# Mathematics Colloquium Fest – January 27, 9:15 am – noon, Stetson Court Classrooms

## Mon, January 27th, 2020

9:15 am - 12:00 pm

- This event has passed.

A total of 11 math colloquiums, starting at 9:15 and going until noon, in three parallel sessions, all in Stetson Court Classrooms. This is a time to learn some new math. And for junior and senior math majors, it is an easy way to get four colloquium credits. Pizza lunch to follow at noon.

Here is the schedule of speakers:

Session 1:

9:15 – 9:45 am:

Clara Hathorne ‘20

Polynomial Resultants and Applications

The resultant of polynomials is typically used to show whether two polynomials share a root. But interestingly, resultants are closely related to discriminants as well as have interesting implications for Bezout’s Theorem.

10 – 10:30 am:

Emma Herrmann ‘20

The Damage Number of a Graph for the Game of Cops and Robbers

The game of Cops and Robbers is played on a reflexive graph with a set of cops and a set of robbers, but this talk will mostly analyze the game played with one cop and one robber. The Cop wins if, after a finite number of time steps, he is able to catch the robber. Else, the robber wins. In the first time step, the cop picks any vertex on the graph to start followed by the robber. Play continues alternating between the cop, who moves first, followed by the robber. It is assumed that both the cop and the robber play optimally: the cop plays to minimize the number of time steps to capture, and the robber plays to try and not get captured. Sanaei and Cox (2019) introduce the damage number of a graph, defined by the minimum number of vertices on the graph that can be occupied by the robber during the game. And the ‘damage’ is done after the robber completes a move. This concept is interesting because damage can be studied on any type of graph, whether the cop eventually wins or not. Now the cop and robber play optimally by minimizing the damage done and visiting the most vertices so to inflict the most damage, respectively. This talk will cover recent results on the damage number and its impact on the game.

10:45 – 11:15 am

Giebien Na ‘20

Exploring the Square Root of 2

This talk will explore different proofs for the irrationality of the square root of 2. I will begin with proofs that require only simple intuition or insights, and then cycle through progressively more interesting scenarios. In particular, I will highlight the difficulties that arise when we presume an ignorance of the fundamental theorem of arithmetic. My final examination will rest on a geometric proof for the square root of two, of which I will show that its logic may be extended to other numbers, such as the square root of 3 and the square root of 5, and an interesting phenomenon that the geometric proofs for the irrationality of certain square roots align well initially with the triangular numbers, although this must of course eventually fail, since we would have some serious problems if the geometric method could demonstrate the irrationality of the square root of the 8th triangular number, which is a perfect square!

11:30 am – 12:00 pm:

Matt Zappe ‘20

The SIS Epidemic Model and Virus Dynamics on Hub-and-Spoke Graphs

In the field of epidemiology, it is vitally important for mathematicians to develop models that capture the dynamics of disease spread as accurately as possible. Although vaccines exist to prevent the spread of infectious diseases, many of these diseases still cause suffering, mortality, and extreme financial costs, especially in developing countries where sufficient healthcare is rarely available. Epidemic models, such as the SIS or Susceptible-Infectious-Susceptible model, allow us to analyze the transmission interactions within the population, and the results of these models can be used to implement proactive and retroactive strategies to curb the spread of an infectious disease. Specifically, we are able to use the SIS model to study virus propagation on a simplified hub-and-spoke graph for particular infectious diseases, including the common cold and sexually transmitted diseases like gonorrhea. By implementing the SIS model in star-like graphs, we are able to observe long-term phase transition behavior as a function of cure rate, infection rate, and the number of spokes on the graph. Given two parameters representing infection and cure probabilities, we can use the model to prove that a threshold value of these parameters will both exist and determine the steady-state behavior of the system. We find that below this threshold, the virus will die out, and above this threshold, the virus will not die out and node infection probabilities will converge to a unique, non-trivial point. The results of this model can be generalized to more complex hub-and-spoke graphs consisting of multiple levels. We conclude with a discussion on the overall importance of epidemic mathematical modelling in combatting the spread of infectious disease.

Session 2:

9:15 – 9:45 am:

Divya Wodon ‘20

ABC Conjecture and Fermat’s Last Theorem

The colloquium will focus on the ABC conjecture proposed by Oesterle and Masser and its relation to Fermat’s theorem. The ABC conjecture states that with three relatively prime integers a, b, and c that satisfy a + b = c, the size of c is bounded above by the product of the distinct prime numbers dividing a, b, and c (Ghitza, 2012). To note, a number is prime if it has exactly two factors, itself and 1. The conjecture relates to Fermat since while still unproven, if it holds true, it implies Fermat’s last theorem for large powers. Specifically, Fermat’s last theorem states that no positive integer satisfies the equation x^n + y^n = z^n for n greater than 2. This colloquium will explore how if the ABC conjecture is true, it implies Fermat’s last theorem for large n.

10:00 – 10:30 am:

Jack Roche ‘20

A Renormalization Approach to the Zeta Function at -1

The Riemann Zeta Function lies at the heart of the most important unsolved problem in mathematics. Furthermore, it is of particular use in various applications in Physics that when evaluated at -1 it yields -1/12. It is precisely because this result is counterintuitive that finding alternative methods of proof – and thereby discovering additional properties of the Riemann Zeta function – is useful. One such method, based on a technique used by physicists known as renormalization, examines averaging and re-scaling partial sums and finds an alternative continuation in (4U([n/2])-Un)/3, where Uk is the partial sum of the first k terms.

10:45 – 11:15 am:

Isabelle Furman ‘20

Hyperbolic Geometry and Crochet

In my talk I will discuss how the hyperbolic plane can be modeled with crochet, and how such a model can be utilized to better understand concepts of hyperbolic geometry.

11:30 am – 12:00 pm:

Chris Kim ‘20

Lambert’s Proof of the Irrationality of Pi

I will discuss Lambert’s proof that pi is irrational using continued fractions.

Session 3:

10:00 – 10:30 am:

Mia Carroll ‘20

The Singular Value Decomposition and Its Applications to Digital Image Compression

The singular value decomposition (SVD) is the factorization of a real or complex m x n matrix into the product of three simpler matrices. This decomposition can be used to reduce the amount of stored information, while preserving the features of the original matrix. In this talk, we will explore the properties of SVD and its applications to digital image compression.

10:45 – 11:15 am:

Louisa Ebby ‘20

The Recurrence of Random Walks and Polya’s Theorem

Random walks are used to model countless processes in biology, chemistry, physics and economics. For a simple random walk, a particle moves one step in any direction with equal probability. If the particle starts at the origin, will it return to its starting point? It so happens that the answer depends on the dimension of the walk, which we will prove in this talk using Polya’s Theorem.

11:30 am – 12:00 pm:

Hunter Wieman ‘20

Bernstein Polynomials and Applications

We introduce the Bernstein Polynomials and discuss their properties. We use them to constructively prove that any continuous function on a compact interval can be uniformly approximated by a sequence of polynomials. We also discuss applications to computer graphics and animation.

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