Spring 2021 Symposium.
Note: Some rooms with many speakers have large video files that may take longer to load.
Speakers in order of appearance
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Luke Hoffman
- The Fundamental Theorem of Algebra
- Abstract:
- Giving a proof of the fundamental theorem of algebra using basic algebraic topology.
- Mentor: Hunter Vallejos
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Abigail Perryman
- An Introduction to Character Theory
- 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
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Steven Xu
- Representation Theory of Lie Groups
- 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
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John Teague
- Classifying Finite-Dimensional Representations of 𝔰𝔩₃ℂ
- 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
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Simon Xiang
- Bordism and TQFTs
- 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
Speakers in order of appearance
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Altanali Nagji
- Bass-Serre Theory and Schreier's Theorem
- 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
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Kai Bovik
- Braids in Mathematics
- 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
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Ruiqi Zou
- Dehn Twist Equivalence of Simple Closed Curves
- 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
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Ryan McWhorter
- Symplectic Geometry and the Uncertainty Principle
- 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
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Ethan Sollenberger
- Counting Faces with Shellability
- 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
Speakers in order of appearance
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Yitong Yao
- Binomial non-arbitrage pricing model
- 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
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John Nichols
- Black Scholes Model
- Abstract:
- The Black Scholes Model is a way to price options. It is inspired by Brownian motion.
- Mentor: Tristan Pace
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Yuxiang Gao
- Introduction to linear regression models
- 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
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William Yan
- Models in Stochastic Programming
- Abstract:
- Topics on convex optimization, news vendor model, and robust optimization.
- Mentor: Luhao Zhang
Speakers in order of appearance
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Juan Antonio Artero Calvo
- What are Tensors?
- Abstract:
- A basic introduction to tensors.
- Mentor: Michael Hott
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Omar Rodriguez Garcia
- Generating Sums of Powers
- 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
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Lohit Jagarapu
- Lebesgue Spaces and Inequalities
- Abstract:
- Introduction to Lebesgue spaces. Will explain fundamentals of measure theory and Lebesgue integration in the talk.
- Mentor: Ziheng Chen
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Wenting Lu
- Application of Optimal Transport in Goods Operation
- Abstract:
- This talk is about optimal transport, and how it applies in our real life.
- Mentor: Ken DeMason
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Adarsh Hullahalli
- Numerical solutions to PDE's
- Abstract:
- Numerical solutions of the Navier Stokes equation.
- Mentor: Yiran Hu
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Kaushal Patel
- Synchronized Chaos to Transmit Private Communications
- 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
Speakers in order of appearance
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Tien Vo
- Image Style Transfer
- 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
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Reese Feldmeier
- Random Sampling of the Hypersphere
- 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
Speakers in order of appearance
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Abhinav Rachakonda
- Complex Analysis and the Prime Number Theorem
- 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
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Amy Somers
- An Introduction to Affine Schemes
- 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
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Matthew Allen
- Constructing Points on Elliptic Curves
- 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
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Michael Panner
- Moduli Spaces and Geometric Invariant Theory
- 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