DRP

Spring 2021 Symposium.

Note: Some rooms with many speakers have large video files that may take longer to load.

Speakers in order of appearance

Porfirio Rodriguez
- The Alexander Polynomial
- 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

Marilyn Lionts
- Free Actions on Tree
- 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

Omar Aceval
- The Brouwer Fixed Point Theorem
- 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

Anna Groves
- Bridging the Gap Between Algebra and Topology
- 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

Speakers in order of appearance

Grace Mulry
- Rational Points on Conics and the Geometry of Cubic Curves
- 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

Nir Elber
- Putting Primes in Their Place
- 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

Jacob Gutierrez
- The Probability of Two Integers Being Coprime
- 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

Manisha Ganesh
- Burnside's Lemma
- 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

Speakers in order of appearance

Tyler Dean
- Neural Networks and Classification Problems
- 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

Himani Verma
- Matrix Tree Theorem and Cayley's Formula
- 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

Katie Fuller
- Eulerian Graphs
- 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

Gerardo Villalobos
- Isoperimetric Constant of Expander Graphs and Spectral Bounds
- 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

Video of Room 4 Part I
Video of Room 4 Part II

Speakers in order of appearance

David Sarabia
- Discussion on the Navier-Stokes Equations
- Abstract:
  - I will be discussing the importance and work that has been done on the Navier-Stokes equations.
- Mentor: Andy Ma

Reese Feldmeier
- Introduction to Lagrangian Mechanics
- 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

Jeremy Krill
- Lie Groups and Lie Algebras (video)
- 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

Simon Xiang
- An introduction to de Rham cohomology
- 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

Speakers in order of appearance

Jacob Way
- The Universal Approximation Theorem
- 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

Kyle Cox
- Markov chains and card shuffling
- 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

Rob Steve
- Random Walks, Electrical Networks, and MCMC Simulation
- 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

Michaela Petty
- Ising Model
- 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

Louie Wang
- Classification
- Abstract:
  - An introduction of the classification models with a focus on KNN (K-Nearest Neighbors).
- Mentor: Milad Naeini

Video of Room 6 Part I
Video of Room 6 Part II

Speakers in order of appearance

Ethan Roy
- An Intro to Knot Theory & the Jones Polynomial
- Abstract:
  - This talk will be on 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

Veronica King
- Braids are a Group!
- 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