Spring 2024

Abstract: This presentation is an overview of the exploration of metric and norm spaces, delving into their common theoretical foundations and practical applications. Metric spaces, serving as the fundamental framework for measuring distances between points, are examined alongside norm spaces, which extend these concepts to vector spaces. The presentation highlights the significance of these mathematical structures in functional analysis and real-world examples.

Mentor: Justin Toyota 

Abstract: We will give an introduction to what measure theory is. We will then define what a measure is in the realm of the subject. It'll allow us to give an intuitive notion of length, area, and volume. Within this talk, we will talk about measure spaces and measurable sets. As well as, go into sigma algebras. At the end, we will briefly draw connections to the Lebesgue measure and how it's brought together for integration.

Mentor: Ali Mehzer 

Abstract: Will introduce Banach Spaces (and Duals of Banach Spaces) explain why they are cool and useful. Then introduce hilbert spaces and why they are useful, then jump into the proof of the Riesz Representation Theorem, explaining why it is super useful in the study of hilbert spaces.

Mentor: Antonio Farah 

Abstract: In this talk, we begin by playing a topological game which characterizes a Baire space. From here, I will use the Baire Category theorem to prove the Uniform Boundedness Principle. This principle is one out of three important results in Functional Analysis. It shows that if a family of continuous linear functions with the domain of a Banach space is pointwise bounded, then it is uniformly bounded. Lastly, I will discuss a fun application and answer the question: Do Fourier series on the circle converge at every point?

Mentor: Luisa Velasco 

Abstract: Functional analysis provides a powerful framework for the analysis of PDE's--especially with regard to formalizing the notion of weak solutions and allowing us to talk about solutions as a sum of eigenstates of an operator. Additionally, treating ODE's as operators allows us to prove some inequalities which would have otherwise been very lengthy direct proofs.

Mentor: Jeffrey Cheng 

Abstract: In this talk, I'll discuss the mathematical foundations of RSA, explain how it works, and briefly discuss some of the practicalities of its implementation.

Mentor: Jay Lee 

Abstract: Conceptual talk about Cryptographical systems, how keys/locks are generated in public domains. Overview of the Discrete Logarithm Problem and then the Elliptic Curve Discrete Log problem and its advantages along with the real-world example of Bitcoin's security. (Which uses a specific elliptic curve)

Mentor: John Michael Clark 

Abstract: Mathematical process of Shor's Algorithm

Mentor: William Winston 

Abstract: This talk will introduce categories and functors and discuss how homology is a functor and how this is useful in proofs.

Mentor: Amy Bradford 

Abstract: The theory of elliptic curves has many different applications and is a widely studied topic in number theory. In this talk we will explore a curious application of the theory of elliptic curves: namely, solving problems in Facebook memes. We will discuss Fermat's Last Theorem, the group structure on rational cubics, projective space, Weierstrass normal forms, elliptic curves, and the Nagell-Lutz theorem.

Mentor: Abhishek Shivkumar 

Abstract: For a non-cyclic quiver Q, the Krull-Schmidt theorem gives us that the decomposition of its representations is unique up to reordering of indecomposables. Furthermore, Gabriel’s theorem tells us that if Q is connected, then it has finite representation type if and only if it is of type ADE. In this talk, I will apply these theorems to decompose the representations of the quiver A_3.

Mentor: JiWoong Park 

Abstract: Imagine there's a country in which every town is connected directly by a path, but you can only go across every path in one specific direction, then there always exists a universal town, which is a town from which you can get to any other town in the country. Furthermore, there will be a town from which you can get to anywhere else by only passing through one other town. We will also prove that no matter the initial directions of the roads, you can always make every town a universal town by changing at most one direction.

Mentor: Ryan Wandsneider 

Abstract: In my presentation, I will include a basic introduction to algebraic structures (groups, rings, and fields), an explanation of finite field structures, some theorems concerning finite fields, and applications of finite fields within and outside of number theory.

