AWM Research Symposium 2015

Special Session Abstracts

Contents

• Research from the ”Cutting EDGE”
• Many facets of Probability
• Topics in Computational Topology and Geometry
• Low-dimensional Topology
• Number Theory
• Mathematics at Government Labs and Centers
• Symplectic Topology/Geometry
• Recent mathematical advancements empowering signal/image processing
• Algebraic Geometry
• Statistics
• PDEs in Continuum Mechanics
• Discrete Math (and Theoretical Computer Science)
• Mathematical Biology
• Sharing the Joy: Engaging Undergraduate Students in Mathematics

Research from the ”Cutting EDGE”

(organized by Ami Radunskaya and Kathleen Ryan)

organized by Sandra Cerrai and Elena Kosygina)

“Intermittency for the stochastic wave and heat equations with fractional noise in time”
Raluca Balan, University of Ottawa

Abstract: Stochastic partial differential equations (SPDEs) are mathematical objects that are used for modeling the behaviour of physical phenomena which evolve simultaneously in space and time, and are subject to random perturbations. A key component of an SPDE which determines the properties of the solution is the underlying noise process. An important problem is to study the impact of the noise on the behavior of the solution. In the study of SPDEs using the random field approach, the noise is typically given by a generalization of the BIn this talk, we consider the stochastic heat and wave equations driven by a Gaussian noise which is homogeneous in space and behaves in time like a fractional Brownian motion with index $H > 1/2$. We study a property of the solution $u(t,x)$ called intermittency. This property was introduced by physicists as a measure for describing the asymptotic behaviour of the moments of $u(t,x)$ as $t \rightarrow \infty$. Roughly speaking, $u$ is “weakly intermittent” if the moments of $u(t,x)$ grow as $\exp(ct)$ for some $c>0$. It is known that the solution of the heat (or wave) equation driven by space-time white noise is weakly intermittent. We show that when the noise fractional in time and homogeneous in space, the solution $u$ is “weakly $\rho$-intermittent”, in the sense that the moments of $u(t,x)$ grow as $\exp(ct^{\rho})$, where$\rho>0$ depends on the parameters of the noise. This talk is based on joint work with Daniel Conus (Lehigh University).

“On the rate of convergence of the 2-D stochastic Leray-α model with multiplicative noise”
Hakima Bessaih, University of Wyoming – Laramie

Abstract: We study the convergence of the solution of  the two dimensional (2-D) stochastic Leray-$\alpha$ model to the solution of the 2-D stochastic Navier-Stokes equations both driven by a  with multiplicative noise.  We are mainly interested in the rate of convergence of the error function as $\alpha$ converges to 0. We show that when properly localized the error function converges in mean square and the convergence is of order $O(\alpha)$. We also prove that the error function converges  in probability to zero with order at most $O(\alpha)$.

“A regular binary Stochastic Block Model”
Ioana Dumitriu, University of Washington

Abstract: The Stochastic Block Model (SBM) is widely used in the study of clustering in large, complex networks; its building unit is the random Erdos-Renyi (E-R) graph, which grants it many nice properties (e.g., edge independence) and allows for the use of many deep, specialized tools (since the E-R random graph is a well-studied and well-understood model). Despite decades of effort, though, even the binary SBM (which has only two communities) has been completely solved only recently (by Mossel, Neeman, and Sly, and independently by Massoulie). By completely solved we mean that all parameter regimes have been characterized in terms of recovery/approximation/detectability. In particular, when the average degree is constant, one can do no better than approximate the correct labeling (up to an upper-bounded fraction of the vertices).  Inspired by these results, we have considered a regular binary SBM, where the building unit is the random regular graph, rather than the E-R graph. On the one hand, such a model loses edge independence; on the other, the constraints imposed by regularity are very rigid and there are no high-degree vertices (which is one of the difficulties in the classical binary SBM study). We show that, in high contrast to the classical binary SBM, complete recovery (and even efficient complete recovery) is possible for almost all cases when the degrees are constant. This is joint work with Gerandy Brito, Shirshendu Ganguly, Chris Hoffman, and Linh Tran.

“Construction of multivariate distributions with given marginals and correlation “
Nevena Maric, University of Missouri- St. Louis

Abstract:  I will talk about existence (through an explicit construction) of multivariate distributions when given marginal distributions and correlation matrix only. This problem naturally addresses issue of attainable correlations in different distributions which, in general, is very little known about. I will also discuss our recent results in this direction with special reference to minimum correlations in the bivariate problem.

“Hypoellipticity in infinite dimensions”
Tai Melcher, University of Virginia

Abstract: It is well known that “nice” geometries allow smooth diffusion of particles, in the standard sense that the transition probability measure of such a diffusion is absolutely continuous with respect to the volume measure and has a strictly positive smooth density. There are also well-known conditions on certain degenerate geometric settings that allow smooth diffusion. These smoothness properties are important in the study of degenerate geometries appearing in certain physical models.  Smoothness results of this kind in infinite dimensions are typically not known, the first obstruction being the lack of an infinite-dimensional volume measure. We will discuss a particular class of infinite-dimensional spaces equipped with a natural degenerate geometry where we may prove the associated diffusion is smooth in a strong sense, in that it has strictly positive smooth density with respect to an appropriate reference measure.
This is joint work with B. Driver and N. Eldredge.

“Excited random walks in random cookie
environments”
Elena Kosygina, Baruch College and the CUNY
Graduate Center

Abstract: We consider a nearest neighbor random walk on the integer lattice whose probability w(x,n) to jump to the right from site x depends not only on x but also on the number of prior visits n to x. The collection w(x,n), where x is an integer and n is a positive integer, is sometimes called a “cookie environment” due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the probability to jump to the right according to the “flavor” of the cookie eaten. Assume that the cookie stacks are i.i.d. and that only the first M cookies in each stack may have a “flavor”. All other cookies are assumed to be “plain”, i.e. after their consumption the walker makes unbiased steps to one of its neighbors. The “flavors” of the first M cookies within the stack can be different and dependent.  We discuss recurrence/transience, ballisticity, and limit theorems for such walks. Time permitting we shall also explain the most recentresults for the case of infinite Markovian cookie stacks. The talk is based on joint papers with D.\ Dolgopyat (University of Maryland), T.\ Mountford (EPFL, Lausanne), J. Peterson (Purdue University), M. Zerner (Tuebingen University).

“Large deviation principles for random projections of L^p balls and the atypicality of
Cramer’s theorem”

Kavita Ramanan, Brown University

“Conformal Restriction: the chordal and the radial”
Hao Wu, MIT

Abstract: When people tried to understand two-dimensional statistical physics models, it is realized that any conformally invariant process satisfying a certain restriction property has corssing or intersection exponents. Conformal field theory has been extremely successful in predicting the exact values of critical exponents describing the bahvoir of two-dimensional systems from statistical physics. The main goal of this talk is to investigate the restriction property and related critical exponents. First, we will introduce Brownian intersection exponents. Second, we discuss Conformal Restriction—the chordal case — and the relation to halp-plane Browinian intersection exponents. Finally, we discuss Conformal Restriction—the radial case —and the relation to whole-plane Brownian intersection exponents.

Topics in Computational Topology and Geometry

(organized by Erin Chambers and Elizabeth Munch

“PCA of persistent homology rank functions with case studies in point processes, colloids and sphere packings”
Kate Turner

Abstract: We introduce a method of performing functional PCA using the persistent homology rank function. We then explore what this method highlights in various simulated and real world examples including pairwise interaction point processes, colloid data and sphere packings. This is joint work with Vanessa Robins.

“Multiple Principal Components Analysis in Tree Space”
Megan Owen

Abstract: Data generated in such areas as medical imaging and evolutionary biology are frequently tree­shaped, and thus non­Euclidean in nature. As a result, standard techniques for analyzing data in Euclidean spaces become inappropriate, and new methods must be used. One such framework is the space of metric trees constructed by Billera, Holmes, and Vogtmann. This space is non­positively curved (hyperbolic), so there is a unique geodesic path (shortest path) between any two trees and a well­defined notion of a mean tree for a given set of trees. Algorithms for finding a first principal component for a set of trees in this space have also been developed, but they cannot be used in an iterative fashion. We present the first method for computing multiple principal components, demonstrate its robustness, and apply it to a variety of datasets.

“Comparing Graphs via Persistence Distortion”
Yusu Wang

Abstract: Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph­like, such as the cosmology web. A metric graph offers one of the simplest yet still meaningful way to represent the non­linear structure hidden behind the data. In this talk, I will describe a new distance between two finite metric graphs, called the persistence­distortion distance, that we (together with T. K. Dey and D. Shi) recently introduced. The development of this distance measure draws upon the idea of topological persistence. This topological perspective along with the metric space viewpoint provide a new angle to the graph matching problem. Our persistence­distortion distance has two novel properties not shared by previous methods: First, it is stable against the perturbations of the input graph metrics. Second, it is a continuous distance measure, in the sense that it is defined on an alignment of the underlying spaces of both input graphs, instead of merely their nodes. This makes our persistence-distortion distance robust against different discretizations of the same underlying graph. Furthermore, despite considering the input graphs as continuous space, that is, taking all their points into account, we can compute the persistence­distortion distance in polynomial time. This is joint work with T. K. Dey and D. Shi.

