Fall 2023 Schedule

Recordings and slides of talks are available on Echo360.

November 30
Who: Mayla Boguslav
Title: Revealing and Exploring the Literature's Known Unknowns: Ignorance and How it Drives Science
Abstract: Scientific discovery progresses by exploring new and uncharted territory. More specifically, it advances by a process of transforming unknown unknowns first into known unknowns, and then into knowns. Over the last few decades, researchers have developed many knowledge bases to capture and connect the knowns, which has enabled topic exploration and contextualization of experimental results. But recognizing the unknowns is also critical for finding the most pertinent questions and their answers. Little work has focused on how scientists might use them to trace a given topic or experimental result in search of open questions and new avenues for exploration. We present methods and tools to help researchers automatically uncover these unknowns through the illumination of specific goals for scientific knowledge.

October 26
Who: Vlad Kokushkin
Title: Accounting for Working Memory in models of students' mathematical cognition
Abstract: Working memory (WM) is a psychological construct for modeling a human’s ability to simultaneously store and process incoming information. In this talk, I discuss two models that describe the role of students’ WM in doing mathematics. The first model is known as Unit Transformation Graphs (UTGs) and has been proven to be effective in modeling the cognitive load of fractional tasks. The second model applies to multidimensional mathematical contexts, such as mathematical proofs. I further present two empirical studies supporting these models and dive into some WM-related concepts, such as cognitive load, cognitive overload, and cognitive offloading.

September 28
Who: Jake Kettinger
Title: New Perspectives on Geproci Sets
Abstract: The geproci property is a recent development in the world of geometry. We call a set of points Z ⊆ P3 k an (a, b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011 and nondegenerate non-grids have been known only since 2018. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in my thesis, a procedure was known for creating specific nondegenerate non-grid (a, b)-geproci sets for 4 ≤ a ≤ b, but it was not known what other examples there can be. Furthermore, before the work in my thesis, almost all examples of geproci sets that were known were contained in unions of b disjoint lines (known as half grids) and there was no known way to generate new examples of non-half grids. Here, I will discuss how to use geometry in positive characteristics to find new methods of producing geproci half grids and non-half grids.

August 31
Who: Evan Camrud
Title: A story of two hypo-s: hypocoercivity and hypoellipticity for two degenerate stochastic dynamics
Abstract: We describe the asymptotic stability for two nonlinear stochastic dynamics via hypocoercive estimates in the first, and direct pathwise probabilistic estimates in the second. In the second setting, we establish minimum conditions on the additive noise for hypoellipticity, which guarantees the existence of the invariant measure.

Spring 2023 Schedule

Recordings and slides of talks are available on Echo360.

May 04
Who: Jocelyn Rios
Title: My journey developing mathematical tasks related to equity
Abstract: In this informal talk, I share my experiences developing mathematical tasks that aim to support students of color and equity in the classroom (particularly in the context of an HSI). I reference different perspectives from the literature and results from my own research that have informed my evolving thoughts on equitable task designs. The goal of this talk is to share ideas and give attendees opportunities to engage with some of the tasks.

April 06
Who: Marc Fehling
Title: Adaptive methods for finite elements
Abstract: Partial differential equations are used to model natural phenomena. Oftentimes they cannot be solved directly, but numerical methods can be used to find approximate solutions. Finite element methods present a popular way to solve these equations using piecewise polynomials. The required computational effort grows with the problem size, so it is important to apply efficient algorithms. With adaptive methods for example, one can increase resolutions locally only where it is needed. Combined with parallelization, it is a powerful tool to reduce runtimes. In this talk I will explain how finite element methods work, point out the benefits of hp-adaptive methods, and show how our algorithms scale on supercomputers.

March 09
Who: Yajing Liu
Title: ReLU Neural Networks, Polyhedra Decomposition, and Application for Adversarial Image Detection
Abstract: The weights and biases of a rectified linear unit (ReLU) feedforward neural network decompose the input space into polyhedra and determine a continuous piecewise linear map F on each polyhedron. In this presentation, we first define binary vectors by recording the ReLU activation/non-activation at each node to represent each polyhedron. Then, we show how these vectors can be used to determine the hyperplanes of the polyhedron and an affine linear function that agrees with F on the polyhedron. Finally, we illustrate how one can utilize the binary vectors to detect and analyze adversarial attacks in the context of digital images.

