Fall 2024

Abstract: Measure theory is an area of analysis that provides a robust way to give subsets an area, or a measure. But what is a measure? What are measures used for, and why are they important? In this talk, we will discuss the motivations behind measure theory, the properties of measures, how to construct them, and how they are used in other areas of mathematics and in real life.

Mentor: Zach Richey 

Abstract: In this talk, I will discuss the Fourier Transform and its application in solving partial differential equations. Then, as an explicit example of using the Fourier Transform, we will solve the Schrödinger Equation.

Mentor: Luisa Velasco 

Abstract: The Laplace equation is one of the most important examples of partial differential equations. In this talk, I will explore many interesting properties of harmonic functions -- functions that satisfy the Laplace equation.

Mentor: Zachary Lee 

Abstract: The wave equation stands as a fundamental equation in mathematical physics, modeling phenomena such as sound, light, and vibrating strings. This talk delves into the pivotal role of Fourier Analysis in interpreting and solving the wave equation, focusing on the framework provided by Fourier Series. We will explore the propagation of waves and key results, including D'Alembert's solution, highlighting the interplay between mathematics and physical intuition. The presentation will conclude with an introduction to higher-dimensional generalizations and further insights into the Fourier Transform. This talk is tailored for those with a solid foundation in calculus and curiosity about analysis or mathematical physics.

Mentor: Justin Toyota 

Abstract: I will be discussing Erdös and Rényi's paper that introduces a basic model of random graphs, treating graph evolution as a stochastic process. They analyze the structural properties of these graphs as they evolve, identifying distinct phases characterized by specific substructures. The authors also often refer to Poisson distributions to describe the occurrence of various graph components throughout the evolution process.

Mentor: Suraj Dash 

Abstract: Generating functions are an important tool in proving combinatorial identities, finding closed forms for recurrence relations, and more. We give several basic examples of generating functions that illustrate their use. No experience with combinatorics is needed.

Mentor: Addie Duncan 

Abstract: Young tableaux serve as a vital combinatorial framework connecting representation theory and combinatorics, particularly in the study of symmetric and general linear groups. This talk will explore how Young tableaux facilitate the construction and classification of irreducible representations by translating algebraic problems into combinatorial ones. We will examine the correspondence between partitions, Young diagrams, and the modules they represent, highlighting how these tools simplify the understanding of complex representation structures. Through illustrative examples, the presentation will demonstrate the elegance and effectiveness of using Young tableaux to bridge the gap between abstract algebraic concepts and tangible combinatorial methods.

Mentor: JiWoong Park 

Abstract: Fermat's Two square theorem states that every prime congruent to 1 modulo 4 can be expressed as a sum of two squares. The original proof consists of a multiplicativity argument and a descent argument. In this talk we present a different proof using certain combinatorial objects called windmills.

Mentor: Abhishek Koparde 

Abstract: I will show how we can derive an inequality on the numbers of trees within a graph by computing volumes of certain convex bodies.

Mentor: Jayden Wang 

Abstract: In this talk, I will discuss the basics of group theory. I will define some basic terms including group, subgroup, homomorphism, isomorphism, coset, normal subgroup, and give some interesting examples of each.

Mentor: Ryan Wandsnider 

Abstract: This talk will explore the fascinating world of continuous but nowhere differentiable functions through the lens of analysis and its surprising connections to randomness and machine learning. We will begin by revisiting the foundational concepts of continuity and differentiability before introducing the Weierstrass function—a pathological function that is continuous everywhere but differentiable nowhere. From there, we'll pivot to real-world applications, including Brownian motion in stochastic processes and the surprising role of non-smoothness in modern machine learning.

Mentor: Sara Ansari 

Abstract: This talk will present an introduction to group theory, as well as the presentation of groups, and their metric spaces. We will also use this to introduce the concept of growth functions for infinite groups.

Mentor: Aru Mukherjea 

Abstract: An introduction to representation theory focusing on representations of finite groups.

Mentor: Ryan Wandsnider 

Abstract: Burnside’s Lemma offers a powerful method for counting distinct objects under symmetry, treating objects related by rotation or reflection as equivalent. The proof leverages concepts from group theory, including orbits and stabilizers, to establish the counting formula. This talk concludes with an example illustrating the application of the lemma.

Mentor: Sudharshen K V 

Abstract: The Thompson group is a finitely presented group which is still large in the sense that it is torsion-free, has subgroups isomorphic to all n-dimensional integer lattices, and notably contains countably many copies of itself. We introduce an algebraic definition of the Thompson group using orientation-preserving piecewise-linear homeomorphisms on [0,1] to show how finitely presented groups can contain unexpected properties given their size construction.

Mentor: Erin Bevilacqua 

Abstract: I will introduce the concept of a representation, explaining how a group can act on a vector space, which gives us insight into the structure of the group. sl2(C) is an important Lie algebra with interesting representations. In this talk, I will explore how sl2(C) acts on a vector space, focusing on the roles of eigenvalues and eigenspaces.

Mentor: John Michael Clark 

Abstract: A discussion of David Hilbert's Nullstellensatz (Theorem of Zeros), which establishes a connection between algebra and geometry. In this talk, we provide a relevant background in ring theory that leads into a discussion of the proof of the Theorem of Zeros and its implications.

Mentor: Daniel Koizumi

Abstract: Hilbert schemes are an important construction in algebraic geometry, and a classical example of a moduli space, defined as a smooth parametrization of subschemes. In this talk, I first explore the construction and definition of hilbert schemes. Then, I consider the special cases of the hilbert scheme of a plane and for projective curves, using the combinatorial young's tableaux to give an explicit description.

Mentor: Suraj Dash 

Abstract: Discussion about an interesting function that yields a surprising identity relating to prime factorization. We will draw a connection to the radical/power-free component of an integer. Finally, we show that this function has a finite number of cycles of certain lengths.

