Spring 2026

Abstract: This talk introduces two classical invariants in knot theory: the unknotting number and the bridge number. I will begin with the basic idea of a knot invariant and explain why such quantities are useful for distinguishing knots and understanding their complexity. Then I will use standard examples, especially the trefoil knot and the figure-eight knot, to show how these invariants are computed or estimated in practice. I will also briefly discuss why the unknotting number can be difficult to determine, and how different projections of the same knot may reveal different information.

Mentor: Aru Mukherjea 

Abstract: This talk will be a brief introduction to the ideal membership problem for polynomial rings and how Gröbner bases are effective tools for building algorithms which solve it.

Mentor: Sai Sivakumar

Abstract: To determine a webpage's relative importance, one can compute its pagerank via random walks across the web. Spammers often desire to inflate their pagerank to direct traffic to their webpages. This talk discusses both spam methods, and how to design a pagerank system to detect and address such methods.

Mentor: Addie Duncan 

Abstract: This talk presents a brief overview of measure theory and integration theory before exploring basic probability theory.

Mentor: Dillon Grannis 

Abstract: Heegaard splittings provide a concrete way to build and analyze 3-manifolds. In this talk, we introduce the idea of decomposing 3-manifolds into two handlebodies glued along a common boundary surface. Emphasizing intuition, examples, and visualizations, we show how familiar topological spaces admit a natural Heegaard splitting and how classical constructions can be understood through this decomposition. This goal is to develop geometric intuition for 3-manifolds and illustrate why Heegaard splittings serve as a fundamental tool in low-dimensional topology.

Mentor: Justin Chen 

Abstract: Introductory Category theory, including Yoneda Embedding, homotopy, fundamental groups, and Van Kampen's Theorem. Pictures and art demo included!

Mentor: Peter Seong 

Abstract: The Lebesgue measure extends the ordinary notion of length on intervals to a much larger class of subsets of the real line, providing the foundation for modern integration and probability theory. In this talk, I will explain why such an extension is needed, beginning with the limitations of measuring sets using only interval length. I will then outline the construction of Lebesgue measure through outer measure, the Carathéodory measurability criterion, and the resulting class of measurable sets. Finally, I will discuss why this construction is so important in probability theory, where probability measures are built on measurable spaces and many random phenomena are modeled on the real line using Borel and Lebesgue measurable sets. The goal is to give a conceptual picture of both the construction itself and the reason it plays such a central role in analysis and probability.

Mentor: Iris Jiang 

Abstract: The Genus G surface (multi-hole torus) has the universal cover of a hyperbolic 4G gon. We can directly define a distribution on a Genus G surface by using a distribution that starts out on the hyperbolic plane. Then, defining a distribution on a Genus G surface is equivalent to summing over the Fuchsian groups for points in the hyperbolic 4G gon. This construction gives simple sampling and reduces some boundary artifacts.

Mentor: William Ghanem

Abstract: The talk will introduce various foundational concepts for Classical Differential Geometry such as the pushforward, sections, and mappings. We will also explore the different frames of Differential Geometry up to the Frenet Formulas and Frenet Frame.

Mentor: Adrian Flores 

Abstract: The Gauss–Bonnet theorem relates the Euler characteristic, a global topological invariant, to an integral of curvature. Many results in geometry share a similar flavor, relating topological invariants to integrals of geometric quantities, such as the Riemann–Roch and Hirzebruch signature theorems. In this talk, I will explain how these examples are connected through the notion of the index of a differential operator. After briefly introducing the concept of symbols and elliptic operators, I will state the Atiyah–Singer index theorem, which relates analytic and topological quantities. The goal is to give a general picture of how differential operators can be used to study topology and to illustrate the interplay between geometry, analysis, and topology. If time permits, I will also outline the main idea behind the proof of the Atiyah–Singer index theorem.

Mentor: Adrian Flores 

Abstract: Markov Chain And Its Application in Allele

Mentor: Jake Wellington 

Abstract: The Law of Large Numbers says that if you repeat a random experiment enough times, your average result will converge to the true expected value. This talk covers why that happens, what it breaks down, and how Monte Carlo simulation uses this idea to estimate things that have no clean formula.