Mentor: Aaron Benda 

Abstract: In this talk, we will explore elliptic curves, beginning with their definition as non-singular cubic curves and the addition law that provides a group structure. We'll then discuss their behavior over finite fields and introduce relative essential theorems. Finally, we will examine Elliptic Curve Cryptography, highlighting its advantages for security and efficiency in modern cryptography.

Mentor: Abhishek Koparde 

Abstract: After highlighting several useful properties of one variable polynomials, the talk will demonstrate the ways in which they do not directly extend to the multivariable case. Then, we will discuss how to bridge this gap by introducing Gröbner bases and the Buchberger algorithm. Furthermore, the presentation will showcase a set of educational interactive software tools that can be used to visualize various examples of these polynomials and algorithms.

Mentor: Laney Bowden 

Abstract: This talk introduces unsupervised clustering methods. We will learn about 3 types of clustering algorithms, how they work, and their advantages/drawbacks. This presentation will include lots of pictures and gifs comparing the 3 clustering methods. Come explore the strange world of high-dimensional data and why shapes matter in data science!

Mentor: Addie Duncan 

Abstract: Neural networks are a common structure for generating AI. In this talk I will introduce the concept of different types of neurons as well as introduce the gradient descent algorithm.

Mentor: Erin Bevilaqua 

Abstract: I will cover some basics on Markov chains and random walks on groups, then analyze mixing times in the setting of riffle shuffles. No need to have prior knowledge of group theory.

Mentor: Hunter Vallejos 

Abstract: In this talk I will explore the mathematical foundations of Probability Theory, Brownian Motion, and Stochastic Differential Equations. I will then show how these concepts can be applied in options pricing to derive the Black-Scholes equation.

Mentor: Cooper Faile 

Abstract: This talk will go through basics of knot theory such as Reidemeister moves and knot invariants. At the end, I will give context to how these concepts can be used to understand knots in DNA and their implications.

Mentor: Allie Embry 

Abstract: The presentation will focus on the works of Differential Topology by Guillemin and Pollack. The central point of the presentation is the definitions of an open ball in correlation to open and closed sets. Furthermore, the subject will build upon the mathematical concepts of a Smooth Map, Diffeomorphism, and most importantly Manifolds. An example will be presented towards the objective of Manifolds being locally diffeomorphic to R^n.

Mentor: Alberto San Miguel Malaney 

Abstract:

Mentor: Toby Aldape 

Abstract: The divergence theorem, Green's theorem, and Stokes's theorem are all major integration theorems seen in a vector calculus course. In this talk, we will use the language of differential forms to unify and generalize these theorems into the generalized Stokes's theorem.

Mentor: Jacob Gaiter

Abstract: Noether's Theorem is a statement about the connection between the symmetries of the system and its conserved quantities. This is almost never proved when brought up in a typical undergraduate physics class, and as it turns out the proof is very simple when cast under the lens of symplectic geometry.

Mentor: Riccardo Pedrotti 

Abstract: The braid representations of knots offer a simplified approach to knot construction by confining crossings to a compact portion of the braid, thereby providing a clearer understanding of the knot's structure. Central to our discussion is the concept of the word of the braid, a descriptor facilitating the analysis of the knot representation. We delve into finding the word of a simple braid, elucidating the process involved in determining this descriptor as well as relevant theorems relevant to the braid notation.

Mentor: Abhishek Shivkumar 

Abstract: analyze the uncertainty of where will you have lunch with the use of markov chain

Mentor: Jayden Wang 

Abstract: We will be describing a stochastic control problem involved an inverted pendulum. We will introduce necessary background material to understand a solution of the problem and then provide a solution.

Mentor: Ziheng Chen

Abstract: Concentration inequalities have their advantages of being nonasymptotic over CLT. In this talk, I will provide a brief introduction to some common concentration inequalities and demonstrate their interesting applications in probability puzzles, such as coin tossing and the coupon collector problem.

Mentor: Lewis Liu