“Categorification in applied topology”
Radmila Sazdanović

Abstract: Categorification can be thought of as a way of realizing various classical objects as shadows of new, algebraically richer objects–a perspective which often leads to beautiful and structurally deep mathematics. We will introduce the notion of categorification and provide several examples in pure and applied mathematics

“Statistical Estimation of Random Field Thresholds Using Euler Characteristics”
Anthea Monod

Abstract : We introduce Lipschitz­Killing curvature (LKC) regression, a new method to produce (1­\alpha) thresholds for signal detection in random fields that does not require knowledge of the spatial correlation structure. The idea is to fit the observed empirical Euler characteristics to the Gaussian kinematic formula via generalized least squares, which quickly and easily provides statistical estimates of the LKCs ­­ complex topological quantities that are otherwise extremely challenging to compute, both theoretically and numerically. With these estimates, we can then make use of a powerful parametric approximation of Euler characteristics for Gaussian random fields to generate accurate (1­\alpha) thresholds and p­values. Furthermore, LKC regression achieves large gains in speed without loss of accuracy over its main competitor, warping. We demonstrate our approach on an fMRI brain imaging data set. This is joint work with Robert Adler (Technion), Kevin Bartz (Renaissance Technologies), and Samuel Kou (Harvard); the speaker’s research is supported by TOPOSYS (FP7­ICT­318493­STREP).

“Layered Separators with applications”
Vida Dujmovic

Abstract: Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Omega(sqrt n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor­closed families. I will talk about special type of graph separator, called a “layered separator”. These separators may have linear size in n, but has bounded size with respect to a different measure, called the “breadth”. We use layered separators to prove O(log n) bounds for a number of problems where O(n^{1/2}) was a long standing previous best bound. This includes the nonrepetitive chromatic number and queue­number of graphs with bounded Euler genus.

“Parametrized homology & Parametrized Alexander Duality Theorem”
Sara Kalisnik

Abstract: An important problem with sensor networks is that they do not provide information about the regions that are not covered by their sensors. If the sensors in a network are static, then the Alexander Duality Theorem from classic algebraic topology is sufficient to determine the coverage of a network. However, in many networks the nodes change position over time. In the case of dynamic sensor networks, we consider the covered and uncovered regions as parametrized spaces with respect to time. I will discuss parametrized homology, a variant of zigzag persistent homology, which measures how the homology of the level sets of a space changes as the parameter varies. I will show also how we can extend the Alexander Duality theorem to the setting of parametrized homology. This approach sheds light on the practical problem of ‘wandering’ loss of coverage within dynamic sensor networks.

“Using Statistics in Topological Data Analysis”
Brittany Terese Fasy

Abstract: Persistent homology is a method for probing topological properties of point clouds and function. The method involves tracking the birth and death of topological features as one varies a tuning parameter. Features with short lifetimes are informally considered to be “topological noise.” I am interested in bringing statistical ideas to persistent homology in order to distinguish topological signal from topological noise and to derive meaningful, yet computable, summaries of large datasets. In this talk, I will define some of the existing topological summaries of data, and show how we can provide statistical guarantees of these summaries.

Low-dimensional Topology

organized by Elisenda Grigsby and Shelly Harvey

“Invariants and Legendrian Graphs”
Danielle O’Donnol (Oklahoma State University)

Abstract: A Legendrian graph is a graph embedded in such a way that its edges are everywhere tangent to the contact structure. We have extend the classical invariants Thurston-Bennequin number and rotation number to Legendrian graphs. I will talk about some of our recent results. This is joint work with Elena Pavelescu.

“Surgery obstrutions and Heegaard Floer homology”
Jen Hom (Columbia University)

Abstract: Using Taubes’ periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples. This is joint work with Cagri Karakurt and Tye Lidma.

“Legendrian knots, augmentations, and rulings”
Caitlin Leverson (Duke University)

Abstract: A Legendrian knot in R^3 with the standard contact structure is a knot for which dz-ydx=0. Given a Legendrian knot, one can associate the Chekanov-Eliashberg differential graded algebra (DGA) over Z/2. Fuchs and Sabloff showed there is a correspondence between augmentations to Z/2 of the DGA and rulings of the knot diagram. Etnyre, Ng, and Sabloff showed that one can define a lift of the Chekanov-Eliashberg DGA over Z/2 to a DGA over Z[t,t^{-1}]. This talk will give an extension of the relationship between rulings and augmentations to Z/2 of the DGA over Z/2, to a relationship between rulings and augmentations to a field of the DGA over Z[t,t^{-1}]. No knowledge of the Chekanov-Eliashberg DGA will be assumed.

“Heegaard Floer techniques and cosmetic crossing changes”
Allison Moore (Rice University)

Abstract: The cosmetic crossing conjecture asserts that the only crossing changes which preserve the oriented isotopy class of knot are nugatory. We will discuss techniques in Heegaard Floer homology which can be used to address the cosmetic crossing conjecture for certain classes of knots. This is joint with Lidman.

“Applications of open book foliations”
Keiko Kawamuro (University of Iowa)

Abstract: An open book foliation is a singular foliation on a surface in 3-manifolds induced by an open book decomposition of the 3-manifold. In this talk I will discuss some applications of open book foliations to topology and contact topology.

“Three manifold mutations and Heegaard Floer homology”
Corrin Clarkson (Indiana University)

Abstract: Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three-manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if the gluing is not isotopic to the identity, then there exists an embedding of S into a three-manifold M such that the rank of the non-torsion summands of the Heegaard Floer homology of M differs from that of its mutant.

“Sutured Khovanov Homology of Braids and the Burau Representation”
Diana Hubbard (Boston College)

Abstract: I will discuss a connection between the Euler characteristic of the Sutured Khovanov Homology (SKH) of braids and the classical Burau Representation. This yields a straightforward method for distinguishing, in some cases, the SKH of two braids. As a corollary I will explain why SKH is not necessarily invariant under braid axis preserving mutation.

“A combinatorial proof of the homology cobordism classification of lens spaces”
Margaret Doig (Syracuse University)

Abstract: Two lens spaces are homology cobordant iff they are homeomorphic by an orientation-preserving homeomorphism. This result is implicit in recent work in Heegaard Floer theory, and we provide a new proof using the Heegaard Floer d-invariants. The d-invariants may be defined combinatorially for lens spaces, and our proof is entirely combinatorial. (joint with Stephan Wehrli)

Number Theory

organized by Wei Ho, Matilde Lalin and Jenny Fuselier

“p-adic Deligne-Lusztig constructions and the local Langlands correspondence”
Charlotte Chan, University of Michigan

Abstract: The representation theory of SL2(Fq) can be studied by studying the geometry of the Drinfeld curve. This is a special case of Deligne-Lusztig theory, which gives a beautiful geometric construction of the irreducible representations of finite reductive groups. I will discuss recent progress in studying Lusztig’s conjectural construction of a p-adic analogue of this story. It turns out that for division algebras, the (etale) cohomology of the p-adic Deligne-Lusztig (ind-)scheme gives rise to supercuspidal representations of arbitrary depth and furthermore gives a geometric realization of the local Langlands and Jacquet-Langlands correspondences. This talk is based on arXiv:1406.6122 and forthcoming work.

“Explicit construction of Ramanujan bigraphs”
Brooke Feigon, CUNY- The City College of New York

Abstract: In this talk I will explain how we explicitly construct an infinite family of Ramanujan graphs which are bipartite and biregular. This talk is based on joint work with Cristina Ballantine, Radhika Ganapathy, Janne Kool, Kathrin Maurischat and Amy Wooding.

“Sierpinski and Riesel Numbers in Sequences”
Carrie Finch, Washington and Lee University

Abstract: A Sierpinski number is an odd positive number $k$ with the property that $k \cdot 2^n + 1$ is composite for all natural numbers $n$. A Reisel number is an odd positive number $k$ with the property that $k \cdot 2^n – 1$ is composite for all natural numbers $n$. The smallest known Sierpinski number is 78557, and the smallest known Riesel number is 509203. In this talk, we explore the intersection of Riesel numbers and Sierpinski numbers with well-known sequences, such as the Fibonacci numbers, Lucas numbers, polygonal numbers, and others.

“Quantum modular and mock modular forms”
Amanda Folsom, Amherst College

Abstract: In 2010, Zagier defined the notion of a “quantum modular form,” and offered several diverse examples. Here, we construct infinite families of quantum modular forms, and prove one of Ramanujan’s remaining claims about mock theta functions in his last letter to Hardy as a special case of our work. We will show how quantum modular forms underlie new relationships between combinatorial mock modular and modular forms due to Dyson and Andrews-Garvan. This is joint work with Ken Ono (Emory U.) and Rob Rhoades (CCR-Princeton).

“Counting Simple Knots via Arithmetic Invariant theory”
Alison Miller, Harvard University

Abstract: Certain knot invariants coming from the Alexander module have natural number-theoretic structure: they can be interpreted as ideal classes in certain rings. In fact, these invariants fit into the structure of arithmetic invariant theory established by Bhargava and Gross. In this context, we can ask the following asymptotic question: how many different possible values can these invariants take for knots whose Alexander polynomial has bounded size? Answering this question also lets us count simple (4q+1)-knots, a family of high-dimensional knots which are completely classified by these invariants. We will focus on the case of knots of genus 1, and mention possible extensions to higher genus knots.

“Class numbers of quadratic number fields: a few highlights on the timeline from Gauss to today”
Lillian Pierce, Duke University

Abstract: Each number field (finite field extension of the rational numbers) has an invariant associated to it called the class number (the cardinality of the class group of the field). Class numbers pop up throughout number theory, and over the last two hundred years people have been considering questions about the growth and divisibility properties of class numbers. We’ll focus on class numbers of quadratic extensions of the rational numbers, surveying some key results in the two centuries since the pioneering work of Gauss, and then turning to very recent joint work of the speaker with Roger Heath-Brown on averages and moments associated to class numbers of imaginary quadratic fields.

