exactly one white neighbor, then that neighbor is changed from white to blue. Any initial set of blue vertices that can color the entire graph blue is called a zero forcing set. The zero forcing number is the cardinality of a minimum zero forcing set. A well known result is that the zero forcing number of a simple graph is an upper bound for the maximum nullity of the graph (the largest possible nullity over all symmetric real matrices whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise). A variant of zero forcing, known as power domination (motivated by the monitoring of the electric power grid system), uses the power color change rule that starts by initially coloring vertices white and blue and then applies the following rules: 1) In step 1, for any white vertex w that has a blue neighbor, change the color of w from white to blue. 2) For the remaining steps, apply the color change rule. Any initial set of blue vertices that can color the entire graph blue using the power color change rule is called a power dominating set. We present results on the power domination problem of a graph by considering the power dominating sets of minimum cardinality and the amount of steps necessary to color the entire graph blue.

“Interactions of Elastic Cilia Driven by a Geometric Switch”

Amy Buchmann, Tulane University

Abstract: Cilia, flexible hairlike appendages located on the surface of a cell, play an important role in many biological processes including the transport of mucus in the lungs and the locomotion of ciliated microswimmers. Cilia self-organize forming a metachronal wave that propels the surrounding fluid. To study this coordinated movement, we model each cilium as an elastic, actuated body whose beat pattern is driven by a geometric switch where the beat angle switches between two ‘traps’, driving the motion of the power and recovery strokes. The cilia are coupled to a viscous fluid using a numerical method based upon a centerline distribution of regularized Stokeslets. We first characterize the beat cycle and flow produced by a single cilium and then investigate the synchronization states between two cilia. Cilia that are initialized in phase eventually lock into anti-phase motion unless a additional velocity dependent switch is incorporated.

“Regularization results for inhomogenous ill-posed problems in Banach space”

Beth Campbell Hetrick, Gettysburg College

Abstract: We prove continuous dependence on modeling for the inhomogenous ill-posed Cauchy problem in Banach space. Consider the abstract Cauchy problem du(t) dt = Au(t), u(T) = χ, where t ≤ T, A is a densely-defined linear operator in a Banach space X, and χ ∈ X. This final value problem is a familiar example of an inverse problem that is ill-posed; that is, small differences in observed final data may lead to large differences in solutions. For A = ∆, the Laplace operator, we have the backward heat equation that arises in many applications. Motivated by this, we prove regularization for particular inhomogeneous ill-posed problems. Written as an initial value problem, the problem is given by du(t) dt = Au(t) + h(t), 0 ≤ t < T, u(0) = χ, where −A generates a uniformly bounded holomorphic semigroup {e zA|Re(z) ≥ 0} and h : [0, T) → X. In the model problem, the operator A is replaced by the operator fβ(A), β > 0, which approximates A as β goes to 0. Here we use a logarithmic approximation introduced by Boussetila and Rebbani. Our results extend earlier work of Karen Ames and Rhonda Hughes on the homogeneous ill-posed problem.

“In Pursuit of a Bayesian False Discovery Approach to Syndromic Surveillance”

Deidra Coleman, Philander Smith College

Abstract: We give a procedure to detect outbreaks using epidemiological data while controlling the Bayesian False Discovery Rate (BFDR). The procedure entails choosing an appropriate Bayesian model that captures the spatial dependency inherit in epidemiological data and considers all days of interest, selecting a test statistic based on a chosen measure that provides the magnitude of the maximum spatial cluster for each day, and identifying a cutoff value that controls the BFDR for rejecting the collective null hypothesis of no outbreak over a collection of days for a specified region. We use our procedure to analyze botulism-like syndrome data collected by the North Carolina Disease Event Tracking and Epidemiologic Collection Tool (NC DETECT).

“Topological Symmetry Groups of Graphs in S₃“

Emille Davie Lawrence, University of San Francisco

Abstract: The study of graphs embedded in S₃ has been motivated by chemists’ need to predict molecular behavior. The symmetries of a molecule can explain many of its chemical properties, however we draw a distinction between rigid and flexible molecules. Flexible molecules may have symmetries that are not merely a combination of rotations and reflections. Such symmetries prompted the concept of the topological symmetry group of a graph embedded in S₃ . We will discuss recent work on what groups are realizable as the topological symmetry group for several families of graphs, including the Petersen family and Möbius ladders.

“Regularization of non-linear ill-posed problems with applications to non-autonomous PDE’s of arbitrary even order”

Matthew Fury, Penn State Abington

Abstract: Ill-posed problems have a significant presence in several fields such as thermodynamics, mathematical biology, and environmental science. The classic example is the backward heat equation, i.e. the heat equation with a known final value. Solving this particular problem involves the task of determining heat evolution in reverse time. A similar problem is recovering the source of contamination within an already polluted body of water. Because these problems are ill-posed with no systematic method of obtaining a solution, several authors including Lattes and Lions, Showalter, Miller, and later Ames and Hughes have considered approximation techniques such as the quasi-reversibility method.

In this talk, we show that a non-autonomous, non-linear backward heat equation may be regularized by replacing its model with a closely-defined well-posed model. This model follows one first introduced by Boussetila and Rebbani and later modified by Tuan and Trong. We first apply operator theory to gain a general result in Hilbert space and then apply our findings to the non-linear backward heat equation with non-constant diffusion coefficient in L 2 spaces. Finally, we extend these results to non-autonomous partial differential equations of arbitrary even order.

“Eigenvalue Distributions for the Hermitian Two-Matrix Model”

Megan McCormick Stone, University of Arizona

Abstract: The Gaussian Unitary Ensemble (GUE) is the collection of N × N Hermitian matrices with random entries chosen from a Gaussian normal distribution. As N grows, the scaled eigenvalues of the GUE follow a semicircle distribution. The Hermitian two-matrix model is a generalization of the GUE. This model consists of pairs of Hermitian matrices equipped with some probability distribution. Because the probability distribution for the two-matrix model includes an interaction term, the techniques used to characterize the eigenvalues of the GUE cannot be applied directly to the two-matrix model.

The interaction term can be expressed, via spectral decomposition and a change of variables, in terms of the Harish-Chandra-Itzykson-Zuber (HCIZ) integral. A recent formula due to Golden, Guay-Paquet, and Novak connects the HCIZ integral to monotone Hurwitz numbers, which count a specific class of ramified coverings of the sphere. Using the leading order behavior of this formula, and after making assumptions about the coupling constant used for the interaction term, the asymptotic behavior of the eigenvalues for the two-matrix model can be characterized. In this talk, I will describe the necessary assumptions and explain why they are reasonable assumptions to make for the two-matrix model.

