Spring 2025 schedule

Jan 27, 2025 — Alex Squires

Title: Mapping Class Groups of Infinite Type Surfaces

Abstract: Topologists have long been interested in classifying and understanding surfaces. We know that in the case of closed, orientable surfaces that surfaces of the same genus are homeomorphic, giving a countably infinite number of classes of surfaces. In the case of infinite type surfaces, we also need the space of ends and the space of ends accumulated by genus in order to complete the classification. The introduction of Mapping Class Groups to the area shed new light on how to understand infinite type surfaces. We will begin discussing what infinite type surfaces have coarsely bounded mapping class groups, as set forward by the 2019 Mann-Rafi paper "Large Scale Geometry of Big Mapping Class Groups" 

Feb 3, 2025 — Philip Pampreen

Title: Algorithmic Decidability, the Halting Problem, and Undecidability Proofs

Abstract:I will introduce basic concepts for computability theory including programs, decidable sets, and halting. Then I will use these concepts to present the halting problem and a corollary called the Busy Beaver problem. Then, I will use these techniques and a Turing complete esoteric programming language called 1# to present a proof by Larry Moss that the pointed tiling problem of the quadrant is algorithmically undecidable.

Feb 10, 2025 — Tibor Burdette

Title: Hadamard matrices and biunimodular sequences of length p

Abstract: For p prime, there is a one-to-one correspondence between p x p circulant complex Hadamard matrices and biunimodular sequences of length p. In this presentation, we investigate a result of Haagerup that shows the finiteness of the former by embedding the latter into a complex variety and showing the finiteness thereof. A key ingredient of the proof is Tao’s uncertainty principle for cyclic groups of prime order.

Feb 17, 2025 — Shipra Baranwal

Title: EARLY DETECTION OF DEPRESSION IN SOCIAL MEDIA TEXTS USING ADVANCED NLP TECHNIQUES : LEVERAGING LARGE LANGUAGE MODELS AND BIO-BERT

Abstract: Early detection of depression through social media analysis is critical for timely intervention and improved outcomes. Building on previous eRisk challenges, this study develops text classification models to identify signs of major depressive disorder from online posts. We employ traditional statistical methods, advanced NLP, and deep learning techniques, using a bag-of-words representation with classifiers including Naive Bayes, AdaBoost, Random Forest, Logistic Regression, and SVM. Our findings highlight the efficacy and limitations of these approaches, paving the way for enhanced automated mental health screening.

Feb 24, 2025 — Hyeran Cho (OSU)

Title: Hyperbolicity of Random Branched Coverings

Abstract: For a finitely presented group Γ with a finite presentation ⟨S|R⟩, let X be the presentation 2- complex. We introduce n-fold random branched coverings of X branched over the centers of its 2- cells. We prove that fundamental groups of random branched coverings are asymptotically almost surely Gromov hyperbolic. In other words, for a n-fold random branched covering $X(σ) → X$, the probability that $π_1(X(σ))$ is Gromov hyperbolic goes to 1 in the limit n → $\infinty$. This is a joint work with Jean-Francois Lafont , Rachel Skipper

 Mar 10, 2025 — Michael Logal

Title:  Isogeny-Torsion Graphs of Elliptic Curves

Abstract: An isogeny is a rational map between elliptic curves. An isogeny graph is a graph where nodes represent isomorphism classes of elliptic curves and edges represent isogenies, labeled by degree. An isogeny-torsion graph is an isogeny graph, but the nodes are instead replaced with the torsion of the elliptic curves. A recent paper by Garen Chiloyan and Alvaro Lozano-Robledo states that there are 52 isomorphism types of isogeny-torsion graphs that are associated to elliptic curves defined over ℚ. We will discuss this result and some results over quadratic fields, which is my current research project.

Mar 24, 2025 — Caden Farley

Title: The Allen-Cahn equation and minimal surfaces

Abstract: The Allen-Cahn equation is a semilinear elliptic PDE which arose in the modelling of phase transitions in materials science. Solutions to the Allen-Cahn equation are critical points of an energy functional and are closely related to minimal surfaces and sets of finite perimeter. In recent years, geometric analysts have exploited this relationship to obtain new results about the existence and geometry of minimal surfaces in Riemannian manifolds. In this talk, I'll survey the basics of this connection, discussing the properties of solutions to Allen-Cahn, examples of solutions and methods for constructing solutions, the results linking the Allen-Cahn equation to minimal surface theory, and recent developments and open problems

Mar 31, 2025 — Dimitrios Nikolakopoulos

Title: Counting Matroids on an n-element Set

Abstract: This talk explores the counting of matroids on a ground set [n]={1,2,…,n}, focusing on linear-representable, linear-flock representable, Frobenius-flock representable, and algebraic matroids. We will discuss methods for counting these matroids, emphasizing the combinatorial and algebraic techniques involved.

 Apr 7, 2025 — Shankha Shubhra Mukherjee (Notre Dame)

Title: Directional Transforms in Topology: From Reeb Graphs to Persistent Homology

Abstract: Reeb graphs and persistent homology have emerged as powerful tools in topological data analysis, enabling researchers to capture and classify intricate geometric and topological structures. In this talk, I will show how Reeb graphs reveal a shape’s evolution under slicing and how persistent homology detects loops and holes at various scales. We will then see how directional transforms let us tailor these topological methods to different settings, from high-dimensional data to specialized slicing strategies. We will discuss the concept of Reeb transforms, which refine Reeb graphs for specific definable shapes and offer new insights into structure and connectivity. The talk will also touch on monodromy in persistent homology, a perspective that helps us understand how topological features evolve as we smoothly vary parameters. Throughout, we will look at examples that showcase the broad applications of these ideas, including shape comparison and large-scale data visualization.

Apr 21, 2025 — Hannah Garrett

Title: Integer Matrix Representations of Triangle Groups into SL(3, R) and G2 (tentative)

Abstract: A linear representation of degree n of a group G is a homomorphism of G into a group of nonsingular n x n matrices. A study is made of those representations of triangle groups into the Lie group SL(3,R) and G2 for which the matrix entries are all integers. We will specifically looking at the triangle groups (6,3,3) where n=3 and (4,6,2) where n=7.

Apr 28, 2025 — Vasileios Oikonomou (University of Missouri)

Title: Existence of Periodic Hollow Vortices

Abstract:In this talk, we will present new results on the existence of periodic configurations of hollow vortices. A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a fluid, whose flow is governed by the 2D Euler equations. We will show that there exists a one-parameter family of steady, periodic hollow vortices that bifurcates from a periodic point vortex configurations and extends up to the onset of certain singularities, which we also classify. One notable application is the existence of von Kármán hollow vortex streets. The technique involves reformulating the problem using complex analysis tools, including conformal mappings and layer potential representations, then carrying out a global bifurcation argument.