“Root numbers of hyperelliptic curves”
Maria Sabitova, CUNY-Queens College

Abstract: We analyze the root number of an abelian variety A over a local non-archimedean field K in terms of toric and abelian varieties parts of special fibers of Neron models of A over K and over a finite Galois extension over which A acquires stable reduction. As an application of the obtained results we calculate several cases of global root numbers of Jacobians of hyperelliptic curves of genus 2. This is joint work with A. Brumer and K. Kramer.

“Multiple zeta values: A combinatorial approach to structure”
Adriana Salerno, Bates College

Abstract:Multiple zeta functions are a multivariate version of the Riemann zeta function. There are many open problems concerning these values, for example, it’s not even known if these numbers are rational or even algebraic (although it is strongly suspected that they are transcendental). However, these values satisfy many interesting algebraic relations between them. A new approach to understanding multiple zetas is to study purely their algebraic structure. I will talk about a few spaces (which turn out to have the nice structure of a Lie algebra) that are essentially equivalent to a formal version of these zetas, and where all the interesting questions turn into combinatorial questions.

Mathematics at Government Labs and Centers

organized by Gail Letzter and Carla Martin)

“An In Situ Approach for Approximating Complex Computer Simulations and Identifying Important Time Steps”
Kary Myers, Statistical Sciences Group, Los Alamos National Laboratory. CANCELLED

Abstract: As computer simulations continue to grow in size and complexity, they provide a particularly challenging example of big data. Many application areas are moving toward exascale (i.e. 10^18 FLOPS, or FLoating­point Operations Per Second). Analyzing these simulations is difficult because their output may exceed both the storage capacity and the bandwidth required for transfer to storage. One approach is to embed some level of analysis in the simulation while the simulation is running, often called in situ analysis. In this talk I’ll describe an online in situ method for approximating a complex simulation using piecewise linear fitting. Our immediate goal is to identify important time steps of the simulation. We then use those time steps and the linear fits both to significantly reduce the data transfer and storage requirements and to facilitate post processing and reconstruction of the simulation. We illustrate the method using a massively parallel radiation­hydrodynamics simulation performed by Korycansky et al. (2009) in support of NASA’s 2009 Lunar Crater Observation and Sensing Satellite mission (LCROSS).

“A New Basis for Graph and Tensor Partitioning: Standardizing the Interactions Matrix/Tensor”
Genevieve Brown, Department of Defense Postdoctoral Fellow

Abstract: Spectral graph partitioning involves separating a graph into subsets of nodes based upon the eigenvectors of an adjacency matrix (or transformations thereof). Several popular two­dimensional graph partitioning algorithms attempt to optimize a quantity called “modularity,” but this strategy has an inherent flaw since modularity cannot account for community structures having skewed node degree distributions. Previous work has  corrected this flaw by introducing a standardized factor into the modularity. Motivated by the success of the standardized modularity, our first task is to investigate whether the standardization of an “interactions” matrix also results in improvement to existing two­dimensional algorithms. We derive a formula using a null model of statistical independence and implement the formula numerically in a way that preserves sparsity. Our second task is to consider higher­dimensional generalizations of spectral graph partitioning. In particular, we present an overview of an algorithm for non­negative tensor factorization and describe how the method lends itself well to efficient and parallel versions.

“Cooperative Computing for Autonomous Data Centers”
Cynthia Phillips, Sandia National Laboratories

Abstract: We present a new distributed model for graph computations motivated by limited information sharing. Two autonomous entities have collected large social graphs. They wish to compute the result of running graph algorithms on the entire set of relationships. Because the information is sensitive or economically valuable, they do not wish to simply combine the information in a single location and then run standard serial graph algorithms. We consider two models for computing the solution to graph algorithms in this setting: 1) limited­sharing: the two entities can share only a polylogarithmic size subgraph; 2) low­trust: the two entities must not reveal any information beyond the query answer, assuming they are both honest but curious. That is, they will honestly participate in the protocol, but will then curiously pore over the information received in the protocol to learn whatever it is possible to learn. We believe this model captures realistic constraints on cooperating autonomous data centers. We present results for both models for s­t connectivity: is there a path in the combined graph connecting two given vertices s and t? This is one of the simplest graph problems that requires global information in the worst case. In the limited­sharing model, our results exploit social network structure to exchange O(log^2 n) bits, overcoming polynomial lower bounds for general graphs. In the low­trust model, our algorithm requires no cryptographic assumptions and does not even reveal node names.
This is joint work with Jon Berry (Sandia National Laboratories), Michael Collins (Christopher Newport University), Aaron Kearns (University of New Mexico), Jared Saia (University of New Mexico), and Randy Smith (Sandia National Laboratories).

“Optimization Approach for Tomographic Inversion from Multiple Data Modalities”
Zichao (Wendy) Di, Argonne National Laboratory

Abstract: Fluorescence tomographic reconstruction can be used to reveal the internal elemental composition of a sample while transmission tomography can be used to obtain the spatial distribution of the absorption coefficient inside the sample. In this work, we integrate both modalities and formulate an optimization approach to simultaneously reconstruct the composition and absorption effect in the sample. By using multigrid­based optimization framework (MG/OPT), significant speedup and improvement of accuracy has shown for several examples.

“Distributing linear systems for parallel computation”
Karen Devine, Sandia National Laboratories

Abstract: In parallel computing, partitioning algorithms are used to divide the work of a computation among the parallel processors. Their goal is to minimize processor idle time (by dividing computational work evenly among processors), as well as interprocessor communication costs (by reducing the amount of data that processors share). Just as there are many algorithms for solving linear systems, with the choice of algorithm depending on the structure of the system, there are many strategies for partitioning these systems, with the choice of strategy also depending, in part, on the systems’ structure. In this talk, I will describe some strategies employed to partition linear systems arising in parallel scientific and data­centric computing.

“Nonlinear Solvers for Dislocation Dynamics”
Carol S. Woodward, Center for Applied Scientific Computing, Lawrence Livermore National Laboratory

Abstract: Strain hardening simulations within the Parallel Dislocation Simulator (ParaDiS) require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events and rapidly changing problem size. To reduce simulation run times we are incorporating new nonlinear solvers and higher order implicit integrators from the Suite of Nonlinear and Differential / Algebraic Equation Solvers (SUNDIALS). We compare performance of fixed point, Anderson accelerated fixed point, and Newton’s methods for parallel simulations looking at efficiency and algorithmic robustness. Preliminary results show significant speedup using the acceleration methods.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE­AC52­07NA27344. Lawrence Livermore National Security, LLC. LLNL­ABS­648520.

“Feasibility and Infeasibility­ Hard problems for cryptography”
Lily Chen, National Institues of Standards and Technology

Abstract: Most public key cryptography schemes are based on hard problems. However, not every hard problem can be used to build a public key cryptographic scheme. This presentation will trace the journeys in pursing proper hard problems for cryptographic usages in the past 40 years and discuss the challenges raised by quantum computing. Current proposed post­quantum cryptography (PQC) schemes, a.k.a. quantum computing resistance cryptography schemes, are based on the problems which are believed not to be vulnerable to quantum computing. The presentation will introduce the research in Information Technology Laboratory (ITL), National Institute of Standards and Technology (NIST) on post­quantum cryptography and explore possible migration path in standardizing the cryptographic schemes which can resist to quantum computing.

“The sun and space weather”
Yihua (Eva) Zheng (NASA Goddard Space Flight Center, Heliophysics Science Division

Abstract: The sun, not only provides light and heat sustaining life on Earth, but also is a major driver of space weather. The importance of space weather has been recognized both nationally and globally. Our society depends increasingly on technological infrastructure, including satellites used for communication and navigation as well as the power grid. Such technologies, however, are vulnerable to space weather effects caused by the Sun’s variability. In this presentation, I will show images of solar eruptive events such as solar flares and coronal mass ejections, different types of space weather storms and how they impact Earth and our society.

Symplectic Topology/Geometry

organized by Katrin Wehrheim

Abstract: We show that all closed symplectic 4-manifolds have the packing stability property: there are no obstructions beyond volume to embedding symplectically a collection of sufficiently small balls. This generalizes a theorem of Biran which gives the same result under the assumption that the symplectic form lies in a rational cohomology class. This work is done in collaboration with Richard Hind and Emmanuel Opshtein.

“Symplectic properties of positive modality Milnor fibres”
Ailsa Keating (Columbia University)

Abstract: The Milnor fibre of any isolated hypersurface singularity contains exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. I will explain how to construct exact Lagrangian tori in the Milnor fibres of all non-simple (i.e., positive modality) singularities of real dimension four. Time allowing, I will give applications to the structure of their symplectic mapping class groups.

“Constructing symplectic embeddings”

Dusa McDuff (Columbia)

Abstract: There are rather few known ways to construct symplectic embeddings;  I will discuss two of them, one by folding
and one by inflation along a (union of) codimension two submanifolds.

“Cylindrical Contact Homology: An Abridged Retrospective”
Joanna Nelson (IAS and Columbia University)

Abstract: Cylindrical contact homology is arguably one of the more notorious Floer theoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations have tarnished its claim to being a well-defined contact invariant. However, recent work of Hutchings and Nelson has managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. This talk will highlight our implementation of intersection theory, non-equivariant constructions, domain dependent almost complex structures, automatic transversality, and obstruction bundle gluing, yielding a homological contact invariant which is expected to be isomorphic to $SH^+$ under suitable assumptions, though does not require a filling of the contact manifold. By making use of family Floer theory we obtain a $S^1$-equivariant theory defined over $\mathbb{Z}$-coefficients, which when tensored with $\mathbb{Q}$ yields cylindrical contact homology, now with the guarantee of well-definedness and invariance.

