Speaker: Sang Hyun Kim (mentor: General Ozochiawaeze)
Title: An Introduction to Topological Data Analysis
Abstract: This talk introduces the foundational concepts of Topological Data Analysis (TDA) — a powerful toolkit for uncovering shape-based structures in high-dimensional and noisy data. Starting with the motivation for topological inference, the presentation explores key tools such as homology, Čech and Vietoris–Rips complexes, and Betti curves, culminating in the concept of persistent homology. Through illustrative visualizations and toy datasets, we demonstrate how topology can reveal features like clusters, loops, and voids that traditional methods might overlook. The talk also touches on the Nerve Theorem and Mapper algorithm, showing how these can summarize complex data into interpretable graphs, bridging qualitative insights and quantitative analysis.
Speaker: Shawanwit Poomsa-ad (mentor: Chrisil Ouseph)
Title: Nets: Convergence in General Topological Space
Abstract: Introduced by Moore and Smith in 1922, nets are a useful tool in general topological spaces. In metric spaces, many fundamental properties, such as convergence, continuity, closure, and compactness, can be fully described using sequences. However, beyond metric spaces, sequences may no longer be sufficient to capture these properties. Nets generalize the structure of sequences by allowing their domain to be an arbitrary directed set. By removing the restriction of the natural numbers, nets capture the familiar characteristics of metric spaces in more general topological space. This talk will highlight some examples of how sequences fall short and introduce nets as a powerful tool to amend these failures.
Speaker: Mukul Agarwal (mentor: Mustafa Nawaz)
Title: Dirichlet's Theorem on Primes in Arithmetic Progression
Abstract: The fact that there are infinitely many primes is one of the oldest and most fundamental results in Number Theory. This result extends to arithmetic sequence of the form a, a + q, a + 2q, ... where a and q are coprime. Dirichlet's Theorem states that any such sequence contains infinitely many primes. In this talk, we discuss tools from Analytic Number Theory including Dirichlet characters and L-functions, and sketch a proof of Dirichlet's Theorem.
Speakers: Ephram Chun and Xinyue Gu (mentor: Fabio Capovilla-Searle)
Title: Gauss-Bonnet Theorem
Abstract: What do curves, curvature, and counting triangles have in common? In this talk, we explore how the geometry of surfaces connects deeply to their underlying structure through triangulation and curvature. Starting with intuitive notions of curvature on curves and surfaces, we introduce the Euler number, a topological invariant that counts vertices, edges, and faces in a triangulation. We then connect this to Gaussian curvature via the Gauss-Bonnet Theorem, which ties geometry and topology together in a single elegant formula. A computational example will be presented, showing how to compute total curvature and Euler number for a triangulated surface, making the abstract ideas concrete. This talk requires only basic familiarity with calculus and linear algebra, and is aimed at sparking curiosity in differential geometry and topology.
Speaker: Drew Zlatniski (mentor: Kuan Ting Yeh)
Title: An Introduction to Distributions and the Fourier Transform
Abstract: Distribution theory is a powerful framework with wide applications in mathematics, physics, and engineering. It is deeply connected to the Fourier transform, a fundamental tool in the study of partial differential equations. In this talk, we will present examples that illustrate the core ideas of distribution theory and demonstrate how it can be combined with the Fourier transform to solve the Poisson equation.
Speaker: Priyank Behera (mentor: Zijin Liu)
Title: The Heat Kernel on Riemannian Manifolds
Abstract: The Heat Kernel is the fundamental solution to the heat equation, and it is quite simple to find it in Euclidean Space (RN). But trying to extend the Heat Kernel to a general manifold is a tricky task. In my presentation I will introduce the Riemannian Manifold, define the Laplace Operator on a Riemannian Manifold, and then define the Heat Kernel on a Manifold. Then, I will compute the closed form of the Heat Kernel on a circle.
Speakers: Samantha Goodwill, Yunye Liang (mentor: Estepan Ashkarian)
Title: Nash Equilibrium in Game Theory
Abstract: We will provide a short introduction to game theory, give some basic definition to commonly used terminologies, motivate the need for Nash's theorem through dominance of strategies, then discuss Nash's Theorem with examples of it in practice.
Speaker: Daanish Suhail (mentor: Shahbaz Khan)
Title: What is a Representation?
Abstract: A discussion over the basics of what a representation is. This talk will go over basic definitions in the context of finite representations, equivalence, and build up to Maschke's theorem.
Speakers: Sammith Belur, Havish Goutham-Prageeth (mentor: Nico Bridges)
Title: Can you untie everything?
Abstract: Legend has it that the Phrygians tied a knot so twisted that it became impossible to untie. An oracle declared that the person who untied the so-called Gordian Knot would rule all of Asia. Alexander the Great arrived to the city which held the knot and was told of the legend. The conqueror attempted to untie the impossible knot before realizing that the method of doing so did not matter. He proceeded to slice the knot with a flourish of his sword.
In this talk we will not slice knots, but use a mathematical sword to tell when the impossible knots need to be sliced in order to be untied.
Speakers: Ife-oluwaposimi Ogunbanjo, Dayoon Suh (mentor: Rivkah Moshe)
Title: Reinforcement Learning: Theory and Application Using Flappy Bird
Abstract: Reinforcement learning (RL) is a type of machine learning where an agent learns to make decisions by interacting with an environment, aiming to maximize its cumulative reward over time. It has many applications, including game development, graph theory, robotics, and financial modeling. In our presentation, we explore the theory and mathematics behind reinforcement learning, focusing on key concepts such as Markov Decision Processes, which provide a framework for modeling the agent’s environment, actions, rewards, and transitions, and Q-learning, which helps the agent determine which actions lead to better outcomes over time through trial and error.