“Injective choosability of subcubic planar graphs with girth 6”

Shanise Walker, Iowa State University

Abstract: An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor have distinct colors. A graph G is injectively k-choosable if it has an injective coloring where the color of each vertex v of G can be chosen from any list L(v) of size k. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. We show that a subcubic planar graph with girth at least 6 is injectively 5-choosable, which improves several known bounds on the injective chromatic number of planar graphs

“Classifying unipotent matrices in the symmetric space of SL₂(F_q)”

Carmen Wright, Jackson State University

Abstract: Symmetric spaces for real matrix groups were orgiinally studided by Elie Cartan ang generalized by Berger. A generalized symmetric space is a homogeneous space Q = {gθ(g) −1 |g ∈ G} where θ is an involution, an automorphism of order 2.

A proposed conjecture for the symmetric space Q is that it can be decomposed into semisimple and unipotent components. We will show that this is true for SL2(Fq) and discuss the characterization of the unipotent matrices.

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Special Session: SMPosium: A celebration of the Summer Mathematics Program for Women

Organizers: Alissa S. Crans, Pamela A. Richardson

“Braids in Knot Theory and Contact Geometry”

Diana Hubbard, University of Michigan

Abstract: Braid theory is a rich area of study with many connections to different mathematical fields. As a low-dimensional topologist, I am particularly interested in two perspectives on braids: braids in knot theory and braids appearing in contact geometry. In this talk I will explain how these two points of view have informed each other, and I will discuss some results that lie at their intersection.

“Models of Expander Graphs in terms of Random Maps”

Angelica Gonzalez, University of Arizona

Abstract: Expander graphs are graphs that are both highly connected and sparse (in terms of the number of edges). Satisfying these two conflicting properties has proven to be useful in many mathematical, computational, and physical contexts. For example, expander graphs are useful in the design and analysis of communication networks, theory of error correcting codes, and convergence of Markov chains. The quality of a graphs efficiency in this sense can be directly related to the spectrum of its adjacency matrix. In this talk, we will explain this relationship and explore these notions for a specific class of graphs that are directly related to one-face maps. In particular, we will see how the genus of one-face maps plays a crucial role in this analysis.

“Bounding Lengths in Hahn Fields”

Karen Lange, Wellesely College

Abstract: A standard power series either has finitely many terms or the terms are ordered as the natural numbers. Thus, we can think of any standard power series as having finite length or the length of the first infinite ordinal. Hahn fields, generalizations of standard power series fields, consist of formal power series of arbitrary ordinal length. Given a field K and an ordered abelian group G, let K((G)) be the Hahn field with terms of the form agt g , for g ∈ G and ag ∈ K. If K is an algebraically closed field and G is a divisible ordered abelian group, the Hahn field K((G)) is algebraically closed as well. Thus, given a polynomial over such a Hahn field, it is natural to try to bound the lengths of its roots in terms of the lengths of the polynomial’s coefficients. We discuss bounding results related to this question (after briefly describing Hahn fields) and our motivation for tackling them.

“Unipancyclic Matroids”

Erin McNicholas, Willamette University

Abstract: A uniquely pancyclic (UPC) graph is a graph on n vertices with exactly one cycle of each size from 3 to n, The search for such graphs was first proposed in 1973, and since that time only a handful have been found. In this talk we broaden the search to UPC matroids. Matroids generalize notions of dependence and independence found in various fields including linear algebra and graph theory. Results presented in this talk include work done over three summers as part of the Willamette Math Consortium REU.

“Genus one knots and their derivatives”

Carolyn Otto, University of Wisconsin-Eau Claire

Abstract: In this talk we will discuss the relationship of genus one knots and their derivatives. Specifically, we will prove that if a knot is algebraically slice and genus one, we will always be able to find a derivative of the knot with an arf invariant of zero. Using this result, we will be able to show that there are families of knots, that are created by an infection operator, that admit a derivative with vanishing arf invariant. We will end by showing how this result generalizes to higher genus knots.

“Doppelgangers: Bijections of Plane Partitions”

Rebecca Patrias, Universite du Quebec a Montreal

Abstract: We introduce the K-jeu de taquin algorithm of Thomas and Yong and use it to give a bijection between certain plane partitions, thereby solving a 30-year-old combinatorial mystery. This is joint work with Zachary Hamaker, Oliver Pechenik, and Nathan Williams.

“Quantum Zero-Knowledge Protocols”

Mariel Supina, University of California, Berkeley

Abstract: Zero-knowledge protocols (ZKPs) are interactive protocols in which a prover convinces a verifier of the truth of a statement without revealing any information to the verifier other than the fact that the statement is true. First developed by Goldwasser, Micali, and Rackoff in 1985, such protocols have many practical uses. ZKPs are particularly applicable to cryptography (for example, when one party wishes to prove that they possess a private key without conveying any information about that key). Proofs that a given protocol is zero-knowledge often fail if we allow the verifier to be a quantum computer, since many such proofs involve “rewinding” (reverting the verifier to a previous internal state), which is impossible on a quantum computer. Watrous (2009) and Unruh (2015) have developed techniques to circumvent this issue. In this talk, I will discuss such techniques and explore some open problems related to quantum zero-knowledge protocols.

“Using Data from Longitudinal Observations to Portray How Motor Development Impacts Infants’ Sleep”

Calandra Tate Moore, US Government

Abstract: Sleep functions to consolidate newly learned information and skills into memory. Researchers of motor development aim to understand developmental changes that contribute to adaptive control of motor actions such as sitting, crawling, and walking. Infant sleep patterns vary greatly for many reasons but seldom do we contemplate the influence of sleep on development. Prior work indicates that for some motor skills, sleep measures such as the number of night-time wakings, may indicate changing motor ability.
This talk describes a microgenetic approach to studying the relationship between motor development and sleep using data from a larger longitudinal study to document infants’ sleep experiences and motor abilities on a daily basis. Details of this longitudinal study will be discussed, as well as the role of statistical methods in data-mining, relationship discovery, and time-series analysis to appropriately classify patterns that characterize the complex relationship between sleep and motor development.

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Special Session: The many facets of statistics – applied, pure and BIG

Organizers: Monica Jackson, Jo Hardin

“Measuring Teacher Effectiveness Using Value-Added Models”

Anna Bargagliotti, Loyola Marymount University

Abstract: Measuring quality of teaching is a difficult task. Education policy has pushed to find direct measures of teacher effectiveness. Two such measures include teacher observations and looking at outcomes of a teachers’ students. The latter methods looks to model the value that a teacher has added over the course of a year. This talk will discuss the nuts and bolts of value-added models and, with a toy data set, show how value-added measures for a set of teachers can be estimated.

“Race and causal inference in health disparities research: are we just going around in circles?”

Emma Benn, Icahn School of Medicine at Mount Sinai

Abstract: While researchers examining racial/ethnic disparities in health are often well-intentioned, the underlying conceptual framework used to conduct their studies do not always get us closer to finding a mechanistic mode of intervention. This stems from the fact that the common approach to these types of studies may lack some core statistical principles of causal inference, thus resulting in circular conclusions that merely reinforce the notion that racial/ethnic differences in health outcomes exist. In this talk, I intend to discuss some important statistical considerations for health disparities research that might move us away from describing racial/ethnic differences and closer to eradicating them.