“The topology of toric origami manifolds
Ana Rita Pires (Fordham University)

Abstract: The topology of a toric symplectic manifold can be read directly from its orbit space (a.k.a. moment polytope), and much the same is true of the (smooth) topological generalizations of toric symplectic manifolds and projective toric varieties. An origami manifold is a manifold endowed with a closed 2-form with a very mild degeneracy along a hypersurface, but this degeneracy is enough to allow for non-simply-connected and non-orientable manifolds, which are excluded from the topological generalizations mentioned above. In this talk we will see how the topology of an (orientable) toric origami manifold, in particular its
fundamental group, can be read from the polytope-like object that represents its orbit space.These results are from joint work with Tara Holm.

“Symplectic fillings”
Laura Starkston (UT Austin)

Abstract: The classification problem for symplectic manifolds with a given contact boundary has been solved in a number of cases for relatively simple three manifolds. The proofs rely at their core on pseudoholomorphic curve arguments, but the detailed classification problems can be interpreted in different topological ways. I will discuss results on the symplectic filling classification problem when the boundary 3-manifold is a Seifert fibered space.

“Non-Hamiltonian actions with isolated fixed points”
Sue Tolman (UIUC)
Abstract: Let a circle act symplectically on a closed symplectic manifold  M. If the action is Hamiltonian, we can pass to the reduced space; moreover, the fixed set largely determines the cohomology and Chern classes of M.  In particular, symplectic circle actions with no fixed points are never Hamiltonian. This leads to the following important question: What conditions force a symplectic action with fixed points to be Hamiltonian? Frankel proved that Kahler circle actions with fixed points on Kahler manifolds are always Hamiltonian.  In contrast, McDuff constructed a non-Hamiltonian symplectic circle action with fixed tori.
Despite significant additional research, the following question is still open: Does there exists a non Hamiltonian symplectic circle action with isolated fixed points?  The main goal of this talk is to answer this question by constructing a non-Hamiltonian symplectic circle action with exactly 32 fixed points on a closed  six-dimensional symplectic manifold.

“Non-Orientable Lagrangian Endocobordisms”
Lisa Traynor (Bryn Mawr College)

Abstract: Lagrangian cobordisms between two Legendrian submanifolds are known to have significant topological rigidity.  For example, in the symplectization of the standard contact 3-space, the genus of an orientable Lagagrangian endocobordism for a Legendrian knot must vanish.  I will describe how for non-orientable Lagrangian endocobordisms of a Legendrian knot there is some, yet restricted, topological flexibility.  This is joint work with Orsola Capovilla-Searle.

Recent mathematical advancements empowering signal/image processing

organized by Julia Dobrosotskaya and Weihong Guo)

“Row action methods and its relation to potential theory”
Xuemei Chen (Univ. of Missouri, Columbia)

Abstract: The Kaczmarz algorithm is an iterative algorithm to solve overdetermined linear systems. We will investigate in a randomized version of it and analyze the recovery error in the mean square sense and in the almost sure sense. The question of which probabilitydistributions on a random fusion frame lead to provably fast convergence is addressed.   In particular, it is proven which distributions give minimal Kaczmarz bounds, and hence give best control on error moment upper bounds arising from Kaczmarz bounds. Uniqueness of the optimal distributions is also addressed.

“An Iterative Algorithm for Large-scale Tikhonov Regularization”
Julianne Chung (Virginia Tech), Katrina Palmer (Appalachian State University)

Abstract:  In this talk, we describe a hybrid iterative approach for computing solutions to large-scale inverse problems via Tikhonov
regularization.  We consider a hybrid LSMR approach, where Tikhonov regularization is used to solve the subproblem of the LSMR approach. One of the benefits of the hybrid approach is that semiconvergence behavior can be avoided.  In addition, since the regularization parameter can be estimated during the iterative process, the regularization parameter does not need to be estimated a priori, making this approach attractive for large scale problems.  Numerical examples from image processing illustrate the benefits and potential of the new approach.

“A PDE-free variational model for multiphase image segmentation”
Julia Dobrosotskaya, Weihong Guo (Case Western Reserve University)

Abstract: We introduce a PDE-free variational model for multiphase image segmentation that uses a sparse representation basis (wavelets or
shearlets) instead of a Fourier basis in a modified diffuse interface context.  This model uses such features of diffuse interface behavior as coarsening and phase separation to merge relevant image elements(coarsening) and separate others into distinct classes(phase separation). To balance these two tendencies, one can adjust the diffuse interface parameter $\epsilon$, just as in the classical diffuse interface models that arise in material science. However, in the new spatial derivative-free set-up, the interface width is no longer proportional to $\epsilon$ (due to the well-localized elements in the chosen sparse representation systems, and thus a completely different diffusive nature of the model), allowing to combine the advantages of non-local information processing with sharp edges in the output. Numerical experiments confirm the effectiveness of the proposed method.

“Denoising an Image by Denoising its Curvature”
Stacey Levine (Duquesne University)

Abstract:  In this work we argue that when an image is corrupted by additive noise, its curvature image is less affected by it. In
particular, we demonstrate that for sufficient noise levels, the PSNR of the curvature image is larger than that of the original image. This leads to the speculation that given a denoising method, we may obtain better results by applying it to the curvature image and then reconstructing from it a clean image, rather than denoising the original image directly. Numerical experiments confirm this for several PDE-based and patch-based denoising algorithms.

“A Weighted Difference of Anisotropic and Isotropic Total Variation Model for Image Processing”
Yifei Lou (UT Dallas)

Abstract: We propose a weighted difference of anisotropic and isotropic total variation (TV) as a regularization for image processing tasks, based on the well-known TV model and natural image statistics. Due to the difference form of our model, it is natural to compute via a difference of convex algorithm (DCA). We draw its connection to the Bregman iteration for convex problems, and prove that the iteration generated from our algorithm converges to a stationary point with the objective function values decreasing monotonically. A stopping strategy based on the stable oscillatory pattern of the iteration error from the ground truth is introduced.  In numerical experiments on image denoising, image deblurring, and magnetic resonance imaging (MRI) reconstruction, our method improves on the classical TV model consistently, and is on par with representative start-of-the-art methods.

“An MBO Scheme on Graphs for Classification and Image Processing”
Ekaterina Merkurjev (UCLA)

Abstract:  In this talk, we present a computationally efficient algorithm utilizing a fully or semi nonlocal graph  Laplacian for
solving a wide range of learning problems in data classification and image processing. In  their recent work “Diffuse Interface Models on Graphs for Classificaiton of High Dimensional Data”,  Bertozzi and Flenner introduced a graph-based diffuse interface model utilizing the Ginzburg-Landau  functional for solving problems in data classification. Here, we propose an adaptation of the classic  numerical Merriman-Bence-Osher (MBO) scheme for minimizing graph-based diffuse interface functionals,  like those originally proposed in the paper by Bertozzi and Flenner. A multiclass extension is introduced using the Gibbs simplex. We also make use of fast numerical solvers for finding eigenvalues and eigenvectors of the graph Laplacian, needed for the inversion of the operator. Various computational examples on benchmark data sets and images are presented to demonstrate the performance of our algorithm, which is successful on images with texture and repetitive structure due to its nonlocal nature. Image processing results show that our method is multiple times more efficient than other well known nonlocal models. Classification experiments  indicate that the results are competitive with or better than the current state-of-the-art algorithms.

“Detecting Plumes in LWIR Using Robust Nonnegative Matrix Factorization Method”
Jing Qin (UCLA)

Abstract:  We consider the problem of identifying chemical plumes in hyperspectral imaging data, which is challenging due to the diffusivity of plumes and the presence of excess noise. We propose a robust nonnegative matrix factorization (RNMF) method to segment hyperspectral images considering the low-rank structure of the data and sparsity of the noise. Because the optimization objective is highly non-convex, NMF is very sensitive to initialization. We address the issue by using the fast Nystrom method and label propagation algorithm (LPA). Using the alternating direction method of multipliers (ADMM), RNMF provides high quality segmentation results effectively. Experimental results on real hyperspectral video sequence of chemical plumes show that the proposed approach is promising in terms of detection accuracy and computational efficiency.

“New spectral filters for a statistical approximation of corrupted images”
Viktoria Taroudaki (speaker): Applied Mathematics and Scientific
Computation Program, University of Maryland, Dianne P. OLeary: Computer
Science Department, Institute for Advanced Computer Studies, University
of Maryland

Abstract: Blur and noise alter images recorded by various devices. One way to reconstruct those images is using spectral filters. Assuming a
known blurring matrix, the filters weigh different components of the image depending on the singular values of the matrix. Since the noise is unknown, the reconstruction problem is an ill-posed inverse problem and we seek a solution with minimal expected error. New filters are
presented here and shown to give good solutions compared with old ones.

Algebraic Geometry

“Orbits in Affine Flag Varieties”
Elizabeth Milićević (Haverford College)

Abstract: Flag varieties are often studied by decomposing them into orbits of various special subgroups. This principle is also fruitful in the case of the affine flag variety, which is the quotient of a reductive algebraic group over a field of Laurent series. In this talk, we will discuss a combinatorial tool for visualizing the unipotent orbits inside of the complete affine flag variety. This alcove walk model due to Parkinson, Ram, and Schwer has applications to questions in algebraic geometry as well as analytic number theory.

“The double
ramification cycle and tautological relations”
Emily Clader (ETH Zurich)

Abstract: The double ramification cycle is an element of the Chow ring of the moduli space of
curves, defined by studying curves that admit a map to the projective line with prescribed ramification.  Pixton has recently proposed a conjectural formula for this cycle in terms of well-known classes.  While the double ramification cycle on M_{g,n} lies in codimension g, Pixton’s formula a priori has contributions in all degrees.  I will discuss a proof that the components in degrees past g vanish, which lends support to Pixton’s conjecture and also yields a family of interesting relations in the Chow ring.  This is joint work with Felix Janda.

