Graduate Student Presentations

M.S. Talk

November 19, 2024

12:15pm

Daniels K-12 Room

Title: The Classification of 4-Manifolds is Computationally Undecidable

Abstract: A long-standing goal in topology has been to classify all manifolds of a given dimension up to homeomorphism. This has led to historically significant results such as Classification of Closed Connected Surfaces and the Prime Decomposition Theorem for 3-Manifolds. A computational variant of this problem is known as the Homeomorphism Problem for n-manifolds, which asks whether an algorithm could determine if two manifolds are homeomorphic. We will give an overview of the homeomorphism problem for manifolds in each dimension and look at some of the topological and algebraic details surrounding the problem. Lastly, we will give a proof of Markov’s Theorem for 4-manifolds, showing that the Homeomorphism Problem for 4-manifolds is computationally undecidable.

M.S. Talk

November 19, 2024

12:45pm

Daniels K-12

Title: RSA vs Elliptic Curve Cryptography

Abstract: This talk compares RSA and Elliptic Curve Cryptography, two of the most commonly used asymmetric encryption systems. The primary topic is the underlying algebra of these encryption schemes: while both rely somewhat on the same "trapdoor" function of discrete logarithms, which are easy to calculate, but hard to reverse, there are some key differences that lead to elliptic curve cryptography being the preferred system today. Additionally, we will discuss some practical concerns when implementing these systems.

M.S. Talk

April 26, 2024

12:15-1:15pm

ENG 247

M.S. Talk

May 2, 2024

12:30pm

OSB B138

TITLE: Adaptive Spectral Numerical Method for Solving PDEs via Conformal Maps

ABSTRACT: PDEs can be spontaneous and difficult to solve, especially when they involve poles or branch cuts. In this talk, we introduce an adaptive spectral numerical method that is intended to solve such PDEs. We also analyze the use of conformal maps and their effect on reducing error around error-prevalent regions. Abstract

M.S. Talk

May 2nd, 2024

1:00pm

OSB B138

River VanIwaarden

M.S. Talk

November 30, 2023

12:30pm

OSB B215

TITLE: "Evolutionary Algorithms and their Applications"

Abstract

M.S. Talk

April 27, 2023

12:30-1:00pm

OSB B136

TITLE: “Equivalent Codes From Finite Fields”

ABSTRACT: Equivalency of codes, in regards to, encryption becomes a different question when we remove ourselves from the typical alphabet. Historically, people would take messages and use maps on the alphabet to translate the messages into equivalent statements that look different on the surface. In the context of fields, this is where MacWilliams did her work and showed that, under certain circumstances, we can still show certain codes are equivalent even if they look different. We will explore her work and show with examples how this can be done, and then look to what lies beyond MacWilliams' initial work.

M.S. Talk

Nov. 29, 2022

12:30-1:00pm

OSB B136

M.S. Talk

Nov. 29, 2022

1:00-1:30pm

OSB B136

M.S. Talk

Dec. 1, 2022

12:30-1:00pm

OSB B134

TITLE: Numerical Approximation of a Helmholtz Equation on a Generalized Domain Using a Meshfree Method

ABSTRACT:

The Helmholtz equation is the eigenvalue problem associated with the Laplace operator.  We often encounter this equation in the study of natural phenomenon such as wave propagation.  To numerically approximate the eigenvalues of a Helmholtz equation we may use a meshfree method.  An advantage of meshfree methods is that we may define our problem on a domain with complex geometry.  In particular, we can apply a meshless method to approximate the solution of a Helmholtz equation on a two-dimensional generalized domain.

M.S. Talk

Dec. 1, 2022

1:00-1:30pm

OSB B134

TITLE: Neural Networks and Universal Approximation: A Fundamental Result

ABSTRACT:

Artificial neural networks have been gaining in popularity over the past few decades due to their ability to solve problems arising in various areas of applied mathematics and engineering. More recently a new application in particular interest of the scientific computing community has materialized, and that is the ability for neural networks to solve partial differential equations. In general, finite element methods (FEMs) are used to solve PDEs in practice, but these methods have some limitations due to the required mesh size. Neural networks offer a promising alternative to FEMs due to their ability to bypass some of these limitations. However, the ability of neural networks to solve partial differential equations hinges upon the fundamental result that neural networks are universal approximators. That is, given any arbitrary function, there exists a neural network that approximates this function to any desired degree of accuracy. In this talk, we will rigorously prove this result.

M.S. Talk

Dec. 1, 2022

4:45-5:15pm

OSB B134

TITLE: An Introduction to Homology

ABSTRACT:

In this talk, I will motivate the study of homology and introduce different homology theories. Along the way, I will provide examples to illustrate concepts and demonstrate basic calculations. I will conclude the talk by mentioning a few big theorems that can be proven using homology, and I will discuss both categories and functors arising from this study and a path in the opposite direction.