Mentor: Abhishek Shivkumar 

Abstract: Quadratic forms with discriminants of -4n can be used along with quadratic reciprocity and genus theory in order to find congruences such that primes are of the form x^2+ny^2.

Mentor: Zimao Tian

Abstract: This talk will cover the main ideas and terms that can help clarify in the statement of the monotone convergence theorem for sequences. After defining convergent sequences, boundedness, and monotone, the talk will explain and prove the monotone convergence theorem. There will be an additional portion that covers a numerical example that represents the nature of the monotone convergence theorem, putting more emphasis on what that monotone convergence theorem means.

Mentor: Martha Hartt 

Abstract: This paper explores the intersection of computer-assisted proofs and functional programming, with a particular focus on the Lean Theorem Prover. Lean, a proof assistant, is used to build formal mathematical proofs interactively and efficiently. The paper introduces the concept of proof assistants, highlighting their role in formalizing mathematical reasoning, and discusses the functionality of Lean, an open-source proof assistant. Through an exploration of the Natural Number Game, the paper illustrates how Lean can be applied to construct foundational proofs, such as demonstrating the properties of natural numbers using tactics like induction and commutativity. By examining specific examples, the paper emphasizes how Lean allows for both complicated and succinct proof strategies. This study illustrates how proof assistants, powered by functional programming, can enhance both the learning and practice of mathematics, making complex proofs more manageable.

Mentor: Elise Brod 

Abstract: Proof assistants use computer science to help write formal proofs. The talk will introduce a proof assistant, Coq, and examine an internal tactic of proof by simplification to compare the proofs generated by Coq and the proof we usually write. There will be examples showing when the proof assistant could be more intelligent than us, when the opposite, and when they share the same problems.

Mentor: Elise Brod 

Abstract: The talk mainly covers concepts of bifurcation, chaos, and strange attractors, examining concepts under different examples.

Mentor: Aaron Benda 

Abstract: Introduce Brownian motion, simple process, and the definition of Ito's Integral.

Mentor: Mark Abate 

Abstract: This talk briefly introduces some optimize algorithms, including traditional gradient descent method, conjugate gradient method, and Polyek momentum. I will focus mainly on the convergence analysis of those methods. If time applies, I will give a brief introduction to the Nesterov momentum.

Mentor: Lewis Liu 

Abstract: Using numerical analysis for solving second derivatives

Mentor: Jeffrey Cheng

Abstract: Ensuring accurate model choice and fit is imperative to extracting meaningful statistical conclusions from data. In particular, when we have limited data sets, other model fitting methods, such as those based on machine learning approaches may fail due to insufficient training data. Consequently, traditional maximum likelihood approaches for determining model fit are discussed, in the context of understanding how viral particles, which we model as random particles, spread at various concentrations.

Mentor: Alexandra Embry 

Abstract: Introduces the basics of neural networks, including the basic design, the mathematical intuition behind it, and implementation in Python. Includes an application in terms of a digit recognition program using neural networks.

Mentor: William Winston 

Abstract: In this talk, we will investigate the topology of surfaces using the analogy of flatlanders and tic-tac-toe. We will see how this applies to higher dimensional tori.

Mentor: Remy Bohm 

Abstract: My talk will focus on an introduction to knot theory and the Seifert Algorithm to prove that there are infinitely many distinct knots.

Mentor: Ioannis Karagiorgis 

Abstract: This talk will focus on Topological Data Analysis (TDA) using algebraic topology and the Nerve Theorem. We will first set up the basics of sets, topology, and algebraic topology before defining concepts like simplices, geometric simplicial complexes, good open covers, and nerves. We'll cover the Nerve Theorem and then dive into Mapper, a computational method utilizing algebraic topology to explain datasets. This topological view of data analysis provides a unique, fresh look at otherwise tricky datasets.

Mentor: Nathan Louie 

Abstract: Topics include definition of smooth map, tangent space, immersion, submersion, proper map, regular value, Sard's theorem, and weak Whitney immersion theorem.

Mentor: Luis Torres 

Abstract: We will start by introducing manifolds and general notions of geometry. After which we will define Riemannian and (G,X) structures on manifolds, before building up to a proof to under what conditions these two structures are compatible.

Mentor: Aaron Benda 

Abstract: Braids consist of a set of n strands that are arranged vertically, and each strand starts and ends in parallel, following specific over-and-under crossing patterns. We will discuss how braids have a group structure and areas that braid groups appear.

Mentor: Aru Mukherjea 

Abstract: This talk dives into the basics of knot theory, starting with the unknot and how it differs from more complex, composite knots. It also looks at tricolorability, a simple way to tell knots apart. The idea of the unknotting number is explored, showing how many steps it takes to untangle a knot into a simple loop. Finally, it touches on surfaces without edges and the Euler characteristic.

Mentor: Jemma Schroder 

Abstract: Using Seifert van Kampen theorem to compute the fundamental group of genus n surfaces

Mentor: Toby Aldape 

Abstract: This presentation will introduce the audience to the basics of working with and constructing surfaces, or 2-dimension manifolds. We will explore what objects can be made by gluing the edges of polygons together, which will lead into a discussion of the cell structures which underlie the construction. Then, we will introduce the Euler Characteristic to show an interesting result to provide an intuitive proof for a theorem pivotal to the classification of surfaces.

Mentor: John Teague

Abstract: In this talk, we will first define what it means for two knots to be equivalent using Reidemeister moves. Then, we will briefly describe the intuition for a knot invariant. Finally, we will calculate the Jones Polynomial for an example knot using two methods: the recursive bracket polynomial with the writhe, and the skein relation for the bracket polynomial.

Mentor: Abhishek Shivkumar