“Repairing tropical curves by means of linear tropical modifications”
Maria Angelica Cueto (Columbia University)

Abstract: Tropical geometry is a piecewise-linear shadow of algebraic geometry that preserves important geometric invariants. Often, we can derive classical statements from these (easier) combinatorial objects. One general difficulty in this approach is that tropicalization strongly depends on the embedding of the algebraic variety. Thus, the task of funding a suitable embedding or of repairing a given “bad” embedding to obtain a nicer tropicalization that better reflects the geometry of the input object becomes essential for many applications. In this talk, I will show how to use linear tropical modifications and Berkovich skeleta to achieve such goal in the curve case. Our motivating example will be plane elliptic cubics defined over a non-Archimedean valued field. This is joint work with Hannah Markwig (arXiv:1409.7430).

“Quadrics over Function Fields”
Julia Hartmann (University of Pennsylvania)

Abstract: We discuss the existence of rational points on quadrics over function fields, via the study of the so-called $u$-invariant of a field. Our focus is on function fields over $p$-adic fields.

“A family of type A conformal block bundles of rank one on $M_{0,n}”
Anna Kazanova (University of Georgia)

Abstract: First Chern classes of conformal block vector bundles produce nef divisors on the moduli space of stable n-pointed rational curves. I will explicitly describe the infinite set of all $S_n$-invariant $sl_n$ conformal blocks vector bundles of rank one on $M_{0,n}$. We will see that the cone generated by their base-point free first Chern classes is a polyhedral subcone of the nef cone of $M_{0,n}$, and identify the morphism given by each element of the cone.

“The Craighero-Gattazzo surface is simply-connected”
Julie Rana (Marlboro College)

Abstract: Joint with Jenia Tevelev and Giancarlo Urzua. We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was first conjectured by Dolgachev and Werner. The proof utilizes an interesting technique: to prove a topological fact about a complex surface we use algebraic reduction mod p and deformation theory.

“Geometry of moduli spaces of sheaves on a surface”
Giulia Sacca (Stony Brook and IAS)

Abstract: How much of the geometry of a surface is reflected in the moduli spaces of sheaves on it? In this talk I will give a survey about classical and more recent results answering this question. I will especially focus on the cases of K3, abelian, Enriques and bielliptic surfaces. In the first two cases the geometry of the moduli spaces is very tightly related to that of the underlying surface (so much that, for example, moduli spaces of sheaves on K3 surfaces are considered the higher dimensional analogue of K3 surfaces!). In the last two cases much, but not all, is reflected, making the study of these moduli spaces a very interesting and challenging topic.

“Motivic Gottsche’s curve-counting invariants”
Yu-jong Tzeng (University of Minnesota)

Abstract: On smooth algebraic surfaces, the number of nodal curves in a fixed linear system is universal polynomial of Chern numbers (conjectured by Gottsche, now proven). Recently Gottsche and Shende defined a “refined” invariant which can count real and complex nodal curves and is an invariant in tropical geometry. In this talk I will discuss two “motivic” invariants which generalize the universal polynomials and refined invariant to the algebraic cobordism group and to the Grothendieck ring of varieties.

Statistics

organized by Nancy Flournoy and Mary Gray

PART I: BIOSTATISTICS

“Statistical Change Point Analysis and its Application in Modeling the Next Generation – Sequencing
Data”
Jie Chen, Medical College of Georgia

Abstract:  One of the key features of statistical change point analysis is to estimate the unknown change point location for various statistical models imposed on the sample data.  This analysis can be done through a hypothesis testing process, a model selection perspective, a Bayesian approach, among other methods.  Change point analysis has a wide range of applications in research fields such as statistical quality control, finance and economics, climate study, medicine, genetics, etc. In this talk, I will present a change point model and a Bayesian solution for the estimation of the change point location.  I will provide an application of the proposed change point model for identifying boundaries of DNA copy number variation (CNV) regions using the next generation sequencing data of breast cancer/tumor cell lines.

“Change point estimation: another look at multiple testing problems”
Hongyuan Cao, University of Missouri-Columbia

Abstract:  We consider the problem of large scale multiple testing for data that have locally clustered signals. With this structure, we apply techniques from change point analysis and propose a boundary detection algorithm so that the local clustering information can be utilized. We show that by exploiting the local structure, the precision of a multiple testing procedure can be improved substantially. We study tests with independent as well as dependent p-values. Monte Carlo simulations suggest that the methods perform well with realistic sample sizes and demonstrate the improved detection ability compared with competing methods. The practical utility of our methods is illustrated from a genome-wide association study of blood lipids.

“False discovery rate control of high dimensional TOST tests”
Jing Qiu, University of Missouri

Abstract:  Identifying differentially expressed genes has been an important and widely used approach to investigate gene functions and molecular mechanisms.  A related issue that has drawn much less attention but is equally important is the identification of constantly expressed genes across different conditions. A common practice is to treat genes that are not significantly differentially expressed as significantly equivalently expressed. Such naive practice often leads to large false discovery rate and low power. The more appropriate way for identifying constantly expressed genes should be conducting high dimensional statistical equivalence tests. A well-known equivalence test, the two one-sided tests (TOST), can be used for this purpose. Since the null hypotheses of equivalence analysis (a composite hypothesis) involve an interval of parameters, the null distribution of the p-values of the TOST tests is no longer uniform. Therefore, the existing false discovery rate controlling procedures, which usually assumes uniform distributions for the null distributions of p-values, are very conservative when applied to the TOST tests in high dimensional settings. This work aims to study the performance of the existing FDR controlling procedures and construct new procedures for the TOST tests in high dimensional settings.

“Applying Statistical Methods to Pfizer New Medicine Process and Product Development”
Ke Wang,  Associate Director, WWPS – PGS Statistics, Pfizer Inc. Groton, CT

Abstract:  The pharmaceutical industry is working to a new paradigm, guided by FDA’s “Pharmaceutical Current Good Manufacturing Practices (CGMPs) for the 21st Century: A Risk-Based Approach”. Part of this initiative includes a focus on Quality by Design (QbD) which has brought new emphasis on statistical techniques in developing, estimating, and monitoring pharmaceutical product performance. Our Statistics group mission is to apply good statistical practice in terms of thinking, design, and modeling to enhance decision making in the context of business, scientific and regulatory constraints. This talk will share statistical consulting and problem solving experience in advancing drug projects to deliver new medicines to the patient and in applying new statistical approaches to improve process workflows for the science and technology lines at Pfizer.

PART II

“Sequentially Constraint Monte Carlo”
Shirin Golchi, Columbia University

Abstract:  Constraint can be interpreted in a broad sense as any kind of explicit restriction over the parameters by enforcing known behaviours. Difficulties in sampling from the posterior distribution as a result of incorporation of constraints into the model is a common challenge leading to truncations in the parameter space and inefficient sampling algorithms. We propose a variant of sequential Monte Carlo algorithm for posterior sampling in presence of constraints by defining a sequence of densities through the imposition of the constraint. Samples generated from an unconstrained or mildly constrained distribution are filtered and moved through sampling and resampling steps to obtain a sample from the fully constrained target distribution.  General and model specific forms of constraints enforcing strategies are defined. The Sequentially Constrained Monte Carlo algorithm is demonstrated on constraints defined by monotonicity of a function, densities constrained to low dimensional manifolds, adherence to a theoretically derived model, and model feature matching.

“A Symbolic Data Approach to Estimating Center Characteristics Effects on Outcomes”
Jennifer Le-Rademacher, Division of Biostatistics, Medical College of Wisconsin, Milwaukee

Abstract:  This talk introduces a symbolic data approach to evaluating the effects of center-level characteristics on center outcomes. The proposed method appropriately treats centers rather than patients as the units of observation when estimating the effects of center characteristics since centers are the entities of interest in the analysis. To adjust for the differences in outcomes among centers caused by varying patient load, the effects of patient-level characteristics are Þrst modelled treating patients as the units of observation. The outcomes (adjusted for patient-level effects from step one) of patients from the same center are then combined into a distribution of outcomes representing that center. The outcome distributions are symbolic-valued responses on which the effects of center-level characteristics are modelled. The proposed method provides an alternative framework to analyze clustered data. This method distinguishes the effects of center characteristics from the patient characteristics effects. It can be used to model the effects of center characteristics on the mean as well as the consistency of center outcome which classical methods such as the fixed-effect model and the random-effect model cannot. This method performs well even under scenarios where the data come from a fixed-effect model or a random-effect model. The proposed approach is illustrated using a bone marrow transplant example.

“Designing Combined Traditional and Simulator Experiments”
Erin R. Leatherman, Department of Statistics, West Virginia University

Abstract: Deterministic computer simulators are based on complex mathematical models that describe the relationship of the input and output variables in a physical system. The use of deterministic simulators as experimental vehicles has become widespread in applications such as biology, physics, and engineering. One use of a computer simulator is for prediction; given a set of system inputs, the simulator is run to find the predicted output of the system. However, when the mathematical model is complex, a simulator can be computationally expensive. Therefore statistical metamodels are used to make predictions of the system outputs. This talk considers settings in which both data from the simulator and data from an associated physical experiment are available. We introduce the Weighted Integrated Mean Squared Prediction Error (WIMSPE) measure for designing a combined simulator and traditional physical experiment. Examples will illustrate that WIMSPE-optimal combined designs provide better prediction than standard designs for the combined traditional and simulator experiments.