“Base Calling, Binning, SNP Calling on Metagenomic Sequencing Data”

Xinping Cui, University of California, Riverside

Abstract: Recently, the emerging new field of metagenomics facilitated by the advent of nextgeneration sequencing (NGS) technology enables genome sequencing of unculturable and often unknown microbes in natural environments, offering researchers an unprecedented opportunity to delineate bio-diversity of any microbial organism. While the sequencing technologies are evolving at unprecedented speed, researchers engaged in this enterprise are facing major computational, algorithmic and statistical challenges in the analysis of the massive metagenomic data. In this talk, I will introduce a new integrated statistical and computational pipeline empowered by high performance computing that consists of (1) base-calling; (2) binning; and (3) SNP detection on NGS sequencing data.

“Model-free Knockoffs for High-dimensional Controlled Variable Selection”

Yingying Fan, University of Southern California

Abstract: Many contemporary large-scale applications involve building interpretable models linking a large set of potential covariates to a response in a nonlinear fashion, such as when the response is binary. Although this modeling problem has been extensively studied, it remains unclear how to effectively control the fraction of false discoveries even in high-dimensional logistic regression, not to mention general highdimensional nonlinear models. To address such a practical problem, we propose a new framework of model-free knockoffs, which reads from a different perspective the knockoff procedure (Barber and Candes, 2015) originally designed for controlling the false discovery rate in linear models. The key innovation of our method is to construct knockoff variables probabilistically instead of geometrically. This enables model-free knockoffs to deal with arbitrary (and unknown) conditional models and any dimensions, including when the dimensionality p exceeds the sample size n, while the original knockoffs procedure is constrained to homoscedastic linear models with n ≥ p. Our approach requires the design matrix be random (independent and identically distributed rows) with a covariate distribution that is known, although we show our procedure to be robust to unknown/estimated distributions. To our knowledge, no other procedure solves the controlled variable selection problem in such generality, but in the restricted settings where competitors exist, we demonstrate the superior power of knockoffs through simulations. Finally, we apply our procedure to data from a case-control study of Crohn’s disease in the United Kingdom, making twice as many discoveries as the original analysis of the same data.

“Correlation induced by missing spatial covariates”

Monica Jackson, American University

Abstract: Residual spatial correlation in linear models of environmental data is often attributed to spatial patterns in related covariates omitted from the fitted model. We connect the nonunique decomposition of error in geostatistical models into trend and covariance components to the similarly non-unique decomposition of mixed models into fixed and random effects. We specify spatial correlation induced by missing spatial covariates as a function of the strength of association and (spatial) covariation of the missing covariates.

“Integrating Mathematics and Statistics into the Data Science Curriculum”

Stacey Hancock, Montana State University

Abstract: With the rise of “big data,” the past few years have seen the rapid growth of undergraduate, graduate, and professional programs in data science. Indeed, there is a need for such programs. The widely quoted McKinsey Global Institute Study on Big Data in 2011 reports that “the United States alone faces a shortage of 140,000 to 190,000 people with deep analytical skills as well as 1.5 million managers and analysts to analyze big data and make decisions based on their findings. The shortage of talent is just beginning.” Since the definition of “data science” is still evolving, the core courses in data science curricula range from business to mathematics and statistics to foundational computer science. This talk will explore how mathematics and statistics play a role in data science curricula and how to leverage the core analytical skills honed in mathematics and statistics to educate the next generation of data scientists.

“Statistical Approaches in Personalized Medicine using Nonparametric Parameter Estimation”

Alona Kryshchenko, California State University Channel Islands

Abstract: Modeling drug behavior is a very complicated task since every person responds to a drug in his or her own unique way. Pharmacokinetics is the study of drug behavior, from the moment that it is administered up to the point at which it is completely eliminated from the body. Pharmacokinetic population models are very complex and high dimensional. Most methods that are developed in this area use parametric approaches to estimate distributions of population mixture models. These methods limit the search for estimates only under assumptions of specific types of distributions. The nonparametric methods do not make any underling assumptions on distributions and allow users to estimate multimodal and long-tailed distributions, which commonly occur in populations with different genotypes. In this talk, I will describe nonparametric methods for estimating distributions of parameters of various population models and their applications.

“Don’t Count on Poisson: Introducing a flexible alternative distribution to model count data”

Kimberly Sellers, Georgetown University

Abstract: The Poisson distribution is a popular model for count data. Its constraining equi-dispersion assumption (where the variance and mean equal), however, limits its usefulness. The Conway-Maxwell-Poisson (CMP) distribution, instead, is a flexible alternative count distribution that accommodates data over- or under-dispersion (where the variance is larger or smaller than the mean), capturing three classical distributions as special cases. This talk will introduce the statistical properties of this distribution, and survey the diverse methods work that has been developed with this model as motivation.

“Time-Dynamic Profiling with Application to Hospital Readmission Among Patients on Dialysis”

Damla Senturk, University of California, Los Angeles

Abstract:Standard profiling analysis aims to evaluate medical providers, such as hospitals, nursing homes or dialysis facilities, with respect to a patient outcome. The outcome, for instance, may be mortality, medical complications or 30-day (unplanned) hospital readmission. Profiling analysis involves regression modeling of a patient outcome, adjusting for patient health status at baseline, and comparing each provider’s outcome rate (e.g., 30-day readmission rate) to a normative standard (e.g., national “average”). To date, profiling methods exist only for non time-varying patient outcomes. However, for patients on dialysis, a unique population which requires continuous medical care, methodologies to monitor patient outcomes continuously over time are particularly relevant. Thus, we introduce a novel time dynamic profiling (TDP) approach to assess the time-varying 30-day readmission rate. TDP is used to estimate, for the first time, the risk-standardized time-dynamic 30-day hospital readmission rate, throughout the time period that patients are on dialysis. We develop the framework for TDP by introducing the standardized dynamic readmission ratio as a function of time and a multilevel varying coefficient model with facility-specific time-varying effects. We propose estimation and inference procedures tailored to the problem of TDP and to overcome the challenge of high-dimensional parameters when examining thousands of dialysis facilities.

“Linear mixed effect models and gene-gene interaction: Something old, something new …”

Janet Sinsheimer, University of California, Los Angeles

Abstract: Linear mixed effect models (LMMs) have a long history in genetics, going back at least as far as when R. A . Fisher proposed the polygenic model. However, quite recently LMMs surged in popularity for -omic studies and in particular for genome wide association studies. In my talk, I will review what makes these models so popular now in genomics, discuss my groups’ recent work with LMMs to detect maternal gene by offspring gene interactions, and then touch on some open questions.

“Intuition to Modern Statistical / Machine Learning: An Illustration in a BIG Problem”

Zhaoxia Yu, University of California, Irvine

Abstract: We humans use intuition in our daily life. For example, INTUITIVELY, students with similar performance spend similar effort in learning. This Similar-X-Similar-Y (SXSY) intuition, when blended with rigorous statistical modeling, mathematical derivations, and computational algorithms, can become a powerful modern learning tool to understand the interplay between multiple sets of massive, complexly structured, and high-dimensional data. As an illustration, I will present how the SXSY intuition can be applied to investigate the connection between Brain, Imaging, and Genetics (BIG). Our preliminary results suggest that a person’s neuroimaging profile, like his or her human genome, is a signature and is associated with the person’s genetic profile.

