Fall 2025

Abstract: String tricks are interesting to most elementary school kids and some mathematicians. Specifically, we can study them using knot theory. In this talk, I will be explaining the concept of ambient isotopy and Reidemeister moves and demonstrating how to recreate the Jacob's Ladder string trick. 

Mentor: Aru Mukherjea

Abstract: In this talk, I will start by defining knots and slice knots, then arf invariants of quadratic forms. Then I will give examples of assigning arf invariants to knots, and conclude with the Fox-Milnor theorem. These results, together with Murasugi's condition, give us that slice knots must have arf invariant 0. 

Mentor: Audrick Pyronneau

Abstract: In this talk, we introduce knots and knot invariants, beginning with the crossing number. We then launch into the Jones polynomial and show an example of the computation for a simple knot. Lastly, we touch on Khovanov's categorification of the Jones polynomial, completing a progression of increasingly precise invariants.

Mentor: Nathan Louie 

Abstract: I am going to be giving a brief introduction to knot theory and show a proof for twist knots having a genus of 1. 

Mentor: Hillary Kim

Abstract: I will introduce knots and links and discuss how certain concepts in knot theory are present in knitted textiles. 

Mentor: Elise Brod

Abstract: An exploration of topological spaces and their properties using techniques foundational to Algebraic Topology, focusing on cell complexes and touching upon homology, homotopy, fundamental groups, and Van Kampen's Theorem. Pictures and art demo included! 

Mentor: Adrian Flores

Abstract: In this talk, I explain why a CW complex of type K(G,1), that is, a space whose fundamental group is G and whose higher homotopy groups are all trivial is completely determined by the group G. The key idea is that any map between fundamental groups can be realized by an actual map between the spaces. Using this, we build a map cell-by-cell that induces the desired homomorphism on \pi_1, and because all higher homotopy groups vanish, there is no obstruction to extending the map. This construction shows that any two K(G,1) spaces with the same fundamental group are homotopy equivalent. I will illustrate the method with basic examples and explain why this uniqueness is special to K(G,1) spaces. 

Mentor: Peter Seong 

Abstract: Given the result that every Lie group has Euler Characteristic 0, this presentation will give a geometric argument why the Euler characteristic of the 2-sphere is 2. The key idea is to compute the self-interection of the diagonal of the 2-sphere's product space. A perturbation of the diagonal can be seen as a linear map on the 2-sphere which we claim will fix an axis that intersects the 2-sphere at two points. Both points contribute positively to the intersection number, giving the desired result.

Mentor: John M Teague

Abstract: A surface is a 2-dimensional space that can be classified according to some intrinsic (topological) properties, such as the number of “holes” it contains, which are invariant under continuous transformations. Some surfaces can be cut apart and glued together in specific ways to construct a new, topologically distinct surface. Defining a geometry on a surface gives it additional structure and further interesting properties. Any surface can be equipped with exactly one of three types of geometry: spherical/elliptic, Euclidean, or hyperbolic. Conveniently, the type of geometry that a surface admits is uniquely determined by its classification and topology. Not every type occurs as frequently as the others; in particular, we find that hyperbolic geometry is the most common across all surfaces. 

Mentor: Aaron Benda

Abstract: Intuitively, we like to think of manifolds as squishy objects that we can deform as we like. Technically, squishing a manifold is a homotopy equivalence, and we would like to have a homeomorphism, where possible. The Poincaré Conjecture allows us to do this, whenever a manifold is homotopy equivalent to an n-sphere. 

Mentor: Ansel Goh

Abstract: We will explore the proof of the Brouwer fixed-point theorem using tools such as homotopy, loops, and the fundamental group. Our main proof will focus on the topology of the closed disk D^2 and its boundary S^1, and how there cannot exist a continuous retraction from D^2 onto S^1. Using algebraic properties, we will show how the topology of D^2 and its boundary forces the existence of a fixed point, concluding the proof for the Brouwer fixed-point theorem. 

Mentor: Remy Bohm

Abstract: We will distinguish the fundamental groups of the 2-sphere, the n-fold torus, and the m-fold projective plane, then prove the Classification Theorem for Surfaces, which categorizes polygonal regions with paired pastings of edges as homeomorphic to one of the aforementioned surfaces. 