Fall 2022 Schedule

Recordings and slides of talks are available on Echo360.

December 06
Who: Cigole Thomas
Title: Arithmetic Dynamics of Group Actions on Varieties
Abstract: In this talk, I will give a brief introduction to varieties, in particular, character varieties defined over finite fields. The main goal is to understand the dynamics of the action of a group on these objects by looking at transitivity properties and large orbits. Time permitting, I will briefly mention my projects on Varieties in Data Learning and GLAMS: Graduate Learning Assistants in Mathematical Sciences. I will try to make the talk accessible to a general math audience.

November 01
Who: Jessi Lajos
Title: Planting Seeds through Embodiment to Teach Formal Concepts of Abstract Algebra
Abstract: In this descriptive case study, we explored how an embodied cognition researcher integrated embodiment beyond gesture as she taught a first semester abstract algebra course. We found that this instructor intentionally used embodiment to motivate and ground formal definitions, theorems, and proofs. In addition to her use of gesture, she encouraged students to interact with physical materials and simulate the mathematics with their bodies. Simulations opened communication lines between the instructor and students, who were not fluent in formal language. Furthermore, the instructor’s simultaneous use of various forms of embodiment highlighted and disambiguated referents of student’s speech, positioned students’ contributions as legitimate, and carried forth students' theories. Our results offer practical implications for teaching by illustrating examples of how embodiment can be incorporated into an abstract algebra classroom.

October 04
Who: Alex Elchesen
Title: Optimal Transport and Persistence
Abstract: Optimal transport is concerned with finding the most cost-effective way to transport one distribution of mass to another. Solutions to this problem give rise to the popular Wasserstein distances between probability measures. Variants of the Wasserstein distances have become important in applied topology where they are used to compare the outputs of persistent homology computations. In this talk, I'll introduce the classical optimal transport theory, describe its use in applied topology, and discuss recent work that extends the classical theory.

September 06
Who: Tomojit Ghosh
Title: Convex and Non-Convex Models for Visualization, Classification, and Feature Selection
Abstract: In this presentation, I will introduce some convex and nonconvex algorithms suitable for data visualization, dimensionality reduction, and feature selection when class labels are available. I will show how these algorithms are linked to each other in a way to encode class centroids. At last, I will present results on different bench-marking data sets to establish the benefits of the proposed methods.

Spring 2021 Schedule

April 21
Who: Lian Duan
Title: Bertini's theorem over finite field and Frobenius nonclassical varieties
Abstract: Let X be a smooth subvariety of P^n defined over a field k. Suppose k is an infinite field, then the classical theorem of Bertini asserts that X admits a smooth hyperplane section. However, if k is a finite field, there are examples of X such that every hyperplane H in P^n defined over k is tangent to X. One of the remedies in this situation is to extending the ground field k to its finite extension, and considering all the hyperplanes defined over the extension field. Then one can ask: Knowing the invariants of X (e.g. the degree of X), how much one needs to extend k in order to guarantee at least one transverse hyperplane section? In this talk we will report several results regarding to this type of questions. We also want to talk about a special type of varieties (Frobenius nonclassical varieties) that appear naturally in our research. This is a joint work with Shamil Asgarli and Kuan-Wen Lai.

February 24
Who: Eric Kehoe
Title: Long Short-Term Memory Networks, the Tonnetz Lattice, and Musical Harmony
Abstract: In this talk I present the theory and the findings of our ICCS 2020 conference paper, Exploring Musical Structure using Tonnetz Lattice Geometry and LSTMs. The primary goal is to predict harmonic patterns in musical scores using LSTMs, a neural network architecture designed to learn patterns in time series data. Our main contribution is applying Euler's Tonnetz lattice to embed musical chords in Euclidean space while retaining their harmonic relationships, and use this embedding as the base input to an LSTM network for prediction tasks. Training and testing on a collection Bach chorales, we achieve an accuracy rate of 50.4% on validation data, compared to the random guess rate of 0.2% . This suggests that using Euler’s Tonnetz for embedding chords provides a framework in which machine learning tools can excel in classification and prediction tasks with musical data.