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Special Session: History of Mathematics

Organizer: Janet Beery

“G.H. Hardy and the Reform of Mathematics Education at Cambridge circa 1910”

Brenda Davison, Simon Fraser University

Abstract: Mathematics training at Cambridge prior to 1907 was centered on preparing students to sit examinations called the mathematical Tripos. Highly competitive, students were ranked by their performance and their future career prospects depended on their results. Dissatisfaction with this system – particularly with the order of merit – led to a sweeping reform in 1910, a reform in which Hardy played a major role.

The Tripos system, Hardy claimed, when at its zenith in terms of notoriety, difficulty and complexity, was the very time when English mathematics was at its lowest ebb. I will discuss the historical context of this change, and the role of G.H. Hardy, in abolishing a system that put the mathematical training to equip a student to become a research mathematician completely secondary to examination preparation.

“It’s All for the Best”: Optimization, Theology, Calculus, and Science”

Judith Grabiner, Pitzer College

Abstract: Many problems, from optics to economics, are solved mathematically by finding the highest, the quickest, the shortest – the best of something. This has been true from antiquity to the present. We’ll look at why scientists started looking for such explanations, examples of how the approach progressed from optics, mechanics, economics, and theology, and the roles played by Heron of Alexandria, Fermat, Leibniz, Maclaurin, and Adam Smith.

“Incorporating Contributions of Women and Minorities in Classrooms: David Blackwell, Evelyn Boyd Granville and Mary Gray”

Sarah Greenwald, Appalachian State University

Abstract: Stories of mathematicians and statisticians and their contributions can help students connect to mathematics and inspire them. We’ll discuss how to incorporate these into a variety of classes including linear algebra, senior capstone, and general education courses. We’ll examine the benefits and challenges in addition to student responses as we look at examples related to David Blackwell, Evelyn Boyd Granville and Mary Gray. Interviews abound in the existing literature, and I’ve also personally communicated with each of these individuals (David Blackwell is deceased but I communicated with him in the early 2000s). For more information, see http://cs.appstate.edu/~sjg/history/wmm.html

“Learning and Teaching Mathematics in World War II Poland: Experiences of Three Daring Women”

Emelie Kenney, Siena College

Abstract: Poland is known as having had the largest underground during World War II, with this underground involving vibrant, determined women and men. In the area of mathematics, we find clandestine teaching at all levels of education in addition to a secret focus on earning degrees at the gymnasium, undergraduate, and graduate levels. In this talk, I would like to present the lives and accomplishments of three specific women: Zofia Krygowska, who worked as a teacher, student, and organizer during the war, as well as a founder of didactics afterwards; Zofia Szmydt, a student and later a successful differential equations specialist; and Irena Golab, a teacher of mathematics, who ran clandestine classes in her home for younger students. Everything that such women did during World War II put their lives at risk at the hands of the Nazis, but no such terrible risks prevented them from their quests to educate themselves and others.

“Thou Shalt not envy: A Sperner’s Lemma Guide to fair division”

Deborah Kent, Drake University

Abstract: In 1928, Emmanuel Sperner proved an elegant, graph-theoretic result: Every properly colored simplicial subdivision contains a cell whose vertices have all different colors. Sperner arrived at this surprisingly useful result while studying dimensionality of Euclidean spaces. In this talk, Sperner’s Lemma will provide a neat solution to Game-theoretic questions of equitable and envy-free division. This lemma is also central to a proof of the Nash Equilibrium Theorem and the result that the game Hex will never end in a tie.

“Certain Modern Ideas and Methods: Charlotte Angas Scott’s Philosophy of Mathematics”

Jemma Lorenat, Pitzer College

Abstract: While mostly known for her role in breaking gender barriers at Cambridge, educating doctoral students at Bryn Mawr, and systematizing analytic geometry, Charlotte Angas Scott intersected with many of the most prominent figures in the philosophy of mathematics at the turn of the twentieth century. She vetted Bertrand Russell for his 1896 lecture series at Bryn Mawr. She reported on both David Hilbert and Henri Poincaré’s methodological addresses at the International Congress of Mathematicians in Paris for the American Mathematical Society in 1900. And at her endowed chair celebration in 1922, Alfred North Whitehead delivered the address. Far from a mere bystander to these events, Scott’s outspoken and multifaceted writing delves into philosophical questions. This talk will consider three different manifestations of Scott’s philosophy: standards for women’s mathematics education, the relationship between algebra and geometry, and the ontology of imaginary points. This talk focuses on one individual, but it also raises the broader question of who counts as a philosopher in the history of mathematics.

“The Enduring Legacy of Mario Pieri (1860-1913)”

Elena Marchisotto, California State University Northridge

Abstract: Mario Pieri (1860-1913) has been called “a true bridge” between the two most prestigious Italian schools of mathematics which flourished at the University of Turin at the turn to the twentieth century – the research groups of Corrado Segre and Giuseppe Peano. Pieri left a legacy of results in algebraic and differential geometry, vector analysis, foundations of mathematics (elementary, projective and inversive geometries, as well as arithmetic), logic and the philosophy of science.
In my talk, I plan to present a brief synopsis of the panorama of Pieri’s work, providing a picture of the context in which it was produced. In focusing on some of his more noteworthy results, I hope to convey the depth and breadth of Pieri’s mathematics as well as the challenges I have encountered in researching his work, for more than two decades, and attempting to give it its rightful place in the history of mathematics.

“The Krieger-Nelson Prize Lectureship”

Laura Turner, Monmouth University

Abstract: The Krieger-Nelson Prize Lectureship honours outstanding research by women members of the Canadian mathematical community. First awarded in 1995, it is named after Cecilia Krieger (1894–1974), the first woman to earn a Ph.D. in mathematics from a Canadian university, and Evelyn Nelson (1943–1987), a prolific researcher in universal algebra. In this talk, we explore the origins and early history of this prize, from the contributions of its namesakes to the motivations behind the prize itself.

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Special Session: Commutative Algebra

Organizers: Emily Witt, Alexandra Seceleanu

“Free complexes on smooth toric varieties”

Christine Berkesch Zamaere, University of Minnesota

Abstract: Given a module M over the Cox ring of a smooth toric variety, one can consider free complexes that are acyclic for M modulo irrelevant homology. These complexes have many advantages over minimal free resolutions over smooth toric varieties other than projective spaces. We develop this in detail for products of projective spaces. This is joint work with Daniel Erman and Gregory G. Smith.

“Integrable Derivations of Some Hypersurfaces in Characteristic p > 0”

Eleanore Faber, University of Michigan

Abstract: Let k be a commutative ring and A a commutative k-algebra. A k-linear derivation δ of A is called n-integrable, where n is a positive integer or n = ∞, if it extends up to a Hasse–Schmidt derivation of A over k of length n.

