Spring 2022

Abstract: Giving a proof of the fundamental theorem of algebra using basic algebraic topology.

Mentor: Hunter Vallejos 

Abstract: I will talk about some introductory group theory to explain what LaGrange theorem and a proof. Applications of LaGrange theorem.

Mentor: Maksym Chaudkhari

(not recorded)

Abstract: In the theory of representations of finite groups, characters are an essential tool that greatly simplify the process of finding the irreducible representations that are part of a representation’s unique decomposition. In particular, I will be applying this to the symmetric group of degree four.

Mentor: William Stewart 

Abstract: I'll be covering what representation theory is and what are lie groups. Then I'll talk about the classification of the representations if SL_2 C.

Mentor: Tom Gannon 

Abstract: In this talk, I will introduce and motivate the rich representation theory of semi-simple Lie algebras. We will briefly discuss representations of 𝔰𝔩₂ℂ before exploring one of the main (and generalizable) examples of this theory, the classification of finite-dimensional representations of 𝔰𝔩₃ℂ.

Mentor: Alberto San Miguel Malaney 

Abstract: This talk will give a brief introduction to bordism and topological quantum field theories. I assume familiarity with manifolds and category theory. No physics required!

Mentor: William Stewart 

Abstract: Will cover the basics of simplicial structures and CW complexes. Will discuss the basics of the triangulation conjecture, and the homeomorphic/homotopic invariance of smooth manifolds of dimension <= 3. I will explain the calculation of homology groups and Betti numbers of simplicial complexes/piecewise linear manifolds.

Mentor: Allie Embry

(not recorded)

Abstract: In this talk, we present a brief introduction to Bass-Serre theory and free actions on trees. We go over the theorem equating a group acting freely on a tree and the group being a free group itself. We apply this theorem to prove Schreier's Theorem, which demonstrates that any subgroup of a free group is free.

Mentor: Kyrylo Muliarchyk 

Abstract: This talk will introduce group theory to the audience using braids and braid groups. It is designed for anyone to understand, and will present the importance and ubiquity of groups and group theory.

Mentor: Aru Mukherjea 

Abstract: Introduction of Dehn Twist, Twist Equivalence, and proving the two simple closed curves are twist equivalent if they intersect at 1 point.

Mentor: Neza Zager 

Abstract: We discuss the connection between symplectic geometry and the principle from physics, recovering a limited form of it in a classical setting

Mentor: Riccardo Pedrotti 

Abstract: I will present the Dehn-Sommerville equations: a generalization of the Euler-Poincare relations derived from orderings of faces. To do this, I will describe the foundations of polytope geometry, defining polytopes, faces, and complexes. Finally, I will define shellability and the h-vector, from which the Dehn-Sommerville equations will follow.

Mentor: Jayden Wang 

Abstract: I’m going to talk about how to find the price of option at the beginning time of a non-arbitrage binomial model. I will also explain why any price at any subsequent time in this path dependent model can be located and computed.

Mentor: Joseph S Jackson 

Abstract: The Black Scholes Model is a way to price options. It is inspired by Brownian motion.

Mentor: Tristan Pace 

Abstract: In the talk, I will go through linear regression, subset selection (back/forward selection) and a few about shrinkage methods. I will introduce definitions first and use R to illustrate these ideas.

Mentor: Luhao Zhang 

Abstract: A priori estimates of elliptic partial differential equations. Interior and boundary regularity following the Schauder estimates.

Mentor: Jincheng Yang 

Abstract: A basic introduction to tensors.

Mentor: Michael Hott 

Abstract: Introduces generating functions and their application to sums of powers. The talk consists of methods of solving the sums of n to the a for some constant a, and it ends with a talk about finding the sum of n^a in terms of n and how that is useful to finding n^a in terms of k where k is the upper bound of the sum.

Mentor: Charlie Reid 

Abstract: Introduction to Lebesgue spaces. Will explain fundamentals of measure theory and Lebesgue integration in the talk.

Mentor: Ziheng Chen 

Abstract: This talk is about optimal transport, and how it applies in our real life.

Mentor: Ken DeMason 

Abstract: Numerical solutions of the Navier Stokes equation.

Mentor: Yiran Hu 

Abstract: In this talk I will introduce the fundamental ideas of chaos theory and the concept of synchronized chaos. I will then outline how one can utilize the Lorenz system and synchronized chaos to transmit private communications.

Mentor: Cooper Faile

Abstract: I will go over the basics of a neural network. Then I will explain the process of image style transfer as well as the specific of the VGG19 model.

Mentor: Ziheng Chen 

Abstract: Hyperspheres serve as valuable building blocks for a variety of data science applications. We will discuss the random sampling of hyperspheres and some interesting properties that emerge from their study.

Mentor: Trieste Desautels 

Abstract: This semester I had probability class, then I found the topic Bayesian Networks outside of the class very interesting. Then I decided make an introductory presentation about Bayesian Networks with some show cases and basic rules.

Mentor: Colin Walker

(not recorded)

Abstract: Reinforcement learning (RL) is a machine learning area in which an agent adjusts its decision making process based on the results of its choices. In the first half of this presentation, I will explain the mathematics used to describe this process, and in the latter half, I will explain some of the most common algorithms and some applications of this area.

Mentor: Harrison Waldon

(not recorded)

Abstract: I'll introduce the basic definitions of cryptography and introduce the Discrete Logarithms and Diffie-Hellman Key Exchange. Then talk about the Elgamal Public Key Cryptosystem.

Mentor: Hunter Vallejos

(not recorded)

Abstract: I will discuss calculus in the complex plane. I will introduce residue theory as a way to compute integrals of meromorphic functions. I will then apply these concepts and use the Reimann-Zeta function to prove the prime number theory.

Mentor: Aaron Benda 

Abstract: A basic introduction to affine schemes and some ideas from algebraic geometry. We will look at affine schemes by defining the spectrum of a ring as a set, equipping this set with the Zariski topology, and discussing the idea of the structure sheaf.

Mentor: Yixian Wu 

Abstract: First we will give a brief introduction to elliptic curves, their group law, and isomorphisms of elliptic curves. Then we will explore several methods of constructing new points on elliptic curves using base change, the j-invariant, and an algorithm using splitting fields.

Mentor: Tynan Ochse 

Abstract: The talk will introduce the concept of moduli spaces in algebraic geometry as motivation for geometric invariant theory. The notion of group actions and quotients needed to achieve this will then be introduced. Lastly, the case of the affine GIT quotient will be introduced.

Mentor: Suraj Dash