Mentor: William Winston

Abstract: In this talk, I will use the sphere as an introduction to Morse Theory, a way for us to use smooth functions to analyze the structure of surfaces. Using the Morse Lemma, we will show that any surface with two stable critical points is diffeomorphic (can be smoothly mapped one-to-one) to the 2-sphere: the ball shape we see in objects like oranges and volleyballs

Mentor: Jemma Schroder

Abstract: The hyperbolic plane can be represented as a disk in R2 called the Poincaré disk. A genus-g surface can be represented using a regular 4g-gon with paired sides inside that disk. Each side pairing defines a disk-preserving Möbius transformation or generator. Repeatedly applying these generators moves copies of the polygon around that disk without overlap, eventually covering the entire Poincaré disk. These transformations are the deck transformations of the covering map from the disk to the genus-g surface formed by gluing the polygon’s paired sides. This talk constructs those generators, explains how their repeated action tiles the disk, and considers applications to genus-g surfaces. 

Mentor: William Ghanem 

Abstract: In this talk, I will explain tiling spaces, their substitutions, Collarings, Gahler & Anderson Putnam Constructions for Tiling Spaces, and cohomology of tiling spaces 

Mentor: Jae Hwan Lee 

Abstract: I will be deriving the explicit formula to find specific terms in a sequence using a generating function/Ordinary Power Series.

Mentor: Matt Verheul 

Abstract: Using a part of the explicit formula for the Riemann zeta function on the integers, we interpolate the zeta function continuously over the p-adic integers. Along the way, we develop some theory regarding distributions, measures, and integration on the p-adics. 

Mentor: Sudharshan K V

Abstract: This talk follows the evolution of cryptography from shifts ciphers to modern, mathematically based cryptosystems. After highlighting the weaknesses of historical methods, we introduce the key mathematical ideas: modular arithmetic, primality testing, and factorization, that form the foundation of secure encryption today. We conclude with an overview of RSA, showing how public-key cryptography relies on hard computational problems to protect information. 

Mentor: Ethan Katz

Abstract: Quadratic reciprocity is one of the most celebrated results in elementary number theory, revealing a surprising symmetry in the solvability of quadratic equations modulo prime numbers. In this talk, we will explore the statement and intuition behind the Law of Quadratic Reciprocity, examine its historical significance, and walk through key supporting theorems such as Euler’s Criterion and the properties of the Legendre symbol. Through carefully chosen examples, we aim to illuminate the patterns that govern quadratic residues and non-residues, providing a deeper understanding of the interplay between prime numbers. This presentation is designed to be accessible to undergraduate students while highlighting the elegance and power of one of Gauss’s favorite discoveries. 

Mentor: Aryan Amin

Abstract: In this talk, I introduce the foundations of discrete-time Markov chains and use a simple four-state example to build intuition for how a system evolves step by step under uncertainty. The model serves as a motivating framework for defining stochastic processes, the Markov property, one-step transition probabilities, and the transition matrix. Using this structure, I show how state-distribution vectors and repeated matrix multiplication allow us to compute general n-step transition probabilities. The goal of the talk is to present an accessible and intuitive introduction to Markov chains while demonstrating how these tools help us answer meaningful probabilistic questions. 

Mentor: Jordan Gardiner

Abstract: Matrix multiplication is fundamental to scientific computing, data analysis, and machine learning. However, these fields often require many multiplications of large matrices, which are very expensive to compute. This talk introduces a randomized linear algebra approach that approximates large matrix products by sampling a small number of rank-1 outer products. It further explores how an appropriate choice of sampling probabilities can significantly reduce the variance and computational cost. 

Mentor: Justin Toyota

Abstract: This talk will introduce Markov chains, exploring large-time behavior, different states, and countable Markov chains. 

Mentor: Dillon Grannis

Abstract: Message passing is a simple but powerful framework for computing in probabilistic graphical models. In this talk, I will show how graphs naturally encode variable dependencies, and how this structure leads to message passing algorithms such as belief propagation for efficiently computing marginal distributions. 

Mentor: Addie Duncan

Abstract: This talk gives an overview on how we can summarize information within data and use it to create good estimators of its parameters. We introduce how concepts like sufficiency and completeness can lead us to finding the best unbiased estimators, as well as a lower bound for these estimators and show when the bound can be achieved in practice.

Mentor: Justin Le

Abstract: The seemingly innocent problem of domino tilings of a Chessboard yield surprising connections to the fields of graph theory and linear algebra. In fact, despite having an integer solution, the closed form equation can be written in terms of the products of squares of cosines! In this talk, I will show how we can reformulate this question into the language of perfect matchings of graphs, ultimately solving the problem through the use of the weighted 'Kasteleyn' matrix which utilizes imaginary units for some of its weights! 

Mentor: Abhishek Shivkumar

Abstract: This talk will introduce the Axiom of Choice as well as when it should be used (in particular, when it should NOT be used). We will briefly discuss theorems derived from the Axiom of Choice and some of its weird consequences. 

Mentor: Mark Saving

Abstract: Introduce the definition of universal property and use it to define product space, quotient space, tensor product categorically. 