In this talk let k be a field of characteristic p > 0. While over a field of characteristic 0 any derivation is integrable, this question is much more delicate over positive characteristic fields. We study IDerk(A; n), the module of n-integrable derivations along X = Spec(A), for some classes of quasi-homogeneous hypersurfaces X. We can describe when elements of IDerk(A; n) are no longer n+1-integrable and show that for our classes of singularities the chain of inclusions IDerk(A; n) ⊇ IDerk(A; n+1) always becomes stationary, that is, all n-integrable derivations are ∞-integrable for n . 0. In particular, we can explicitly determine the integers n, for which so-called jumps appear, that is, IDerk(A; n) ) IDerk(A; n + 1). These jumps seem to be interesting new invariants of A. This is joint work with Ang´elica Benito.

“Asymptotic Behavior of Certain Koszul Cohomology Modules”

Patricia Klein, University of Michigan

Abstract: Let (R, m) be a local ring, M a finitely generated module over R, and f1, . . . , fd a system of parameters on M. Lech’s limit formula states that as mini ti → ∞ `(M/(f t1 1 , . . . , ftd d )M) t1 · · ·td −→ e(f1, . . . , fd | M), the multiplicity of (f1, . . . , fd) on M. One may ask whether powers of a fixed sequence of parameters may be replaced in this formula by any sequence of parameter ideals In such that In &supe mn . Recalling that the multiplicity may be realized as the alternating sum of the lengths of Koszul cohomology modules and that Hn (f t1 1 , . . . , ftd d | M) ≅ M/(f t1 1 , . . . , ftd d )M, we may rewrite Lech’s limit formula as follows Pn j=0(−1)n−j `(Hi (f t1 1 , . . . , ftd d ; M)) `(Hn(f t1 1 , . . . , ftd d )) %rarr; 1. From this point of view, it is also natural to ask in the case when dim M = dim R = d for which i < d we have `(Hi (In; M))/`(R/InR) → 0. In this talk, we will consider the latter question. The main result is that when M is faithful, the M satisfying the condition that `(Hi (In; M))/`(R/InR) → 0 for all i < d are exactly those M that are Cohen-Macaulay on the punctured spectrum.

“Trace ideals of modules and algebras over commutative rings”

Haydee Lindo, Williams College

Abstract: I will present some new results regarding trace ideals of modules and algebras over commutative rings. This continues the project begun in arXiv:1603.08576 relating the center of the endomorphism ring of a module M, over a commutative noetherian ring, to the endomorphism ring of the trace ideal of M.

“Adams operations for matrix factorizations and a conjecture of Dao and Kurano”

Claudia Miller, Syracuse University

Abstract: Using an idea of Atiyah from 1966, we develop Adams operations on the Grothendieck groups of perfect complexes with support and of matrix factorizations using cyclic group actions on tensors powers. In the former setting, Gillet and Soule’ developed these using the Dold-Kan correspondence and used them to solve Serre’s Vanishing Conjecture in mixed characteristic (also proved independently by P. Roberts using localized Chern characters). Their approach cannot be used in the setting of matrix factorizations, so we use Atiyah’s approach, avoiding simplicial theory altogether.
As an application, we prove a conjecture of Dao and Kurano on the vanishing of Hochster’s theta pairing for pairs of modules over an isolated hypersurface singularity in the remaining open case of mixed characteristic. Our proof is analogous to that of Gillet and Soule’ for the vanishing of Serre’s intersection multiplicity. This is joint work with Michael Brown, Peder Thompson, and Mark Walker.

“The Frobenius Complexity of Hibi Rings”

Janet Page, University of Illinois at Chicago

Abstract: Cartier algebras and their duals, rings of Frobenius operators, have come up in the study of Frobenius splittings, which have been useful in many topics ranging from singularity theory in algebraic geometry to representation theory. When R is a local ring of characteristic p > 0, the Cartier algebra C(R), which is the ring of all potential Frobenius splittings of R, is dual to the ring of Frobenius operators (p e -linear maps) on the injective hull of the residue field. This ring of Frobenius operators need not be finitely generated over R, which led Enescu and Yao to define Frobenius complexity as a measure of its non-finite generation. In their examples Frobenius complexity is not always even rational, but its limit as p → ∞ is an integer. Few other examples have been computed. In this talk, I will discuss a method to compute limit Frobenius complexity for Hibi rings, which are a class of toric rings defined from finite posets. I will show that this computation can be read directly from the defining poset in nice cases.

“Intersection Algebras of Noetherian Rings and Their Properties”

Sandra Spiroff, University of Mississippi

Abstract: The intersection algebra of a commutative Noetherian ring R with respect to two ideals I, J is BR(I, J) = L _r,s∈N I_r ∩ J_s . It was defined by J. B. Fields in 2002, who showed that it is finitely generated when I and J are monomial ideals in a polynomial ring in finitely many variables over a field. In the case that I and J are principal monomial ideals, we obtain explicit formulæ for certain invariants of BR(I, J), dependent upon families of parameters. Specifically, we will discuss the Hilbert-Samuel and Hilbert-Kunz multiplicities, the divisor class group, and the F-signature. This is joint work with F. Enescu at Georgia State.

“Generalized mixed multiplicities”

Yu Xie, Pennsylvania State, Altoona

Abstract: Mixed multiplicities of 0-dimensional ideals can be traced back to Teissier’s Cargèse paper in 1973, where he and Risler used this notion to interpret the Milnor numbers which were used to control the Whitney equisingularity of families of complex analytic hypersurfaces having only isolated singularities. Kirby and Rees as well as Kleiman and Thorup generalized mixed multiplicities to 0-dimensional modules, namely mixed Buchsbaum-Rim multiplicities, to study families of complex analytic complete intersections having only isolated singularities. In 2001, Trung extended the mixed multiplicities to the case where one ideal is 0-dimensional and the other one is arbitrary, and therefore he generalized the classical Milnor numbers to complex analytic hypersurfaces having non-isolated singularities. In this talk, I will show how we extend the concept of mixed multiplicities to arbitrary ideals and modules, how we computes these numbers, their properties and applications.

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Special Session: Biological Oscillations Across Time Scales

Organizers: Stephanie Taylor, Tanya Leise

“Piecewise smooth maps for the circadian modulation of sleep-wake dynamics”

Victoria Booth, University of Michigan

Abstract: The timing of human sleep is strongly modulated by the 24 h circadian rhythm, and desynchronization of sleep-wake cycles from the circadian rhythm can negatively impact health. We have developed a physiologically-based mathematical model for the neurotransmitter-mediated interactions of sleeppromoting, wake-promoting and circadian rhythm-generating neuronal populations that govern sleepwake behavior in humans. To investigate the dynamics of circadian modulation of sleep patterns and of entrainment of the sleep-wake cycle with the circadian rhythm, we have reduced the dynamics of the sleep-wake regulatory network model to a one-dimensional map. The map dictates the phase of the circadian cycle at which sleep onset occurs on day n + 1 as a function of the circadian phase of sleep onset on day n. The map is piecewise continuous with discontinuities caused by circadian modulation of the duration of sleep and wake episodes and the occurrence of rapid eye movement (REM) sleep episodes.