” Empirical Null using Mixture Distributions and Its Application in Local False Discovery Rate

DoHwan Park, University of Maryland – Baltimore County]

Abstract: When high dimensional data is given, it is often of interest to distinguish between significant (non-null, Ha) and
non-significant (null, H0) group from mixture of two by controlling type I error rate. One popular way to control the level is the false discovery rate (FDR). This talk considers a method based on the local false discovery rate. In most of the previous studies, the null group is commonly assumed to be a normal distribution. However, if the null distribution can be departure from normal, there may exist too many or too few false discoveries (belongs null but rejected from the test) leading to the failure of controlling the given level of FDR. We propose a novel approach which enriches a class of null distribution based on mixture distributions. We provide real examples of gene expression data, fMRI data and protein domain data to illustrate the problems for overview.

PDEs in Continuum Mechanics

organized by Anna Mazzucato, Maria Gualdani

“Higher regularity boundary Harnack inequalities”
Daniela De Silva, Department of Mathematics, Barnard College, Columbia University.

Abstract: We discuss some higher regularity boundary Harnack inequalities and their application to obtain smoothness of the free boundary in obstacle-type problems. This is a joint work with O. Savin.

“PDE-based modeling of coarsening in polycrystalline materials”
Maria Emelianenko, Department of Mathematics, George Mason University

Abstract: Microstructure of polycrystalline materials undergoes a process referred to as coarsening (or grain growth), i.e. elimination of energetically unfavorable crystals by means of a sequence of network transformations, including continuous expansion and instantaneous topological transitions, when the material is subjected to heating. This talk will be focused on recent advances in the field of PDE modeling of this process. Two different strategies will be discussed, one describing the evolution of individual crystals in a 2-dimensional system, and one providing a mean field approximation for the evolution of probability density functions, introduced in the context of a simplified 1-dimensional model. Numerical characteristics and predictions obtained by both strategies will be discussed and contrasted.

“Kolmogorov, Onsager and a Stochastic Model for Turbulence”
Susan Friedlander, Department of Mathematics, University of Southern California   (CANCELLED)

Abstract: We will briefly review Kolmogorov’s ( 41) theory of homogeneous, isotropic turbulence and Onsager’s ( 49 ) conjecture that in 3-dimensional turbulent flows energy dissipation might exist even in the limit of vanishing viscosity. Although over the past 60 years there is a vast body of literature related to this subject, at present there is no rigorous mathematical proof that solutions to the Navier-Stokes equations yield Kolmogorov’s laws. For this reason various models have been introduced that are more tractable but capture some of the essential features of the Navier-Stokes equations themselves. We will discuss one such stochastically driven dyadic model for turbulent energy cascades. We will describe how the very recent Fields Medal results of Hairer and Mattingly for stochastic partial differential equations can be used to prove that this dyadic model is consistent with Kolmogorov’s theory and Onsager’s conjecture. This is joint work with Nathan Glatt-Holtz and Vlad Vicol.

“Passive scalars, moving boundaries, and Newton’s law of cooling”
Juhi Jang, Mathematics Department, University of California Riverside

Abstract: We consider the evolution of passive scalars in both rigid and moving slab-like domains, in both horizontally periodic and infinite contexts. The scalar is required to satisfy Robin-type boundary conditions corresponding to Newton’s law of cooling, which lead to nontrivial equilibrium configurations. We present the equilibration rate of the passive scalar in terms of the parameters in the boundary condition and the equilibration rates of the background velocity field and moving domain. This is joint work with Ian Tice.

“Finite determining parameters feedback control for distributed nonlinear dissipative systems – a computational study”
Evelyn Lunasin, Department of Mathematics, United States Naval Academy

Abstract: We present a numerical study of a new algorithm for controlling general dissipative evolution equations using determining systems of parameters like determining modes, nodes and volume elements.   We implement the feedback control algorithm for the Chafee-Infante equation, a simple reaction diffusion equation and the Kuramoto-Sivashinsky equation, a model for flame front propagation or flowing thin films on inclined surface.  Other representative applications include catalytic rod, chemical vapor deposition and other defense-related applications.  We also discuss stability analysis for the feedback control algorithm and derive sufficient conditions, for the stabilization,  relating the relaxation parameter,  number of controllers and sensors, and other model parameters.  This is joint work with Edriss S. Titi.

“Convergence of the 2D Euler-alpha model to the Euler equations in the no-slip case: indifference to boundary layers”
Helen Nussenzveig Lopes, Mathematics Department, Federal University Rio de Janeiro

Abstract: The Euler-alpha equations are a regularization of the incompressible Euler equations used in sub-grid scale turbulence modeling. Formally setting the regularization parameter alpha to zero we obtain the Euler equations. In this talk we consider the Euler-alpha system in a smooth, two-dimensional, bounded domain with no-slip boundary conditions. For the limiting Euler system we consider the usual non-penetration boundary condition. We show that, if the initial velocities for the Euler-alpha equations approximate the initial data for the Euler equations then, under appropriate regularity assumptions, and despite the presence of a boundary layer, the solutions of the Euler- alpha system converge to the Euler solution, in L^2 in space, uniformly in time, as alpha vanishes. This is joint work with Milton Lopes Filho, Edriss TIti and Aibin Zang.

“Behavior of solutions in the focusing nonlinear Schrodinger equation”
Svetlana Roudenko, Mathematics Department, George Washington University

Abstract: One of the important but at the same time simplest evolution equations is the Schrodinger equation, which governs quantum mechanics. When considering other physical fields such as laser optics, plasma, fluid dynamics or Bose-Einstein condensate, one finds the same Schrodinger equation with added nonlinear terms. In this talk, I will consider the focusing case of the nonlinear Schrodinger equation in various spacial dimensions and with a simple form of power nonlinearity and will discuss behavior of solutions depending on given initial data. This equation has several conserved quantities (such as mass or energy), which are important for classifying different types of solutions. Another important object in this equation is the ground state and the relative `size’ of initial data to that of the ground state. I will explain some known cases of thresholds and dichotomies, and will show a recent result (joint with T. Duyckerts) on classifying the behavior of solutions including solutions with arbitrarily large mass and energy.

“On reconstruction of the dynamic tortuosity functions of poroelastic materials”
Miao-Jung Yvonne Ou, Department of Mathematical Sciences, University of Delaware

Abstract: Poroelastic materials are composites of elastic frame with pore space filled with fluid, eg. rock, sea ice and cancellous bone. The dynamic tortuosity is an effective property which quantifies the effective friction arising from the interaction between the solid frame and the viscous fluid in the tortuous pore space; it plays an important role in the energy dissipation of the poroelastic wave equations, which have been used to model ultrasound propagation in cancellous bones. However, dynamic tortuosity is difficult to measure. In this talk, I will present the recent results on using the dynamic permeability, which is easier to measure, at different frequencies to  reconstruct the dynamic tortuosity function for poroelastic materials with any pore space geometry. The key ingredient in the reconstruction is the integral representation formula (IRF) of tortuosity and its analytical structure.  The mathematical structure of the reconstructed tortuosity leads to an effective numerical treatment of the memory term appearing in the high-frequency poroelastic wave equations.The IRF, the reconstruction scheme with numerical results, together with the relations between pore space geometry and moments of the measure in the IRF will be presented. This research is partially sponsored by NSF-DMS-1413039.

Discrete Math (and Theoretical Computer Science)

organized by Blair Sullivan

“Sampling Single Cut-or-Join Scenarios”
Heather Smith

Abstract: Single cut-or-join is perhaps the simplest mathematical model of genome rearrangement, prescribing a set of allowable moves to model evolution.It is reasonable then to ask how the genes of one genome can be “rearranged” so that it evolves into another quickly.To take this one step farther, fix a collection of genomes G = {G_1, G_2, …, G_n}. Label the leaves of a star tree with the genomes in G. The middle of the star will be labelled with a genome G_M which is “close” to G. The number of rearrangements admitted by G_M is the product of the number of ways
one can evolve G_M into each G_i. Over all possible G_M, we would like to uniformly sample from the admitted rearrangements.Mikl\’os, Kiss, and Tannier (2014) examined this same question for binary trees, discovering that no polynomial-time randomized algorithm exists which will sample the rearrangements almost uniformly unless RP=NP. In this talk, I will present some complexity results for the star tree. We also explore similar computational complexity questions for mathematically motivated problems which arose from this project. This is joint work with Istv\’an Mikl\’os.

“Combinatorial algorithms for the Markov Random Fields problem and implications for ranking, clustering, group decision making and image segmentation”

Dorit S. Hochbaum

Abstract: One of the best known optimization models for image segmentation is the Markov Random Fields (MRF) model.   The MRF problem involves minimizing pairwise-separation and singleton-deviation terms.  This model is shown here to be powerful in representing classical problems of ranking, group decision making and clustering.  The techniques presented are stronger than continuous techniques used in image segmentation, such as total variations, denoising, level sets and some classes of Mumford-Shah functionals.  This is manifested both in terms of running time and in terms of quality of solution for the prescribed optimization problem.

We will sketch the first known efficient, and flow-based, algorithms for the convex MRF (the non-convex is shown to be NP-hard).  We then
discuss the power of the MRF model and algorithms in the context of aggregate ranking. The aggregate ranking problem is to obtain a ranking that is fair and representative of the individual decision makers’ rankings. We argue here that using cardinal pairwise comparisons provides several advantages over score-wise or ordinal models. The aggregate group ranking problem is formalized as the MRF model and is linked to the inverse equal paths problem. This combinatorial approach is shown to have advantages over other pairwise-based methods for
ranking, such as PageRank and the principal eigenvector technique.

“Differentially Private Analysis of Graphs and Social Networks”

Sofya Raskhodnikova

Abstract: Many types of data can be represented as graphs, where nodes correspond to individuals and edges capture relationships
between them. It turns out that the graph structure can be used to infer sensitive information about individuals, such as romantic ties. This talk will
discuss the challenge of performing and releasing analyses of graph data while protecting personal information. It will present algorithms that satisfy a rigorous notion of privacy, called differential privacy, and compute accurate approximations to network statistics, such as subgraph counts and the degree sequence. The techniques used in these algorithms are based on combinatorial analysis, network flow, and linear and convex programming.