Mentor: Jake Wellington 

Abstract: Stochastic Gradient Descent (SGD) is the primary optimization method used to train neural networks, but its behavior depends strongly on the architecture of the given network. In this talk, we study how architectural choices influence the geometry of gradients and therefore the dynamics of SGD. We begin by introducing the basic setup of neural networks and SGD, along with standard assumptions such as Lipschitz continuity of gradients and unbiased stochastic gradients. We then discuss why these assumptions can fail in deep neural networks, particularly due to vanishing and exploding gradients. Finally, we examine how design choices such as activation functions, initialization schemes, normalization, and skip connections help stabilize training by improving gradient flow. The main takeaway is that neural network architecture plays a central role in determining whether SGD behaves in a stable and approximately convergent way in practice.

Mentor: Aaron Benda

Abstract: What does it mean to have the best unbiased estimator for a parameter, and when can we be sure that it's existence is unique? In this talk, we develop the tools needed to precisely answer this question. We introduce the notion of sufficient statistics: functions of the sample that capture maximal information about the parameter alongside complete statistics, which rule out any redundancy. Taken together, these two conditions form the hypothesis of the Lehmann-Scheffe theorem, which guarantees that any unbiased estimator that is a function of a complete sufficient statistic is the unique uniformly minimum variance unbiased estimator (UMVUE). We illustrate the theorem through examples, identifying complete sufficient statistics and constructing UMVUEs explicitly.

Mentor: Justin Le 

Abstract: This presentation explores the concept of martingales within the broader study of probability. It introduces their defining properties and highlights their role in modeling fair and balanced stochastic processes. The discussion emphasizes how expectations evolve over time under uncertainty. Key theoretical results like the Optional Stopping Theorem are presented to show how these processes behave under situations that could occur in the real world, like a casino.

Mentor: Mark Abate 

Abstract: A graph is strongly unipositive if its distance matrix has exactly one positive eigenvalue and n-1 negative eigenvalues for any distance weighting. We will show that trees and unicyclic graphs of girth 3 are strongly unipositive.

Mentor: Jeffrey Cheng 

Abstract: I will explore the proof and motivation behind how the irreducible characters of  representations of finite groups form an orthonormal basis for the complex-valued class functions of the finite group. I will also discuss some implications of this theorem.

Mentor: Ryan Wandsnider

Abstract: In Linear Algebra, we notice that finite-dimensional vector spaces have the same underlying structure as coordinate vector spaces. This is, furthermore, unique for a given vector space. When we consider modules, however, our field of scalars is now a ring. Is it possible to still preserve this kind of structure after such a generalization? The Structure Theorem for Finitely Generated Modules guarantees such a decomposition, provided the ring of scalars is a Principle Ideal Domain.

Mentor: Henry Glunz 

Abstract: The Hilbert Basis Theorem asserts that every ideal in the polynomial ring ​is finitely generated; that is, each ideal can be expressed using a finite collection of polynomials whose combinations generate all its elements.The proof proceeds in four main steps. First, we establish a division algorithm for multivariable polynomials by fixing a monomial ordering, ensuring that leading terms are well-defined. Second, we examine monomial ideals—ideals generated by monomials—where membership reduces to a simple divisibility condition, often visualized as a staircase pattern in a lattice. Third, Dickson’s Lemma guarantees that every monomial ideal is finitely generated, via induction on the number of variables. Finally, the key step connects general ideals to monomial ideals: for any ideal I, the ideal generated by its leading terms is itself a monomial ideal. By Dickson’s Lemma, this leading term ideal is finitely generated, and the division algorithm then shows that a corresponding finite set of polynomials generates 𝐼 I. Any nonzero remainder in this process would yield a contradiction, as its leading term would have to lie both inside and outside the leading term ideal. As a consequence, the polynomial ring satisfies the Ascending Chain Condition on ideals, and every affine variety can be defined by finitely many polynomial equations.

Mentor: Daniel Koizumi 

Abstract: We discuss how the class number of a number field measures the failure of unique factorization over its ring of integers, discuss some details of the proof, and give an example of its computation in a nontrivial case.

Mentor: Sudharshan K V 

Abstract: This talk will cover the basics of topological manifolds, spaces that locally resemble Euclidean space but can differ greatly in their global structure. We'll explore how manifolds can be glued together via adjunction spaces, featuring the Lickorish-Wallace theorem as a central example in the study of 3-manifolds. I'll end with a brief discussion of the Poincaré conjecture and the possible shapes of the universe.