For a practical approach, we trained an agent to play the game Flappy Bird using a Deep Q-Network (DQN) algorithm implemented in Python. We also utilized Gymnasium- a toolkit for developing and comparing RL algorithms. This hands-on implementation helped us better understand how RL algorithms learn and improve performance over time.
Speaker: Aiden Cullen (Kale Stahl)
Title: Comparing and Contrasting Riemann and Lebesgue Integrals
Abstract: We seek to prove a theorem regarding the equivalence of Riemann and Lebesgue Integrals. This Theorem, found in Folland's Book on Real Analysis, shows that all Riemann integrable functions are Lebesgue integrable, but not all Lebesgue integrable functions are Riemann integrable. To this end, we will explore the differences in the construction of the Riemann and Lebesgue Integrals and touch on important theorems such as Dominating Convergence and Monotone Convergence.
Title: A Flash of Algebraic Geometry, A Look at Localization
Speaker: Aaron Boes
Mentor: Bek Chase
Abstract: The objects of interest for Algebraic Geometry are affine varieties. The Nullstellensatz, one of the most fundamental theorems in the field, creates a correspondence between varieties and ideals, the analogous object in commutative algebra. This creates a strong connection between algebraic geometry and commutative algebra. Many problems in commutative algebra, and hence algebraic geometry, can be simplified using localization. Localization is the generalization of quotient fields to allow for non-integral domains. However, it has an additional interpretation in algebraic geometry, as it allows the representation of some non-affine varieties as projections from higher dimensional varieties.
Title: Proof of the Law of Quadratic Reciprocity
Speaker: Mary Wang
Mentor: Ben Doyle
Abstract: The Law of Quadratic Reciprocity is a fundamental theorem in number theory. In this presentation, we will briefly go over the necessary background and provide a proof of the theorem.
Title: An Introduction to the Analysis of Boolean Functions
Speaker: Edward Kelley
Mentor: Atal Bhargava
Abstract: In this presentation, we will discuss what are boolean functions, what is the fourier expansion of a boolean function, what is the expectation and variance of a boolean function, some foundational theorems about boolean functions, what is influence and total influence, and the application of boolean functions in social choice problems.
Title: The Infinitude of Primes of the Form 4k+1, via Fourier Analysis
Speaker: Jack Rookstool
Mentor: Nick Gismondi
Abstract: One of the crown jewels of elementary analytic number theory is the Dirichlet theorem on arithmetic progressions. This theorem states that, for q and l coprime, there are infinitely many primes in the form p = qk + l. We discuss the particular case where p = 4k + 1 which will utilize various topics in Fourier Analysis such as characters over the finite abelian group Z*(4) and the discrete Fourier transform and then briefly discuss how the method described can be modified to prove the general case.
Title: Brouwer's Fixed Point Theorem with a Proof from Algebraic Topology
Speaker: Daniel Armeanu
Mentor: Daniel Tolosa
Abstract: Brouwer's Fixed Point Theorem states every continuous function from the unit disk to the unit disk as a fixed point. This theorem is used in many areas of math including differential equations, differential geometry, and perhaps unexpected game theory where it's used to show every zero-sum game has an equilibrium strategy (a strategy in which no player benefits if they all adhere to the strategy). In this talk, I'll present an elegant proof of the theorem using tools from Algebraic Topology particularly homology groups.
Title: Fundamental Group and Relations
Speaker: Minseung Son
Mentor: Lvzhou Chen
Abstract: We introduce some basic notions of general topology and algebraic topology (connectedness, compactness, homeomorphism, homotopy, etc.) and well-known examples of fundamental groups of different spaces.
Title: Introduction to Category Theory
Speaker: Ricardo D'Avennia
Mentor: Yang Mo
Abstract: The purpose of the presentation is to introduce category theory. An overview is given of the relevance of the subject in mathematics, while rigorously defining its structure and elements. The presentation offers historical context for the creation of category theory and showcases an example of it in the field of algebraic topology through the description of homology theory. A general proof of the uniqueness of the identity morphism is shown to display how category theory connects different branches of mathematics. Finally, the presentation offers a brief description of current research on the subject.
Title: Exploring Graph Connectivity Through the Adjacency Matrix in Spectral Graph Theory
Speaker: Junhyeok Kil
Mentor: Anurag Sahay
Abstract: Spectral Graph Theory is the study of the properties of a graph by analyzing the eigenvalues and eigenvectors of matrices associated with the graph. This presentation explores the adjacency matrix of a graph, focusing on its significance and utility in understanding the connectedness of k-regular graphs. We first define basic graph terminologies and introduce the adjacency matrix. Next, we delve into the eigenvalues, particularly the largest eigenvalue of the adjacency matrix. Lastly, we will prove a theorem stating that for a k-regular graph G with connected components X1, X2, …, Xn, the largest eigenvalue of its adjacency matrix (λ=k) has a multiplicity equal to the number of its connected components. By examining these properties, the presentation highlights the powerful role of the adjacency matrix in spectral graph theory and the insights it provides.
Title: Algebraic Number Theory and Quadratic Reciprocity
Speaker: Sukrith Raman
Mentor: Nico Diaz-Wahl
Abstract: A classic problem in number theory is to find all numbers can be written in the form x^2+y^2. This question can be generalized to classifying numbers of the form x^2+ny^2 where the choice of n showcases the need for using different techniques.