“Graph theoretical approaches in cyber security”

Emilie Hogan

Abstract: With recent cyber-attacks on the front pages we realize that secure and resilient cyber systems are necessary. Using graphs as models for
cyber systems is a clear choice since these systems are made up of different types of connections (edges) between computers (vertices). Our recent work has focused on developing new graph theoretical measures for labeled directed graphs and using them to discover patterns of behavior in the graphs. In this talk I will introduce our measures which generalize degree distribution in the case of labeled graphs and show how we have used them to discover events in simulated cyber data. I will also mention a dimension reduction technique mapping graphs to points in R^n (for some n) using these measures, and how we use that to track evolution of dynamic graphs. This is joint work with Cliff Joslyn, Chase Dowling, and Bryan Olsen.

“The Language Edit Distance Problem”
Barna Saha

Abstract:  Given a string s over an alphabet ∑ and a grammar G defined over the same alphabet, how many minimum number of repairs: insertions, deletions and substitutions are required to map s into a valid member of G? We consider this basic question, the language edit distance problem, in this talk. The language edit distance problem has several applications ranging from error-correction in databases,
compiler optimization, natural language processing to computational biology etc. In this talk we show (i)  a near-linear time algorithm for this problem with respect to one of the fundamental context free languages, the Dyck language and its variants, (ii) the first sub-cubic algorithm for the language edit distance problem when any arbitrary context free grammar is considered, and its connection to many fundamental graph problems.

“Grid Minor Theorem and Routing in Graphs”
Julia Chuzhoy

Abstract: One of the key results in Robertson and Seymour’s seminal work on graph minors is the Grid-Minor Theorem (also called the Excluded Grid Theorem). The theorem states that for every fixed-size grid H, every graph whose tree width is large enough, contains H as a minor. This theorem has found many applications in graph theory and algorithms.  Let f(k) denote the largest value, such that every graph of tree width k contains a grid minor of size f(k). Until recently, the best known bound on f(k) was sub-logarithmic in k. In this talk we will survey new results and techniques that establish polynomial bounds on f(k). We will also survey some connections between the Grid-Minor Theorem and graph routing problems, and discuss the major open problems in the area of graph routing. Partly based on joint work with Chandra Chekuri.

“Searching for Structure in Network Science”
Blair Sullivan

Abstract: As complex networks grow increasingly large and available as data, their analysis is crucial for understanding the world we live in, yet graph algorithms are only scalable when limited to relatively simplistic queries (those with low-degree polynomial computational complexity). In order to enable scientific insights, we must be able to compute solutions to more complex questions. To enable this, we turn to parameterized algorithms, which exploit non-uniform complexity to give polynomial time solutions to NP-hard problems when some parameter of the instance is bounded. The theoretical computer science community has been developing a suite of powerful algorithms that exploit specific forms of sparse graph structure (bounded genus, bounded treewidth, etc) to drastically reduce running time.  On the other hand, the (extensive) research effort in network science to characterize the structure of real-world graphs has been primarily focused on either coarse, global properties (e.g., diameter) or very localized measurements (e.g., clustering coefficient) — metrics which are insufficient for ensuring efficient algorithms.

We discuss recent work on bridging the gap between network science and structural graph algorithms, answering questions like: Do real-world networks exhibit structural properties that enable efficient algorithms?  Is it observable empirically? Can sparse structure be proven for popular random graph models? How does such a framework help? Are the efficient algorithms associated with this structure relevant for common tasks such as evaluating communities, clustering and motifs? Can we reduce the (often super-exponential) dependence of these approaches on their structural parameters? This talk includes joint work with E. Demaine, M. Farrell, T. Goodrich, N. Lemons, F. Reidl, P. Rossmanith, F. Sanchez Villaamil & S.
Sikdar.

“General auction mechanism for online advertising”

Gagan Aggarwal

Abstract: In the online advertising market, advertisers compete to show their ads on a webpage. A single webpage might have several slots available to show ads and this gives rise to a bipartite matching market that is typically cleared by the way of an auction. Several auction mechanisms have been proposed, with variants of the Generalized Second Price (GSP) auction being widely used in practice. Motivated by the variety of goals pursued by different advertisers, we consider the problem of designing an auction involving bidders with differing goals. We model this problem using an assignment model with linear utilities, extended with bidder and item specific maximum and minimum prices. We show that, under a non-degeneracy condition, a bidder-optimal stable matching is guaranteed to exist in this model, and use it to design an auction mechanism that is simultaneously truthful for all bidders whose preferences can be expressed in the model. In particular, this mechanism generalizes GSP, is truthful for profit-maximizing bidders, implements features like bidder-specific minimum prices and position-specific bids, and works for rich mixtures of advertiser goals. (Joint work with S. Muthukrishnan, David Pal and Martin Pal)

Mathematical Biology

organized by Erika Camacho and Talitha Washington

“Mitigating Effects of Vaccination on Influenza Outbreaks Given Constraints in Stockpile Size and
Daily Administration Capacity”

Mayteé Cruz-Aponte,
Departamento de Matemática – Física Universidad de Puerto Rico en Cayey

Abstract: Influenzaviruses are a major cause of morbidity and mortality worldwide. Vaccination remains a powerful tool for preventing or mitigating influenza outbreaks. Yet, vaccine supplies and daily administration capacities are limited, even in developed countries. Understanding how such constraints can alter the mitigating effects of vaccination is a crucial part of influenza preparedness plans. Mathematical models provide tools for government and medical officials to assess the impact of different vaccination strategies and plan accordingly. However, many existing models of vaccination employ several questionable assumptions, including a rate of vaccination proportional to the population at each point in time.  We present a SIR-like model that explicitly takes into account vaccine supply and the number of vaccines administered per day and places data-informed limits on these parameters. We refer to this as the non-proportional model of vaccination and compare it to the proportional scheme typically found in the literature.  The proportional and non-proportional models behave similarly for a few different vaccination scenarios. However, there are parameter regimes involving the vaccination campaign duration and daily supply limit for which the non-proportional model predicts smaller epidemics that peak later, but may last longer, than those of the proportional model. We also use the non-proportional model to predict the mitigating effects of variably timed vaccination campaigns for different levels of vaccination coverage, using specific constraints on daily administration capacity.
The non-proportional model of vaccination is a theoretical improvement that provides more accurate predictions of the mitigating effects of vaccination on influenza outbreaks than the proportional model. In addition, parameters such as vaccine supply and daily administration limit can be easily adjusted to simulate conditions in developed and developing nations with a wide variety of financial and medical resources. Finally, the model can be used by government and medical officials to create customized pandemic preparedness plans based on the supply and administration constraints of specific communities.

“Dynamic Networks: From Connectivity to Temporal Behavior”
Anca Radulescu, Department of Mathematics, SUNY New Paltz

Abstract: Many natural systems are organized as networks, in which the nodes (be they cells, individuals or populations) interact in a time-dependent fashion. We illustrate how the hardwired structure (adjacency graph) can affect dynamics (temporal behavior) for two particular types of networks: one with discrete and one with continuous temporal updates. The nodes are coupled according to a connectivity scheme that obeys certain constrains, but also incorporates random aspects.

Using phase diagrams, probabilistic bifurcations and entropy, we compare the effects of different ways of increasing connectivity (by altering edge weights versus edge density versus edge configuration). We determine that the adjacency spectrum is a poor predictor of dynamics when using nonlinear nodes, that increasing the number of connections is not equivalent to strengthening them, and that there is no single factor among those we tested that governs the stability of the system. We discuss the significance of our results in the context of real brain networks. Interpretation of the two models, both with long history of applications to neural coding, may increase our understanding of synaptic restructuring and neural dynamics.

“An Examination of Social Migration within a Cholera Outbreak”

Evelyn Thomas, Department of Mathematics, University of Maryland Baltimore County

Abstract: We present a system of ordinary differential equations that models the spread of Cholera between two populations: one containing healthcare resources, the other deficient of such services. We examine the affect migration based on social factors; specifically the fear of becoming infected and possible mortality when infected, has on the spread of the disease in this system. We utilize such factors to determine intervention strategies for the control and eradication of the disease.

“The Effects of Alcohol Availability on Contagious Violence: A Mathematical Modeling
Approach”

Shari Wiley, Department of Biostatistics and Epidemiology University of Pennsylvania

Abstract: Numerous violence prevention programs are moving towards a broader public health contagion paradigm in understanding and interrupting community violence. The novelty of these paradigms is their use of infectious disease prevention methodologies to interrupt and prevent community violence through changing social norms. However, unlike the public health approach to interrupting infectious disease diffusion, these violence prevention paradigms have yet to be informed by traditional deterministic mathematical models of contagions. We attempt to forge this connection through formulating a mathematical model of contagion and applying it to the spread and interruption of gun violence in Philadelphia. We adopt the classic susceptible-infectious-recovered (SIR) contagion model to describe the relationship between alcohol availability and contagious violence, using data on gun assaults in Philadelphia. In our analysis, we examine distinct populations (non-gun owners, legal gun owners, and illegal gun owners) as factors related to violence transmission. We also include gun assault victim populations to estimate the occurrence of violence over time. To include the role of alcohol availability in the diffusion of gun violence, we applied a random labeling simulation to geospatial data of alcohol outlets and gun assaults in Philadelphia to identify neighborhoods with significant correlation between gun assaults and alcohol outlets. Model results support that a targeted intervention could significantly reduce incidences gun violence, where increasing violence intervention rates by 30% among gun owning (both legal and illegal) violence transmitters could result in a 13% reduction in gun violence, and targeting all high-risk violence transmitters could results in a 40% reduction in gun violence over 10 years. Using results from the random labeling simulations, we identified a high risk region, where the alcohol outlet density was six times greater than other regions and corresponded to a 10% increase in the per-capita gun assault rate.

