MAA  Mathematical Association of America

Seaway Section

Location: 1305

1. Time:

   1:15 pm – 1:35 pm

   Title:

   Seeing the World Through the Lens of Mathematics

   Speaker:

   Duncan Miller (Niagara University)

   Abstract

   Have you ever gone outside, taken in your surroundings, and wondered “Is this mathematics?” If you have, then this presentation is for you! We will explore the concept of moduli spaces in the context of the work of Karen Vogtmann and Michael Borinsky, highlighting the basic geometric and topological properties of these structures. Moduli spaces serve as a crucial framework in analyzing the behavior of complex systems and their underlying metrics. We will delve into the foundational elements of moduli spaces and discuss the implications of Vogtmann and Borinsky's findings for future research in mathematics and physics. Specifically, we will cover topics such as types of moduli spaces, cohomology classes, Feynman diagrams, and quantum field theory. By examining key examples and applications, this talk aims to provide an overview of moduli spaces, fostering a deeper appreciation of their significance in contemporary mathematics.

2. Time:

   1:45 pm – 2:05 pm

   Title:

   The Kuratowski Closure-Complement Problem

   Speaker:

   Sophia Tiffany (SUNY Brockport)

   Abstract

   If $A\subset\mathbb{R}$ under the usual topology, how many unique sets can be obtained from taking closures and complements of $A$? Kuratowski famously proved that at most 14 distinct sets can be obtained in this manner. This talk will delve into the historical background and significance of the problem, the methods used to derive the solution, and the topological concepts underlying the closure and complement operations. We will also examine special cases where fewer than 14 sets are produced and discuss the problem’s connection to other areas of mathematics.

3. Time:

   2:15 pm – 2:35 pm

   Title:

   The Erdös Distance Problem on Riemannian Manifolds

   Speaker:

   June Terzioglu (University of Rochester)

   Abstract

   The Erdös distance problem considers the set $\Delta(P)$ of distinct distances between points in a finite set $P$ and asks for lower bounds on its size (in terms of $|P|$). These finite sets are traditionally considered to be in the plane - in this setting it was proven by Guth and Katz in 2015 that $|\Delta(P)| \gtrsim |P| / \log(|P|)$. But the problem can be considered in any metric space; in particular, it can be considered on a compact connected Riemannian 2-manifold. Nathan Skerrett proved in his undergraduate thesis that $|\Delta(P)| \gtrsim |P|^{1 / 2}$ in this setting; this talk is to discuss strengthening this result to $|\Delta(P)| \gtrsim |P|^{2/3}$.

Location: 2300

1. Time:

   1:15 pm – 1:35 pm

   Title:

   A Quadratic Penalty and Moreau Envelope Descent Algorithm for Constrained Global Minimization

   Speaker:

   Stephanie Wang (University of Rochester)

   Abstract

   In 2022, Heaton, Fung, and Osher proposed an algorithm called Hamilton-Jacobi Moreau Adaptive Descent (HJ-MAD) that guarantees convergence to a global minimum by using the evolution of Hamilton-Jacobi PDEs. Our work closely adapts their algorithm and improves on aspects of its accuracy, efficiency, and/or computational requirements. We have found that the effects of time evolution used in Heaton et al.'s algorithm are negligible and time can rather be taken as a constant, thereby simplifying the algorithm. Furthermore, we modified a version of HJ-MAD, “HJ-MAD Twice-Run,” that is able to significantly reduce runtime. Finally, we also propose our own algorithm, “Quadratic Penalty with HJ-MAD,” and demonstrate it using two constrained inequality functions, which we plan to apply to data-fitting problems in engineering.

2. Time:

   1:45 pm – 2:05 pm

   Title:

   Distribution of Hitting Times on Random Walks of Graphs

   Speaker:

   Anuraag Kumar (University of Rochester)

   Abstract

   This presentation explores distributions of hitting times (or the length of a random walk from one node to another) on finite graphs. We use spectral graph theory and markov methods to analyze these distributions. Understanding these distributions is crucial for applications in network theory, where random walks model processes such as information spread, epidemic dynamics, and Markov Chain Monte Carlo algorithms. While traditional analyses focus on the expected value of hitting times and some higher moments, our study of distributions provides a novel perspective. We present an algorithm that computes the exact distribution based on the spectrum of induced subgraphs, offering a tool for predicting and analyzing these processes. This method is particularly valuable for complex networks where traditional approaches may fail. We also extend the analysis to the continuous-time case. Additionally, we apply Fourier methods to Cayley graphs, a class of highly symmetric graphs constructed from groups, and derive recurrence relations that yield an exact distribution. Our findings provide new insights into the interplay between graph structure and random walk dynamics, opening avenues for further exploration in both theoretical and applied settings.

Location: 2305

1. Time:

   1:15 pm – 1:35 pm

   Title:

   Effective support, Dirac combs, and signal recovery

   Speakers:

   Kelvin Nguyen (University of Rochester ), Karam Aldahleh (University of Rochester)

   Abstract

   A classical result due to Matolcsi and Szucs, and, independently, to Donoho and Stark says that if $f: {\mathbb Z}_N^d \to {\mathbb C}$ is a signal and the Fourier transform $$ \widehat{f}(m)=N^{-d} \sum_{x \in {\mathbb Z}_N^d} \chi(-x \cdot m) f(x), \ \chi(t)=e^{\frac{2 \pi i t}{N}}$$ is transmitted with frequencies ${\{\widehat{f}(m)\}}_{m \in S}$ unobserved due to noise or other interference, then $f$ can be recovered exactly and uniquely provided that $$ |E| \cdot |S| < \frac{N^d}{2},$$ where $E$ is the support of $f$, i.e $E=\{x \in {\mathbb Z}_N^d: f(x) \not=0\}$. In this paper, we consider signals that are Dirac combs of complexity $\gamma$, meaning they have the form $f(x)=\sum_{i=1}^{\gamma} a_i 1_{A_i}(x)$, where the sets $A_i \subset {\mathbb Z}_N^d$ are disjoint, $a_i$ are complex numbers, and $\gamma \leq N^d$. We will define the concept of effective support of these signals and show that if $\gamma$ is not too large, a good recovery condition can be obtained by pigeonholing under additional reasonable assumptions on the distribution of the values.

2. Time:

   1:45 pm – 2:05 pm

   Title:

   Minimal Heights of a Modified Sisyphus Function

   Speaker:

   Ziqian Zhao (Rochester Institute of Technology)

   Abstract

   In this presentation, we explore the modified Sisyphus function of order 6, focusing on how many iterations it takes to reach a fixed point or a cycle. Specifically, we determine the smallest nonnegative integer, $n$, that requires a given number of iterations before the sequence stabilizes. Our findings cover up to 11 iterations, where we establish the minimal value of $n$ for each case. This talk emphasizes the function's behavior without diving too deep into the formal proofs. The focus will be on the function’s properties, observations from the generated sequences, and some intriguing patterns that emerge, rather than heavy theoretical arguments.

3. Time:

   2:15 pm – 2:35 pm

   Title:

   Parametric Integral Transforms and their Applications

   Speaker:

   Bhattarabodin Sonrod (Ithaca College)

   Abstract

   In this study, we present parametric variations of popular integral transforms, such as the Laplace transform and Stieltjes transform. We demonstrate iteration identities and Parseval-Goldstein-like relationships that involve these parametric integral transforms. Moreover, we utilize these findings to compute improper integrals of established functions, such as the MacDonald function and the Tricomi function.