Analysis of map structure reveals changes in sleep patterning, including REM sleep behavior, as sleep occurs over different circadian phases. In this way, the map provides a portrait of the circadian modulation of sleep-wake behavior and its effects on REM sleep patterning. Using the map, we can analyze bifurcations of the sleep-wake regulatory network model to understand how variations in REM sleep propensity and the homeostatic sleep drive affect human sleep patterning.

“Clocks in Mice and Flies and Bears, Oh My!”

Tanya Leise, Amherst College

Abstract: Circadian clocks track internal time in most organisms on earth and are generated by feedback loops of clock gene expression. We’ll take a look at analysis of circadian oscillations in behavioral and molecular records of mice, fruit flies, and brown bears, employing a variety of methods ranging from autocorrelation to wavelet transforms. In mice and flies, we can track expression of a key clock gene, while in brown bears we have records of activity and body temperature rhythms. Our data for these noisy biological oscillators often include relatively few cycles, so that reliable estimation of period can be quite challenging. The phase relationships between different rhythms in the same organism, e.g., between temperature and activity or between intracellular calcium levels and clock gene expression, are also of interest, as well as transient changes in relative phase following a disruption, potentially yielding insight into how such rhythms might be coupled.

“Hippocampal sleep rhythms and memory reactivation: a computational study”

Paola Malerba, University of California San Diego

Abstract: During slow-wave sleep, memories are consolidated in a dialogue between cortex and hippocampus. Although the specific mechanisms of sleep-dependent consolidation are not known, the reactivation of specific neural activity patterns – replay – during slow wave sleep has been observed in both structures and is believed to represent a neuronal substrate of consolidation. In the hippocampus, replay happens during sharp wave – ripples, short bouts of excitatory activity in area CA3 which induce high frequency oscillations in area CA1. Despite the importance of replay within the broader phenomenon of sleep-mediated memory consolidation, the neural mechanisms underlying hippocampal sequence replay are still unknown.
In this work, we develop a model of hippocampal spike sequence replay during sleep. We represent CA3 and CA1 activity with a simplified network model of synaptically coupled pyramidal and basket cells. Noise-induced activation of CA3 pyramidal cells triggered an excitatory cascade, with size controlled by the spread of recurrence in the network. Sharp waves in CA3 resulted in strong excitatory input to area CA1, inducing local ripples. Sharp-wave ripples occur stochastically in the model, and their location and size depend on the convergence between pyramidal cells connections within CA3. The projections from CA3 to CA1 – Schaffer Collateral – induce coordination between spiking regions in CA1 and CA3, so that localized sharp-wave CA3 events produce consistently localized CA1 ripples. In our model, we study the spontaneous reactivation of CA1 and CA3 pyramidal cells.

“Data-driven models of human brain dynamics”

Sarah Muldoon, University at Buffalo, SUNY

Abstract: Understanding the brain as a complex network of interacting components allows for useful insights into brain function, and computational modeling provides a controlled environment to test theoretical predictions of brain network structure. In this talk, I’ll describe work using data-driven computational modeling of brain dynamics to examine individual differences in brain activation and task performance. The computational model is built on structural brain networks derived from diffusion spectrum imaging data, and regional brain dynamics are modeled using biologically motivated nonlinear Wilson-Cowan oscillators. We find that, based on the global versus local spread of activation throughout the brain and/or task-specific sub-networks, we are able to predict individual performance across three different language tasks. Thus, by emphasizing differences in the underlying structural connectivity, our model serves as a powerful tool to examine structure-function relationships in dynamic brain networks.

“A Model for Menstrual Cycle Follicle Waves with Applications”

Nicole Panza, Francis Marion University

Abstract: Ovarian follicle waves have been reported in women by Baerwald et al. (2003). Typically two or three waves occur per cycle. Two nonlinear differential equation models which represent the hormonal regulation of the menstrual cycle for two and three follicle waves per cycle are presented. The model exhibits waves of antral follicles during a woman’s cycle using a Follicle Stimulating Hormone threshold function. The model is used to explore phenomenon such as superfecundation.

“The flexible coordination of hippocampal neurons in rhythms”

Lara Rangel, University of California, San Diego

Abstract: During successful computation, brain regions must have an efficient method for filtering information from multiple sources and coordinating communication with other regions. A great circuit for examining this is the hippocampus, a brain structure critical for learning and memory that must integrate and associate information arriving from multiple sources. Research conducted by Dr. Lara Maria Rangel suggests that the successful processing of information from multiple afferents in the hippocampus is dependent on coordinated oscillatory activity, and more specifically the engagement of hippocampal cells in their surrounding rhythmic circuits. Dr. Rangel is a systems neuroscientist, whose work characterizes the temporal dynamics of cross-regional oscillatory interactions and the flexible participation of neurons in local rhythmic networks during behavior.

“Analysis and Models of Spontaneous Activity in the Lateral Line of Zebrafish”

Nessy Tania, Smith College

Abstract: Temporal patterns of spontaneous activity may vary between sensory systems such as the auditory, vestibular, and lateral line systems due to differences in physiology at the level of hair cells. In the absence of stimuli, hair cells display spontaneous synaptic vesicle fusion and neurotransmitter release, which lead to action potential (spike) generation in innervating afferent neurons. We will discuss properties of the distribution of interspike-intervals (ISI) from spontaneous spiking data recorded from the lateral line of zebrafish (collected by the lab of Josef Trapani, Biology, Amherst College). Additionally, successive ISI’s in the lateral line recordings tended to have positive serial correlation, i.e., successive ISI pairs were either short/short or long/long. This pattern contrasts previous findings from the auditory system where ISI’s tended to have negative serial correlation presumably due to the effects of synaptic depletion.

We have built a computational model of spike generation that included the calcium dependency of neurotransmitter release at the ribbon synapse of hair cells. The model can generate ISI distributions consistent with experimental data. Numerical simulations suggest that fluctuations in total calcium channel activity, including both the number and cooperativity of channels in the population, are a primary contributor to serial correlations in hair-cell evoked spike trains. Given the difference in innervation pattern between auditory and vestibular/lateral line hair cells, we further modeled the effects of single versus multiple synapses on the temporal patterns of spontaneous spike trains. Altogether, our findings provide evidence for how physiological similarities and differences between the auditory, vestibular, and lateral line systems can account for differences in spontaneous activity.

“Using Augmented Phase-Amplitude Oscillators to Infer Directed Connections between Regions of the Mouse Circadian Clock”

Stephanie Taylor, Colby College

Abstract: The master clock that controls the daily rhythms in mouse behavior is a multi-oscillator composed of thousands of individual oscillators that synchronize via intercellular communication. The oscillators can be separated into two regions — the shell and core. The network which links oscillators within the regions and between the regions is the subject of ongoing study. Data are too sparse to infer connections between individual oscillators, but, with the aid of mathematical modeling, are not too sparse to infer the direction of connections between the two regions. We augmented a traditional phase-amplitude oscillator model to simulate experiments in which the communication network of a mouse circadian clock was destroyed and restored. Using experimental evidence and simulations of the dynamics upon restoration, we infer that the core entrains the shell. In this talk, we describe the model, why the augmentation was necessary, and our results.

