Presentations will take place from 9:30AM to 1:00PM on Saturday, December 6 in Lilly Hall, split between two rooms. Snacks will be provided!
9:30-9:45 - Ryan Wans (mentor: Antoine Prouff)
Title: Spectral Theory and Geometry
Abstract: In this talk, we explore the foundations of spectral geometry, a topic that ties together harmonic analysis, algebra, topology, and some differential geometry, among other fields. We start by thinking of the Laplacian as a linear operator acting over the vector space L^2. By attempting to diagonalize this operator, we elucidate connections between its spectrum and the geometry of its underlying domain. This is formalized in Weyl's law, which we prove in the cases of circles and flat tori. Finally, we connect Weyl's law on the torus to the Gauss circle problem and present estimates of the conjectured remainder.
9:50-10:05 - Mukul Agarwal (mentor: Ruipeng Zou)
Title: Elliptic Curves
Abstract: Elliptic curves are simple algebraic curves with surprisingly deep arithmetic properties and important applications, including in modern cryptography. This talk introduces elliptic curves and the role of projective space in their study, discusses key geometric and arithmetic features such as the group law on points, and concludes with a sketch of Mordell’s theorem showing that the rational points on a curve form a finitely generated abelian group.
10:10-10:25 - Yunda Wang (mentor: Nick McCleerey)
Title: Complex Geometry: Classification of Kähler Manifolds with Finite Automorphism Group
Abstract: In complex geometry, automorphism group is a Lie group that characterizes the symmetry of complex manifolds. Compared to the diffeomorphism group of real smooth manifolds, these automorphism are holomorphic translation preserving the complex structure, resulting in a generally much smaller group. When we specialize to Kähler manifolds, where a compatible metric structure is added, the requirement to preserve the Kähler structure imposes additional constraints, which may or may not shrink the automorphism group further smaller. This leads to a very natural question: when is this group so small that it's actually finite or compact? In this talk, we'll be looking at several interesting types of complex (or Kähler) manifolds for which this is the case.
10:30-10:45 - Dhruv Upreti (mentor: Kale Stahl)
Title: Elementary Lie Theory and Applications to Angular Momentum
Abstract: This talk introduces the basics of Lie theory, beginning with a brief review of group theory. We then develop the core ideas of Lie groups and Lie algebras, the exponential map, and a few theorems essential for a first encounter with the subject. Following that we turn to applications in physics, explaining why angular momentum in quantum mechanics is naturally modeled using a Lie algebra and we conclude by introducing the raising and lowering operators and showing how they arise from this framework.
10:50-11:05 - Jacob Strietelmeier (mentor: Shahbaz Khan)
Title: An Introduction to Representation Theory
Abstract: In this talk we will introduce the concept of a representation over a finite group, discuss the complete reducibility (Maschke's Theorem) of representations, and apply this to S_n by discussing Young Diagrams and the Hook Length Formula.
11:05-11:30 - BREAK
11:30-11:45 - Alan A Mobley Burgos (mentor: Shahbaz Khan)
Title: Weyl’s Theorem - An Overview
Abstract: I will be giving brief motivation for study of Lie algebras as well as a proof overview of Weyl's Theorem.
11:50-12:05 - Xinyue Gu, Casey Ward (mentor: Fabio Capovilla-Searle)
Title: A Tour of Simplicial Complexes: A Method of Modeling Topological Spaces
Abstract: Why do we care about simplicial complexes? What can we learn about their topology from Discrete Morse theory? What are some applications of simplicial complexes?
12:10-12:25 - Sammith Belur, Havish Prageeth (mentor: Nico Bridges)
Title: Quantum Lattice Models
Abstract: Introducing and explaining the use of Lattice Models to encode quantum states.
12:30-12:45 - Josh Culver, Dean Shock (mentor: Matthew Wackerfuss)
Title: Continuous functions on function spaces
Abstract: In this talk, we define the notion of a topology and continuous functions on topological spaces, with a focus on metric spaces. We then introduce a norm, a way to put a topology on vector spaces. Finally, we present the theorem of Hahn-Banach, a vital tool for proving the existence of continuous functionals on vector spaces of uncountable dimension.
9:30-9:45 - Nirek Duma (mentor: Josh Douden)
Title: Patterns, Waves, and Change: An Introduction to PDEs
Abstract: This presentation explores how partial differential equations (PDEs) model change, diffusion, waves, and geometric patterns across physics, biology, and art. We introduce core ideas such as fundamental solutions, symmetry, and boundary-value problems, then connect them to real phenomena—from heat flow and vibrating systems to pattern formation and the mathematical structure underlying visual textures in nature and paintings. Emphasis is placed on how PDEs capture complex behavior with simple local rules and how solutions reveal deep connections between mathematics and the world around us.
9:50-10:05 - Dhruv Jain (mentor: John Sterling)
Title: An Introduction to Algebraic Topology and Topological Data Analysis
Abstract: We give a brief introduction to fundamental ideas in algebraic topology, focusing on homotopy and simplicial complexes. Building on this foundation, we introduce the Vietoris–Rips complex and discuss a concrete example.
10:10-10:25 - Riley TerBush (mentor: Ben Doyle)
Title: An Introduction to Chip-Firing
Abstract: In this talk, we introduce the abelian sandpile model. After describing the basic mechanics of the model, we discuss Dhar's Burning Algorithm and the Cori-Le Borgne bijection, which pairs each superstable configuration on a graph with a spanning tree for the graph.
10:30-10:45 - Tejomay Marathe (mentor: Mateo Matijasevick)
Title: A Random Path to the Understanding SDEs
Abstract: This talk gives a concise introduction to stochastic differential equations from a theoretical standpoint. It starts by grounding probability in its measure-theoretic foundations, then moves through the essential definitions needed to formalize randomness in continuous time. Along the way, it highlights the core mathematical obstacles that arose in modeling SDEs, such as the irregularity of Brownian paths and the failure of classical calculus. It explains the key ideas that resolved them, including filtration, martingales, and Itô’s framework.
10:50-11:05 - Jose Gutierrez Perez (mentor: Daniel Flores)
Title: Game Theory
Abstract: This talk introduces game theory with a mathematics focus. My mentor and I used the textbook Game Theory by Giacomo Bonanno to further expand our understanding of the subject. I will explore how game frames are structured, some of the history behind game theory, and concepts such as dominant strategies, Nash equilibria, and von Neumann-Morgenstern utility functions. Through some examples of games and game frames, I will discuss how decision making can be analyzed using these tools and highlight some of the mathematics used to study such behaviors. The goal of the talk is to share what I’ve learned over the semester and illustrate how game theory can be applied to mathematical problems and real-world scenarios.