Mentor: Ansel Goh 

Abstract: The Lebesgue integral is a very important building block of modern mathematics. This talk builds this integral from the ground up. We start with the intuitive problem of measuring the "size" of sets, which leads into the basics of measure theory. With this foundation, we make the Lebesgue integral and compare it with the traditional Riemann approach, doing so highlights the Lebesgue's advantage (powerful convergence theorems that allow us to exchange limits and integrals). From there, we will build intuition for how this will be used in probability theory; talking about the triplet used to define a probability space. This allows us to use measures to represent probabilities and integrals as expectations. We then explore how these abstract concepts are fundamental for the Wasserstein distance, an optimal transport metric to stabilize training Generative Adversarial Networks (GANs).

Mentor: Luisa Velasco 

Abstract: We develop some measure-theoretic foundations of martingale theory, covering filtrations, conditional expectation, and stopping times, with the optional stopping theorem as the main result. We close with an application to risk-neutral pricing in mathematical finance.

Mentor: Sara Ansari 

Abstract: Continued fractions consist of a standard, algebraic algorithm for the set of real numbers. Dynamical systems, on the other hand, provide us with a visualization of the algebra as a moving orbit or cycle. This talk will examine the connection between the two topics by looking at the modulo 1 operation. We will introduce simple maps and the map related to the continued fraction algorithm.

Mentor: Martha Richey 

Abstract: The Schrodinger equation has many applications in physics. In this talk we will discuss conditions for which the non-linear Schrodinger equation is "well-posed" (in the L^2 subcritical sense), meaning, a solution exists, is unique, and depends on the initial data continuously.

Mentor: Zachary Lee 

Abstract: In this talk, I will introduce the p-adic numbers and explain why they matter in number theory. They give a different way to measure size and distance, one that depends on divisibility by a prime number p rather than the usual absolute value. Because of that, they are a natural tool for studying arithmetic questions and also bring in ideas from analysis and topology. I will start with valuations on fields, especially the distinction between Archimedean and non-Archimedean valuations, and then move to the p-adic valuation on Q. After that, I will explain how Q_p is built by completing Q with respect to the p-adic valuation. If time permits, I will also say a little about p-adic expansions, p-adic integers, and some of the basic ways Q_p differs from the real numbers.

Mentor: Aryan Amin Kazi 

Abstract: In this talk, I will give a brief introduction to the idea of homology, an algebraic tool used to study the shape of topological spaces. The main goal of homology is to be able to distinguish spaces by counting the number of “holes" they have in different dimensions. I will outline the basic construction and illustrate how homology distinguishes between spaces such as the sphere and torus.

Mentor: Jemma Schroder 

Abstract: I will show how to use character theory to study representation theory of Symmetric Group 4. I will do so by using theorems of character theory related to trace, orthogonality, normality, and tensor products to find irreducible representations of S4.

Mentor: Matthew Verheul 

Abstract: We discuss how knot polynomials are used to distinguish between knots up to isotopy.

Mentor: Hillary Kim 

Abstract: We discuss the dollar game on a graph, using a formal sum of vertices (known as a divisor) to represent the number of dollars per person. We may represent lending/firing moves with the Laplacian matrix (of a graph). Then, we develop the relation of this matrix to linearly equivalent divisors, which formalizes the available states of the game. We develop the Picard and Jacobian groups, and measure the winnability of a divisor via defining the rank of a divisor. This leads to the Baker-Norine Riemann-Roch theorem, a discrete analogue of the Riemann-Roch theorem on Riemann surfaces. The dollar game naturally connects topics from combinatorics/graph theory, algebra, and geometry.

Mentor: Suraj Dash 

Abstract: In this talk, we will use the principles of point-set topology to describe exactly why a compact connected 1-manifold is homeomorphic to S^1. In our proof, we will use the definitions of compact and connected, the properties of open and closed sets, and topological principles to prove this idea.

Mentor: Jay Lee 

Abstract: In this talk, we explore some invariants of knots including the Alexander polynomial, knot determinant, and their connection to colorability of knots.

Mentor: Ioannis Karagiorgis 

Abstract: In this talk, we will define a tangent space of a surface in R^3. Next, we will demonstrate the usefulness of the first fundamental form for treating vectors in the tangent space.

Mentor: Viktoriia Ipatenkova 

Abstract: The essence of Gödel's first incompleteness theorem is surprisingly simple, especially since computer programs are so common today. By using math to talk about itself and creating a clever logical statement that refers to itself, Gödel revealed a shocking limitation which affects all reasonable mathematical systems. This result suggests that many real-world mathematical problems we would really like to solve may, in fact, actually be undecidable.

Mentor: Mark Saving 

Abstract: Overview of the counterexample to a famous knot theory conjecture (2025). And then some basics.

Mentor: William Winston