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Special Session: Geometric Group Theory

Organizers: Pallavi Dani, Tullia Dymarz, Talia Fernos

“Understanding some reducible outer automorphisms of the free group”

Radhika Gupta, University of Utah

Abstract: In analogy to the action of the mapping class group on the curve complex, the group of outer automorphisms of the free group acts on the free factor complex. A fully irreducible outer automorphism acts with positive translation length on the free factor complex but a reducible element acts elliptically. I will discuss some spaces on which the action of reducible elements is more interesting.

“Splittings of mapping tori of linearly growing automorphisms of free groups”

Natasa Macura, Trinity University

Abstract: Guirardel and Levitt Guirardel defined a tree of cylinders Tc for a tree T with an action of a finitely generated group G. This tree only depends on the deformation space of T, and is invariant under the automorphisms of G if T is a JSJ splitting. We discuss cyclic splittings of mapping tori of linearly growing automorphisms of free groups and describe trees of cylinders for these splittings. This is joint work with C. Cashen.

“Quasi-isometric Boundary Swapping”

Molly Moran, Colorado College

Abstract: Bestvina formalized the concept of a group boundary by introducing the notion of a Z-structure on a group. In his initial paper, Bestvina proved a boundary swapping theorem that can be applied to a group G with a finite K(G, 1). He also suggested that a generalized version of boundary swapping should hold for two groups that are quasi-isometric. We will present a generalization of this result and discuss some of the implications. This is joint work with Craig Guilbault.

“Normal Forms for Diestel-Leader Groups”

Anisah Nu’Man, Ursinus College

Abstract: Diestel-Leader graphs where intially introduced in 2001 by Diestel and Leader as a potential answer to the following question posed by Woess: “Is every connected, locally finite, vertex transitive graph quasi-isometric to some Cayley graph? Let Γd(q) denote the group whose Cayley graph, with respect to a certain finite generating set Sd,q, is the Diestel-Leader graph as constructed by Bartholdi, Neuhauser and Woess. In the case when d = 2 these groups are the well known lamplighter groups Lq = Zq o Z whose Cayley graph is the horocyclic product of two trees of valence q + 1. Metric properties of Diestel-Leader groups have been studied by Stein and Taback, in which they provide a method for computing word length in Γd(q) with respect to the generating Sd,q. In this discussion, we will build upon Stein and Taback’s use of word length to construct a set of normal forms for the Deisetel-Leader group Γd(q) with respect to the generating set Sd,q.

“Algebraic and topological properties of big mapping class groups”

Priyam Patel, University of California, Santa Barbara

Abstract: The mapping class group of a surface is the group of homeomorphisms of the surface up to isotopy (a natural equivalence). Mapping class groups of finite type surfaces have been extensively studied and are, for the most part, well-understood. There has been a recent surge in studying surfaces of infinite type and in this talk, we shift our focus to their mapping class groups, often called big mapping class groups. In contrast to the finite type case, there are many open questions regarding the basic algebraic and topological properties of big mapping class groups. Until now, for instance, it was unknown whether or not these groups are residually finite. We will discuss the answer to this and several other open questions after providing the necessary background on surfaces of infinite type. This work is joint with Nicholas G. Vlamis.

“Geodesics in Outer Space”

Catherine Pfaff, University of California, Santa Barbara

Abstract: Outer automorphisms of free groups are studied via their action on CullerVogtmann Outer Space. I will introduce Outer Space and how geodesics in Outer Space resemble and differ from geodesics in hyperbolic spaces and Teichm¨uller space.

“Obstructions to Riemannian smoothings of a locally CAT(0) manifold”

Bakul Sathaye, The Ohio State University

Abstract: In this talk I will discuss obstructions to Riemannian smoothings of a locally CAT(0) manifold. I will focus on obstructions in dimension = 4 given by Davis-Januszkiewicz-Lafont and show how their methods can be extended to construct more examples of locally CAT(0) 4-manifolds that do not support Riemannian metric with nonpositive sectional curvature. Further, the universal cover of such a manifold satisfies the isolated flats condition and contains a collection of 2-dimensional flats with the property that their boundaries at infinity form a non-trivial link in the boundary of the universal cover.

“Coarse and fine geometry of the Thurston metric”

Jing Tao, University of Oklahoma

Abstract: I will present results of a recent collaboration with Anna Lenzhen, David Dumas, and Kasra Rafi in which we study the geometry of Thurston’s metric on Teichmuller space. This is an asymmetric metric based on the Lipschitz constants of maps between hyperbolic surfaces. We study the coarse properties of Thurston metric geodesics in general, and some finer properties (local isometric rigidity, quantitative non-uniqueness of geodesics) in the case of the punctured torus.

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Recent progress in Several Complex Variables

Organizers: Purvi Gupta, Loredana Lanzani

“On the dimension of the Bergman space for some unbounded, pseudoconvex domains”

Anne-Katrin Gallagher, Oklahoma State University

Abstract: A sufficient condition for the infinite dimensionality of the Bergman space of a pseudoconvex domain is given. This condition holds on any pseudoconvex domain that has at least one smooth boundary point of finite type in the sense of D’Angelo. This is joint work with T. Harz and G. Herbort.

“Studying Hyperbolicity of Complex Domains”

Samangi Munasinghe, Western Kentucky University

Abstract: We are studying some aspects of hyperbolicity related to certain complex domains. Our goal is to find some useful information about the domains.

“On the instability of an identity involving the Menger curvature”

Malabika Pramanik, University of British Columbia

Abstract: The Menger curvature is a geometric quantity that has proved to be surprisingly useful in analytical problems. For instance, a symmetrization identity involving the Menger curvature has led to new connections with the Cauchy transform (a widely studied operator in real and complex analysis) and analytic capacity (a concept in geometric measure theory). On the other hand, it is believed that such identites are rare, and extremely sensitive to changes in the underlying symmetrized kernel. In this talk, I will report on ongoing work with Loredana Lanzani, where we provide a quantification of this instability.

“Choquet-Monge-Ampère Classes”

Sibel Sahin, Ozyegin University

Abstract: In this talk we will consider a special class of quasi-plurisubharmonic functions, namely Choquet-Monge-Ampère classes on compact Kähler manifolds. These classes become a useful intermediate tool in the analysis of Complex Monge-Ampère operator with their small enough asymptotic capacity. We will first characterize these classes through Choquet energy and then compare them with the finite energy classes. We will see that over different singularity types the comparison between Choquet-Monge-Ampère classes and the finite energy classes yields totally different characteristics.