“A Mathematical Model of Nutrients-Phytoplankton-Oysters in a Bay Ecosystem”

Najat Ziyadi, Department of Mathematics, Morgan State University

Abstract: In this talk, we will introduce a simple mathematical model that describes the interactions of nutrients, phytoplankton and oysters in a bay ecosystem. Using the model, we will derive verifiable conditions for the persistence and extinction of phytoplankton and oysters in the bay system.  In addition, we will illustrate how human activities such as increased oyster harvesting and environmental factors such as increased nutrients inflow and increased oyster filtration can generate phytoplankton bloom with corresponding oscillations in the oyster biomass and nutrients level in the bay ecosystem.

“Model of Tumor-Immune Cells Competing for Glucose Resources

Faina  Berezovskaya*, Department of Mathematics, Howard University; Irina Kareva,
Newman-Lakka Institute for Personalized Cancer Care, Tuffs Medical Center

Abstract:  In the tumor microenvironment there exist competition between cancer cells and the cells of the immune system, which may drive many of the tumor-immune dynamics. Here a model of tumor-immune-glucose interactions is proposed. The model is formulated as a predator-prey type model where tumor population is a prey and immune cell population is a predator; both populations compete for shared resources, i.e., glucose, that are necessary for survival and growth of both populations. It is assumed that immune cells die when tumor is not present, and immune cells undergo clonal expansion depending on how much of the tumor cells they have been able to eliminate. The model allows investigating possible dynamical behaviors that may arise as a result of competition for glucose, including tumor elimination, tumor dormancy and unrestrained tumor growth. A full bifurcation analysis is performed to establish a sequence of regimes that can occur as predator (immune system) and prey (cancer cells) compete for shared resources that are necessary for survival of both. The model behaviors was studied in dependence on parameters and the values of coefficients were estimated from the data published in the literature.

“An Integrative Approach to Lamprey Locomotion”

Kathleen A. Hoffman*, Department of Mathematics and Statistics, University of Maryland
Baltimore County with Nicole Massarelli of University of Maryland Baltimore
County, Christina Hamlet of Tulane University, Eric Tytell of Tufts University,
Tim Kiemel of the University of Maryland Baltimore County, Lisa Fauci of Tulane
University, and Geoff Clapp of the University of Maryland College Park

Abstract:  Lampreys are model organisms for vertebrate locomotion because they have the same types of neurons as higher-order vertebrates, but with fewer numbers. Lamprey locomotion requires combining the electrical activity in the spinal cord, that inervates muscle, which in turn contracts the body, propelling the animal through the water. The resulting motion exerts a force on the fluid, and the fluid exerts forces on the body. I will present results of a longterm interdisciplinary collaboration that combines mathematical models and computational fluid dynamics with biological and fluid experiments to understand locomotion through the water.

“A Mathematical Model for Biocontrol of the Invasive Weed Fallopia Japonica”

Jing Li, California State University Northridge

Abstract:  In this paper, we propose a mathematical model for biocontrol of the invasive weed Fallopia japonica using one of its co-evolved natural enemies, the Japanese sap-sucking psyllid Aphalara itadori. This insect sucks the sap from the stems of the plant, thereby weakening it. Its diet is highly specific to Fallopia japonica. The model is developed for studying a single isolated knotweed stand. The plant’s size is described by time dependent variables for total stem and rhizome biomass. As far as the insects are concerned, it is the larvae of Aphalara itadori that do the most damage to the plant and so the insect population is broken down into numbers of larvae and adults, using a standard stage-structured modeling approach. It turns out that the dynamics of the model depends mainly on a parameter h in our model, which measures how long it takes for an insect to handle (digest) one unit of Fallopia japonica stem biomass. If h is too large then the model does not have a positive equilibrium and the
plant biomass and insect numbers both grow together without bound, though at a lower rate than if the insects were absent. On the other hand, if h is sufficiently small then the model possesses a positive equilibrium which appears to be locally stable. The results based on our model imply that
satisfactory long term control of the knotweed of Fallopia japonica using the insect Aphalara itadori is only possible if the insect is able to consume and digest knotweed biomass sufficiently quickly; if it cannot then the insect can only slow the growth, which is still unbounded. (This is joint work with Stephen A. Gourley and Xingfu Zou.)

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Sharing the Joy: Engaging Undergraduate Students in Mathematics

organized by Julie Barnes, Jo Ellis-Monaghan, and Maura Mast

“What is a good question?”

Brigitte Servatius

Abstract:  Chevalier de Mere had one and Pascal answered it. Fermat had one and Wiles answered it.  Cauchy had one and Connelly answered it. Erdos had many and Carl Pomerance, at JMM 2015, shared the story of his collaboration with Erdos. Good questions can get many people
hooked. Good questions are not always formulated by an experienced mathematician, sometimes they come from students. In this talk we discuss how we use good questions to get students interested in math. In WPI’s “Bridge to Higher Mathematics” course, a sophomore course that teaches proof techniques, we are going over famous theorems (= good questions) and discussing several proofs to each one theorem. We ask students to split into two groups, a question group and an answer group.  The question group Q is required to come up with a good question for the answer group A to solve. Strangely enough there usually are far fewer volunteers for Q than for A.  We also discuss the use of good questions in our REU program.

“Helping Students in a Proofs Course Develop Metacognitive Skills” 

Connie Campbell

Abstract:As part of an NSF funded project, the speaker helped to develop a set of video case studies for use in the teaching
of an introductory proofs course.  These videos show students working in pairs to prove or disprove a statement that is new to them.  When used in
the classroom, together with a well guided discussion, these videos allow students to see peers address obstacles and articulate reasoning as they move toward the proof of a statement (or away from one).     This interactive experience allows the viewer a chance to think about how the students in the video are approaching a problem and challenges the viewer to articulate why a particular approach may or may not be working.  The presenter will show some clips from this video library and demonstrate how one might use these videos in the classroom to enhance student learning.

“Is It Time to Revitalize Your Subject?  A Case Study from Complex Analysis”

Beth Schaubroeck

Abstract:  We often love our subject for its inherent richness and beauty.  However, the teaching of our favorite subject should
reach a wider range of students than just future graduate students.  Many of our students will pursue careers in business, industry, government, or
education, and some upper-level mathematics courses are taken by students with majors other than mathematics.  In this talk, I outline an approach to continually examining the content and teaching of our own subject.  Many of the ideas of this talk will be explored through the lens of a movement to revitalize complex analysis, which started to be formalized with an NSF-funded workshop in 2014.

“When the Taught becomes the Teacher”

Annalisa Crannell, Gülce Tuncer

Abstract:  A preceptor is an upper-class student who acts as a liaison between the professor and the students enrolled in a first-year seminar.
At Franklin & Marshall, preceptors hold office hours, provide feedback on writing assignments, help guide students’ research projects, and
give an occasional lecture.   From the preceptor’s point of view, this experience not only requires a deep knowledge of the mathematics in the
course, but also a nuanced understanding of how students understand (and mis-understand) the material.   This last aspect — delving into the
interaction between the students and the math — turns out to be both the biggest challenge and also the greatest reward of the role of preceptor.

“Using feather boas, Wikki Stix^®, or pipe cleaners to aid in student understanding of functions at all levels of the undergraduate mathematics curriculum.”

Julie Barnes

Abstract:  Students in all levels of mathematics often have trouble visualizing concepts taught about functions.  In this talk, we
discuss using hands-on class activities involving feather boas, Wikki Stix^®, or pipe cleaners to help students understand topics from a variety of undergraduate mathematics courses.  Examples of topics covered in this talk include function transformations, properties of derivatives, epsilon-delta proofs of continuity and uniform continuity, and mappings of complex valued functions.  (Note:  All activities could be done with any of the supplies mentioned: feather boas, Wikki Stix^®, or pipe cleaners.)

“Keychain Ziplines: An engaging introduction to velocity in the calculus classroom”

Audrey Malagon

Abstract:  This talk will discuss an inquiry-based calculus activity to introduce the concepts of average velocity and instantaneous velocity. Using materials that are easy to find, students create a “zipline” for a weighted keychain. Building the ziplines not only introduces important calculus concepts, but also promotes curiosity, sets the tone for an interactive class, and allows students to get to know each other.

“Teaching from the Heart of Mathematical Thought”

Barbara Shipman

Abstract:The mathematics that we teach, use, and understand took years, decades, and centuries of intense, creative, and revolutionary thought to conceive and formulate into what we call mathematics today. Scores of mathematicians dedicated their lives to this work, yet students today are challenged to grasp it in months, weeks, and days filled with distractions. This talk highlights a variety of materials and activities that I have designed on topics including cardinality, convergence, continuity, and mathematical logic and language to guide students in thinking about questions at the heart of the mathematics, before any facts, theorems, formulas, or definitions can be written down. This talk is based in part on material supported by NSF grant #0837810.

“Using Applications to Motivate Differential Equations”

Jessica Libertini

Abstract: The field of differential equations is rich with applications that can be used to drive and facilitate learning.
This talk will present several examples of activities and modeling scenarios that can be used either to introduce and motivate a lesson topic or
to allow students to apply recently acquired skills to a meaningful problem. While using applications has clear benefits for our students, many of whom go on to pursue engineering degrees, adding these components to a course can be challenging, so this talk will also explain some logistical approaches to folding these applications into your course with success!

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