Mentor: Toby Aldape

Abstract: This talk provides a gentle introduction to the foundations of algebraic geometry through the lens of affine space and polynomial equations. We will begin by defining affine space and exploring the structure of polynomial functions on it, emphasizing how these algebraic objects naturally give rise to geometric ones. From there, we will examine vanishing sets of polynomials and highlight some of their key properties. Along the way, we will discuss fundamental correspondences, such as how algebraic relations among polynomials reflect geometric features of their zero sets, and preview the broader framework these ideas enable. The talk is designed to be accessible to newcomers while offering insight into why algebraic geometry is both rich and compelling.

Mentor: Ryan Wandsnider

Abstract: Tropical geometry is a new, combinatorial approach to geometry. We will demonstrate how studying a complex polynomial through a limiting and logarithmic process causes its resulting geometric shape, called the amoeba, to shrink until only its piecewise linear skeleton, the tropical hypersurface, remains. 

Mentor: Zimao Tian

Abstract: Tropical geometry is a recent area of mathematics that turns complicated algebraic curves into simple piecewise-linear shapes made of straight edges and vertices. This makes it possible to study classical geometric ideas through combinatorial pictures. In this talk, we introduce Bézout’s Theorem(one of the central results of algebraic geometry) and explore how it appears in the tropical world. A background in combinatorics or algebraic geometry is not required for this talk. 

Mentor: Amy Bradford

Abstract: For a long time, varieties were the central objects of study in classical algebraic geometry. In EGA, Grothendieck revolutionized the field by defining schemes, which have since become the foundation of modern algebraic geometry. 

In this talk, I will begin with a brief introduction to varieties, followed by definitions and examples of schemes. Finally, I will present a fully faithful functor from varieties over a field k to schemes over a field k. This functor gives us a rigorous way to view schemes as natural extensions of varieties. 

Mentor: Wang Yao

Abstract: This talk provides a structured introduction to the long-run behavior of Markov chains. We begin by reviewing the Strong Markov Property and using it to develop fundamental classifications such as recurrence, transience, and positive recurrence. We then introduce invariant distributions and discuss how they characterize equilibrium behavior. Finally, we present the coupling method and show how it provides an elegant and powerful tool for proving convergence of Markov chains to their equilibrium distribution. 

Mentor: Flavio Argentieri

Abstract: In this talk, I will provide an in depth look at martingales and use conditional expectation to show some key theorems about them. 

Mentor: Jake Wellington

Abstract: This talk will be a discussion of Runge Kutta methods and their application in the approximation of ordinary differential equations, along with a derivation of an important result pertaining to the class of Runge Kutta methods that respect ODEs endowed with quadratic invariants. 

Mentor: Christian Liu

Abstract: For many decades PDE’s (Partial Differential Equations) have plagued modern physics and mathematics with their computational complexity and non analytic, or outright non existent, solutions. In fact, the desire to find more concrete and efficient ways of simulating and identifying potential solutions to these expressions continues to inspire scientific inquiry. One such, highly practical, result of this scientific inquiry is an infinite dimensional inner product space known as Hilbert space. Hilbert spaces in particular are special because the metric imposed by the space requires that every element in the space converges to another element also in the space. Meaning that if an expression meets the requirements to exist in a Hilbert space then there must exist some solution to the expression. This is significant because it allows mathematicians and physicists to prove whether or not something is “possible” without having to use analytic/brute force methods that can take days to years to compute depending on the complexity of the expression, which is why these spaces are incredibly useful when analyzing partial differential equations. 

Mentor: Andy Hale 

Abstract: We will begin with an overview of Schrodinger's equation and Strichartz estimates. When an external potential is present, classical solutions cannot in general be expected, which naturally leads to the construction of mild solutions in Strichartz spaces. We will then study the properties of such solutions. 

Mentor: Zach Lee

Abstract: One of the most basic examples of a group is the dihedral group, which represents the symmetries of a polygon. In this talk, we will explore these symmetries in the context of reflection groups and also introduce the notion of a root system. 

Mentor: Sai Sivakumar 

Abstract: The talk will cover the concepts and insights that allow one to think of a group as a geometric object. More formally, a group as a metric space well-defined up to quasi-isometry. The talk will end with the statement of the Švarc-Milnor Lemma. 

Mentor: Justin Chen

Abstract: This talk will explore some basics of random dynamical systems, define attractors and how attractors translate when the systems become probabilistic. 

Mentor: Mark Abate

Abstract: This talk will explore the geometric properties of parametrized curves and develop tools for measuring length and area, culminating in a formula for enclosed area derived from Green’s theorem. Using this framework, we present a proof of the isoperimetric inequality and identify why the circle uniquely achieves equality. 

Mentor: Iris Jiang