“The Cauchy-Riemann Equations in Complex Manifolds”

Mei-Chi Shaw, University of Notre Dame

Abstract: The purpose of this talk is to discuss the recent progress on the Cauchy-Riemann equations in complex manifolds. We will examine the strong Oka’s Lemma and its role in existence and regularity for ∂¯. Recent results on the L 2 closed range property for ∂¯ on an annulus between two pseudoconvex domains will be reported. In particular, we show the Hausdorff property of the L 2 Dolbeault cohomology group on a domain between a ball and a bi-disc, the so-called Chinese Coin problem. Characterization of Lipschitz domains with holes through their Dolbeault cohomology groups will also be discussed. Thus one can hear pseudoconvexity for domains with holes using L 2 Dolbeault cohomology. (joint work with Debraj Chakrabarti, Siqi Fu, and Christine Laurent-Thiébaut).

“On Torsion and Cotorsion of Differentials on Certain Complete Intersection Rings”

Sophia Vassiliadou, Georgetown University

Abstract: I will discuss some results, old and new, on the vanishing/non-vanishing of torsion and cotorsion of Kähler differentials on certain complete intersection rings. Some geometric consequences of these results will also be discussed. This is joint work with Claudia Miller.

“Parabolic skew-products and parametrization”

Liz Vivas, Ohio State University

Abstract: It is a classical result that parametrizaion of unstable manifolds on hyperbolic holomorphic maps can be obtained by a limit of iterates of the map composed with an appropriate inverse action. In this talk I will generalize this result for skew-product invariant holomorphic maps that are parabolic. I will first give an overview of the results known in one and several complex dimensions.

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Special Session: Research in Collegiate Mathematics Education

Organizers: Shandy Hauk, Pao-sheng Hsu

Specifically designed for people who have advanced degrees in the mathematical sciences, session activities will touch on what research suggests about thinking and learning across the college curriculum —from college algebra to calculus, combinatorics, proof, and more. Speakers will communicate the landscape of current research in undergraduate mathematics education as well as offer useful information for present and future faculty members. The goal is to generate lively conversations about the foundations and implications of collegiate mathematics education research.

“Unearthing students’ problematics through proof scripts”

Stacy Brown, California State Polytechnic University, Pomona

Abstract: In this talk I will share findings from a study that explored students’ reasoning about the “within argument contradictions” that arise from logically degenerate cases by analyzing the problematics noticed in students’ proof scripts. Drawing on the exploratory findings, I will report on a framework for students’ noticed proof problematics and explores the viability of the proof script methodology as a mechanism for identifying difficulties experienced by students but unseen by experts. In the case of logically degenerate cases, findings indicate students held conceptions of proofs by cases that inhibited students’ reasoning about the encountered contradictions, as well as students’ difficulties correctly reasoning with logical conjunctions.

“Leveraging Our Bodies When We Learn”

Nicole Infante, West Virginia University

Abstract: Every concept in mathematics can be represented multiple ways. A function is a graph, a formula, a set of points. A critical aspect of learning and understanding mathematical concepts is the ability to use and move between different representations of a common idea. Connections between varied representations and concepts form the foundations of advanced mathematics. Helping students make these connections is a key component of our profession as teachers. Here, we explore how we can assist students in making these connections and deepening their understanding. In particular, we take an embodied cognition approach: our understanding of concepts is shaped through our bodily experiences such as gesture. We examine how instructor gesture can aid student learning and will showcase at least two student centered activities.

“How and when do high school math teachers have the opportunity to learn to mathematics that benefits their teaching?”

Yvonne Lai, University of Nebraska-Lincoln

Abstract: “The more mathematics a teacher knows, the better” — this truism has dictated teaching licensure exams and mathematics requirements for centuries. However, it was not until the past decade and a half that we have seen studies that show that there is mathematical knowledge that is specifically involved in teaching, and that this knowledge may not be found in typical mathematics major coursework such as abstract algebra or real analysis. I will begin with brief survey of the research that leads to this conclusion, for both elementary and high school teaching. I will then discuss some recent results on how policy tends to be more consistent with these findings at the elementary level than at the high school level, and why this may be. Finally, the talk will conclude with some future directions and open questions about the mathematical preparation of high school teachers, and the potential role of mathematicians in contributing to addressing these questions.

“Examining Students’ Combinatorial Reasoning: The Case of the Multiplication Principle”

Elise Lockwood, Oregon State University

Abstract: Combinatorics is a rich and accessible topic, but counting problems are difficult for students to learn and for teachers to teach. In this talk, I present some of my research interests, focusing on one particular area of study: undergraduate students’ reasoning about the multiplication principle. This principle is fundamental to combinatorics, underpinning many standard formulas and counting strategies. I will present a categorization of statement types found in a textbook analysis, and I will incorporate excerpts from a reinvention study that sheds light on student reasoning. Findings from both studies reveal surprisingly subtle aspects of the multiplication principle. I conclude with a number of potential mathematical and pedagogical implications of the research, as well as some future research directions.

“Findings from a National Study of Calculus Programs”

Chris Rasmussen, San Diego State University

Abstract: In this talk I present findings from a national study of Calculus programs, which included both a national survey and case studies of institutions identified as having a relatively successful calculus program. Based on survey results I first present characteristics of STEM intending students who begin their post secondary studies with Calculus and either persist or switch out of the calculus sequence, and hence either remain or leave the STEM pipeline. I then present case study findings from five doctoral degree-granting institutions, including technical universities and medium to large public institutions. Understanding the features that characterize exemplary calculus programs at doctoral degree granting institutions is particularly important because the vast majority of STEM graduates come from such institutions. Analysis of over 95 hours of interviews with faculty, administrators and students reveals seven different programmatic and structural features that are common across the five institutions, including substantive graduate teaching assistant training, coordination across sections, and the use of active learning. A community of practice and a social-academic integrations perspective are used to illuminate why and how these seven features contribute to successful calculus programs.

“An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering”

Michelle Zandieh, Arizona State University

Abstract: In this presentation we address practical questions such as: How do symbols appear and evolve in an inquiry-oriented classroom? How can an instructor connect students with traditional notation and vocabulary without undermining their sense of ownership of the material? We tender an example from linear algebra that highlights the roles of the instructor as a broker, and the ways in which students participate in the practice of symbolizing as they reinvent the diagonalization equation A = PDP⁻¹.

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Wikipedia Edit-a-thon (Saturday 10:15 am – 12:15 pm and 1:44 – 3:45 pm)

• Organizer:
• Ursula Whitcher
• Assistants:
• Marie Vitulli
• Jami Mathewson

Join us at the AWM Symposium to add content about women in mathematics to Wikipedia! We’ll have coffee, internet access, and experienced Wikipedians who can help you start editing. Please bring a laptop and some curiosity.
If you want to start early, you can create a Wikipedia account now:
https://en.wikipedia.org/wiki/Special:CreateAccount
For updates, or to share ideas for articles, join the edit-a-thon Google group:
https://groups.google.com/forum/#!forum/awm-wiki
If you won’t be attending the Symposium in person, you’re still welcome to join and brainstorm with us.