February 10
Who: Sanwar Ahmad
Title: An Introduction to Electrical Impedance Tomography with Complete Electrode Model
Abstract: In this presentation, I will discuss the concept of inverse problems and its application. I will also introduce Electrical Impedance Tomography (EIT) problems, and discuss the mathematical model used for EIT problems.

Fall 2020 Schedule

November 9
Who: Marc Fehling
Title: Algorithms for massively parallel generic hp-adaptive finite element methods (Link to slides)
Abstract: Efficient algorithms for the numerical solution of partial differential equations are required to solve problems on an economically viable timescale. In general, this is achieved by adapting the resolution of the discretization to the investigated problem, as well as exploiting hardware specifications. For the latter category, parallelization plays a major role for modern multi-core and multi-node architectures, especially in the context of high-performance computing. Using finite element methods, solutions are approximated by discretizing the function space of the problem with piecewise polynomials. With hp-adaptive methods, the polynomial degrees of these basis functions may vary on locally refined meshes. We present algorithms and data structures required for generic hp-adaptive finite element software applicable for both continuous and discontinuous Galerkin methods on distributed memory systems. Both function space and mesh may be adapted dynamically during the solution process. We briefly outline the non-trivial parts of the implementation within the open-source library deal.II, and solve the Laplace problem exemplarily on a domain with a reentrant corner which invokes a singularity. With this example, we demonstrate the benefits of the hp-adaptive methods in terms of error convergence and show that our algorithm scales up to 49,152 MPI processes.

October 26
Who: Andreas Gross
Title: The geometry of metric graphs (Link to notes)
Abstract: Metric graphs are arguably among the simplest combinatorial and geometric objects. On the other hand, the combinatorics of the space of all metric graphs, the 'moduli space of tropical curves' is quite intricate. In my talk I will discuss the geometric aspects of this moduli space. I will use this problem to motivate the basic concepts in tropical geometry, so this talk will be accessible to all.

October 12
Who: Michael Epstein
Title: Lemniscate Trees of Random Polynomials and Asymptotic Enumeration of Morse Functions on the 2-Sphere (Link to slides)
Abstract: We'll consider two problems: first we'll investigate the nesting structure of lemniscate configurations associated to complex polynomials, and in the second part of the talk we'll determine the asymptotics for the number of geometric equivalence classes of Morse functions on the 2-sphere. Both the lemniscate configurations and the equivalence classes of Morse functions are enumerated by classes of trees, and both problems are amenable to the methods of analytic combinatorics. Along the way we'll discuss some of the basic techniques in this fascinating area.

September 28
Who: Michael DiPasquale
Title: Waring problems for polynomials (Link to notes)
Abstract: A well-known problem in number theory, Warings' problem, asks whether each natural number k has an associated positive integer s so that every natural number is the sum of at most s kth powers. For example, Lagrange showed that every integer can be written as the sum of four squares. While this was answered positively by Hilbert, finding the optimal such s for a given k remains open. We'll discuss a version of this problem for polynomials. Given a homogeneous polynomial of degree k, one can ask for the smallest number s of linear forms needed to write the polynomial as a sum of s kth powers of linear forms. This is called the Waring rank of the polynomial. It is not difficult to show that every homogeneous polynomial has a Waring rank. The Waring rank of a `generic' homogeneous polynomial of degree k is a celebrated result due to Alexander and Hirschowitz (which classifies all the `deficient' secant varieties of Veronese varieties). It remains an interesting and widely open problem to study the Waring rank of polynomials of a particular form, since it is difficult to determine if a homogeneous polynomial is `generic.' (the Waring rank of a monomial was only settled within the last decade!) Time permitting, I will give some interesting examples of simple homogeneous polynomials with unexpectedly low Waring rank which came out of joint work with Zachary Flores and Chris Peterson. I will assume minimal background for this talk.

September 14
Who: Everyone
Title: Organizational meeting