Spring 2021

Abstract: In this talk, we discuss a general method to define surfaces as quotient spaces of Euclidean polygons. We then define a metric on the quotient space induced by the Euclidean metric, and prove a theorem determining exactly when the quotient metric is locally Euclidean.

Mentor: Theodore Weisman

(not recorded)

Abstract: The Alexander polynomial is an invariant useful for distinguishing knots. I explain two definitions of the Alexander polynomial and why they are equivalent.

Mentor: Jonathan Johnson

Abstract: This talk discusses the theorem that groups that act freely on trees are free groups. To do this, we use the example of the group SL(2, Z)[m] on the Farey tree.

Mentor: Kai Nakamura 

Abstract: A qualitative look into algebraic topology topics such as homotopy and the fundamental group. The presentation ends after a look at a proof of the Brouwer fixed point theorem in two dimensions.

Mentor: Hannah Turner 

Abstract: By exploring one problem from two different perspectives, this talk seeks to provide a visual intuition for understanding certain connections between algebra and topology. Topics include fundamental groups, graphs of groups, fundamental polygons, free groups and Stallings' foldings.

Mentor: Casandra Monroe 

Abstract: I talk about Principal Component Analysis. I first explain its mathematical background, how we can go about performing it, its pros and cons, as well as an example with real data.

Mentor: Milad Eghtedari Naeini

(not recorded)

Abstract: This talk will discuss rational curves and their intersections with the example of parametrizing a circle. It will also explore the group of points on a rational cubic and identify a closely related theorem.

Mentor: Natalie Hollenbaugh 

Abstract: There is a typical way to measure sizes of rational numbers, where 2 has size 2, and -5 has size 5. However, it turns out that prime factorization provides lots of other ways to measure sizes of rational numbers, and these ways to measure size give us some pretty pictures.

Mentor: Tom Gannon 

Abstract: Introduce the problem which is what is the probability of two numbers being coprime. Walk through the product of the probability the two integers do not share primes. Then lastly talk about the how Euler solved the bazel problem which leads us to our answer.

Mentor: Colin Walker 

Abstract: Introduces basics of group theory as well as offers an intuitive explanation to the Burnside Lemma. We will then utilize this lemma to determine the number of possible colorings of a cube.

Mentor: Kyrylo Muliarchyk 

Abstract: In this project, we study the alternating direction method of multipliers (ADMM) applied to image deconvolution. We compared the method with traditional inverse filtering and Wiener filtering.

Mentor: Tharathep Sangsawang

(not recorded)

Abstract: My talk will first focus on Linear Algebra tools used in machine learning. Then I will transition to talking about machine learning (in particular classification problems), with a focus on neural networks, giving a simple example of this process in action. Finally, I will explain how I intend to use what I have learned in the Directed Reading Program on a current problem the University has.

Mentor: Daniel Restrepo 

Abstract: The talk explores the concept of counting trees in Graph Theory with Matrix Tree Theorem, and extends it to Cayley's formula. Alternatively, Prufer's method is also discussed. Proofs and related examples are worked out.

Mentor: Feride Ceren Kose

Abstract: I will explore the properties of graphs to then talk about the Eulerian Graphs. The properties of Eulerian Graphs have been used to solve a famous 18th century problem.

Mentor: Jonathan Johnson 

Abstract: I go over the proof of how the spectrum of an adjacency matrix of a graph can bound the Isoperimetric or expanding constant of a graph. Furthermore, I discuss how it is used to describe families of expander graphs.

Mentor: Maksym Chaudkhari 

Abstract: I will be discussing the importance and work that has been done on the Navier-Stokes equations.

Mentor: Andy Ma

Video

Abstract: A short introduction to Lagrangian mechanics, providing an overview of the Euler-Lagrange equation, Hamilton's principle of least action, and deriving the equation of the catenary.

Mentor: Jackson Van Dyke

Video (start at 14:00) 

Abstract: Discussing the Inverse Function Theorem as normally defined in calculus and how we can extend it for smooth maps of differentiable manifolds

Mentor: Yixian Wu

(not recorded)

Abstract: In this talk I will give an introduction to the fundamental theorem of Galois theory via multiple examples.

Mentor: Benjamin Michael Nativi 

(not recorded)

Abstract: A dive into the theory of Lie groups through the lens of representation theory. We construct Lie groups by defining their associated Lie algebra and explore this relationship through the exponential map.

Mentor: Will Stewart

Video (unfortunately, the first half of this talk is missing from the recording) 

Abstract: How do calculus and algebra relate to topology? Here we motivate the basics of de Rham cohomology by discussing div, grad, and curl in lower dimensions, and how we can build certain quotient spaces from these operators that measure how many "holes" a space has. Then, we generalize to the abstract de Rham complex, introducing differential forms and the exterior derivative (showing how it generalizes div, grad, and curl), and conclude by using the Mayer-Vietoris sequence to compute the cohomology of the punctured plane.

Mentor: Arun Debray

Video (start at 8:40) 

Abstract: I will be discussing the proof of the universal approximation theorem, using some concepts from functional analysis. This proof implies that a neural network can approximate any function.

Mentor: Hunter Vallejos 

Abstract: How many times do you have to shuffle a deck to make sure it is shuffled? We think about card shuffling as a Markov chain and show there is a guaranteed time at which the deck is shuffled.

Mentor: Hunter Vallejos 

Abstract: Studying the analogues between modeling electrical behavior in circuit networks and random walks across graphs. Also, will be showcasing a simulator program we made that probabilistically computes rudimentary heat transfer across a conductive surface using a Markov Chain Monte Carlo method.

Mentor: Joe Jackson 

Abstract: I will discuss the 2-D modeling of the Ising Model of ferromagnetism using the Metropolis Algorithm. I used code in Python to produce simple figures describing the phenomenon.

Mentor: Ziheng Chen 

Abstract: An introduction of the classification models with a focus on KNN (K-Nearest Neighbors).

Mentor: Milad Naeini 

Abstract: Connectomics is an umbrella term covering all applications of graph theory to neuroscience. We will discuss a number of measures used in structural connectomics, with some consideration as to the results they have enabled.

Mentor: Gillian Grindstaff

(not recorded)

Abstract: This talk will be a brief introduction on what a knot is and how we classify and study them. It touches on one method to study knots , the Jones polynomial, and how to calculate it.

Mentor: Kenny Schefers

Video

Abstract:  An introduction into Knot Theory with basic vocabulary and a brief summary of its contribution to chemistry.

Mentor: Neža Žager Korenjak

(not recorded)

Abstract: In this talk, I will introduce the audience to a few aspects of knot theory. By the end of the talk, audience members will be familiar with knots, braids, and why braids form a group.

Mentor: Allie Embry

Video

Abstract:  A synopsis of knot theory and the properties of mathematical knots using an example on Seifert surfaces. 

Mentor: Riccardo Pedrotti

(not recorded)