Spring 2020

Speaker: Rebecca Rohrlich
Title: An Introduction to Rigidity Theory
Abstract: This talk will be an overview of Rigidity Theory, a kind of geometry that analyzes when structures are rigid or flexible. In my talk, I will be focusing mostly on the rigidity of bar-joint frameworks, which you can think of as mechanical objects built by straight, rigid bars attached at flexible joints. This talk is very basic and has no prerequisites; I am just using graph theory and linear algebra.

Speaker: Alex Semendinger
Title: Meanderings in Mathematical Billiards
Abstract: Imagine an ideal billiard ball bouncing elastically around a frictionless table forever. What kind of paths might it trace out? Will they repeat or not? Will the ball travel all around the table, or spend most of its time in a small area? I will walk through some elementary results in mathematical billiards, addressing these questions and more. I will also describe some applications to other areas and open problems in the field.

Speaker: Aritro Pathak
Title: Sums of powers of elements of the Cantor set
Abstract: Let C be the middle third Cantor set within [0,1]. It is well known that C+C=[0,2], i.e, the set of sum of two elements of the Cantor set is the continuum [0,2]. In a paper from November 2017 published later in the Monthly, Athreya, Reznick and Tyson conjectured that the set {x_1^2+x_2^2+x_3^2+x_4^2|x_1,x_2,x_3,x_4 in C} contains [0,1]. We talk about a seemingly new elementary proof of the fact that C+C=[0,2], and a modification of the approach of this proof that helps to make progress towards this conjecture. We construct large explicit open intervals within [0,4] where each element can be written as the sum of four squares of Cantor set numbers as above. With some numerical work, our method should be able to cover almost all of [0,4]. Analogously, we also construct explicit open intervals where each element can be written as a finite sum of higher exponents (≥ 3) of Cantor Set elements. There has been another claimed proof of the Monthly conjecture that cites some known deep results in Hyperbolic dynamics, however our method is completely elementary.

Speaker: Eric Hanson
Title: A Sampling of Homological Dimensions
Abstract: Open since (at least) 1960, the finitistic dimension conjecture is an extremely important open problem in the representation theory of associative algebras. This talk will develop the theory behind this conjecture and the stronger φ-dimension conjecture (based on work by Kiyoshi Igusa and Gordana Todorov in 2005). We will conclude by presenting a counterexample to the φ-dimension conjecture given in recent joint work with Kiyoshi Igusa. This talk will be accessible to a general mathematical audience.

Speaker: Duncan Levear
Title: Linear Programming and The Theorem You Won't Believe
Abstract: Many important applied problems can be framed as linear programs: optimize a linear function subject to linear constraints. There is a surprising fact, the Fundamental Theorem of Linear Programming, which allows us to "discretize" the problem and leads to a simple solution. In this talk, I will explain some applications of Linear Programming and the ideas that go into proving the Fundamental Theorem. This sheds light on a variety of ubiquitous problems from our daily lives: such as classroom scheduling, shift assignment, and effective dieting.

Speaker: Hao Cui
Title: Martingale, Wiener Process, and Stochastic Integral
Abstract: Martingale, which originated as a model of a fair game, is now one of the most important concepts in probability theory. In this talk, we will briefly introduce the definition of martingale and construct a basic example of continuous martingales—the Wiener process, which is the process describing Brownian motion. By exploring some basic properties of this process, we will further construct heuristically the stochastic integral and actually calculate some examples.

Speaker: Rebecca Rohrlich
Title: The Busy Beaver Problem
Abstract: Suppose we have a Turing machine with a blank input tape. What is the longest amount of time the machine can run without getting caught in an infinite loop? This is the Busy Beaver Problem, initially proposed by Tibor Rado in 1960. The problem has exciting implications, since solving it could allow us to validate several famous conjectures (such as the Goldbach conjecture and the Riemann hypothesis) just by a short, brute-force computer program. I will discuss recent progress on this problem, as well as some surprising connections to dynamical systems and number theory.

Speaker: Abhishek Gupta
Title: Belief propagation in Bayesian Networks
Abstract: A Bayesian network is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph. They use Bayesian inference for probability computations. Belief propagation is a message-passing algorithm for performing inference on Bayesian networks. More precisely, it is used to calculate the marginal distribution for each unobserved variable(node), conditional on any observed variables. In this talk, we shall introduce Bayesian networks and explain the belief propagation algorithm for Bayesian networks. Only knowledge of elementary probability will be assumed.

Speaker: Jill Stifano
Title: Proper Diameter in Edge-Colored Graphs
Abstract: In an edge-colored graph, a properly colored path is a path in which no two consecutive edges have the same color. We can use this notion to measure something similar to the diameter of a graph, called proper diameter, which is the largest distance between two vertices using a properly colored path. This talk will explore upper and lower bounds of proper diameter for different types of graphs. It will also introduce the characterization of all 2-connected graphs in which the maximum proper diameter is achievable with 2 colors. This research was done as part of a summer REU.

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Speaker: Simon T. Huynh
Title: Lattice-based Cryptography and The Shortest Vector Problem in Ideal Lattices
Abstract: In this introductory talk, I will present the study of ideal lattices constructed from number fields, their related cryptosystems, and the Shortest Vector Problem which plays an important role in Post-Quantum Cryptography. Among many potential candidates, lattice-based schemes are attractive for their strong provable security and resistance to quantum attacks. But ARE THEY REALLY? In my short study, I found some examples of cyclic sublattices of Z^n where a shortest vector can be easily computed which proves to be not-reliable to be employed as a basis for the new cryptosystem.

Prerequisite: None but your favorite hot or cold drink of choice; Hawaiian pizza is optional but recommended.

Disclaimer: This study was a part of my senior honors thesis supervised by Dr. Kirsten Eisenträger. You can find my write-up here.