# Graduate Colloquium Schedule Spring 2023

Wednesday 1/25/23 at 4:30 in LC422

Speaker: Thomas Hamori

Abstract: In this talk, I aim to give the audience a flavor of the mathematics of traffic flow. To do so, I will give a short introduction to the relevant components of the theory of differential equations. From here, I will show how one may use fluid dynamics to model traffic flow by comparing the flow of cars along a roadway to that of water in a stream. So called 'macroscopic' models are plagued by nonlinearity which causes shock formations, culminating in the breakdown of the model. Finally, I will describe some mechanisms by which mathematicians hope to overcome this problem, including some from my own research. (Joint work with my advisor, Dr. Changhui Tan). The talk is designed to be accessible to both pure and applied track graduate students, including first years.

Wednesday 2/1/23 at 4:30 in LC422

Speaker: Pat Lank

Title: Frobenius pushforwards generate the bounded derived category

Abstract: Within this talk, I will discuss an approach that introduces new ways to understand the structure of the bounded derived category for Noetherian schemes of prime characteristic. For the sake of background, the scope will be restricted to a familiar class of Noetherian rings. First, I will briefly review what a derived category is, and how to think about it. Then we discuss generation in the derived category. This is an algorithm that allows one to build objects from a single one utilizing three elementary operations. If we are lucky and generate all objects from a suitable choice, then there may exist a finite upper bound to the number of steps required to do so. After this set up, we will see how to canonically choose a generator for a class of Noetherian rings of prime characteristic by iterating the Frobenius endomorphism. Furthermore, and even better, we will know that the number of steps required to generate each object has a uniform upper bound.

Thursday 2/16/23 at 4:15 in LC422

Speaker: Duncan Wright

Title: Mathematics and Policy: The Intersection is Larger Than You Think

Abstract: I will share my experience as the 2022-2023 AMS Congressional Fellow, my role in the office and the impact of Science & Engineering Fellows in Congress. I will argue that the critical skills shared by all mathematicians lie in the titular intersection and I will discuss ways to engage with elected officials in addressing policy issues of concern to the mathematics community, including research funding and education. I will also highlight policy fellowships of potential interest for graduate students and faculty.

Wednesday 2/22/23 at 4:30 in LC422

Speaker: Anirban Bhaduri

Title: Coherent Sheaves on the Projective Line

Abstract: We talk about the category of coherent sheaves over a projective line. Firstly, we talk about a projective line in terms of polynomial rings and their ideals. We also show that the geometric objects called sheaves have a rather familiar definitions in terms of algebraic objects like rings and modules. We show the equivalence between the category of coherent sheaves over the projective line with the quotient category of graded modules. With this equivalence, we can look at the structure sheaves, torsion sheaves, homomorphisms in a rather familiar algebraic way.

Wednesday 3/1/23 at 4:30 in LC422

Speaker: Thomas Luckner

Title: Covering Systems, Newton Polygons, and Other Constructive Proof Techniques

Abstract: This talk is an example of a research talk I have done for a position with a predominately teaching-focused school that is interested in research that can easily translate to undergraduate research projects. This talk will focus on two main proof techniques prevalent in my research; covering systems and Newton polygons. The first technique is often used in problems related to primality and existence of integers. The second technique is often used in problems relating to the irreducibility of polynomials. For each of these techniques I will provide motivation, some demonstration of its usefulness, some of my work, and future work for myself and students. If there is time, I will discuss some other tangential projects I have been exploring and will continue to explore.

Wednesday 3/15/23 at 4:30 in LC422

Speaker: McKenzie Black

Title: Asymptotic Behaviors for the Compressible Euler System with Nonlinear Velocity Alignment

Abstract: Goldfinches fly in flocks to protect themselves from predators, while surgeonfish school together in the hope of finding a mate. These collective behaviors are examples of how animals can utilize their unique sensory capabilities to communicate between groups, leading to alignment and flocking for safety and social activities. The Euler-alignment system can be implemented to better help describe these remarkable asymptotic behaviors. These characteristics are inherited from the archetypal alignment model, the Cucker-Smale model. In this talk we will delve into modeling collective behaviors and the varying effects of the nonlinear velocity alignment. Additionally, we will disclose an analysis of the asymptotic emergent phenomena of the pressureless compressible Euler system with a family of nonlinear velocity alignment and nonlocal communicational protocol. This system is an extension of the Euler-alignment system in collective dynamics. Lastly, the impact of this presentation is highlighted by the recent findings that result in a variety of different asymptotic behaviors.

Wednesday 3/22/23 at 4:30 in LC422

Speaker: Yiqun Li

Title: A viscoelastic Timoshenko beam: Model development, analysis, and investigation.

Abstract: Vibrations are ubiquitous in mechanical or biological systems, and they are ruinous in numerous circumstances. We develop a viscoelastic Timoshenko beam model, which naturally captures distinctive power-law responses arising from a broad distribution of time-scales presented in the complex internal structures of viscoelastic materials and so provides a very competitive description of the mechanical responses of viscoelastic beams, thick beams, and beams subject to high-frequency excitations. We, then, prove the well-posedness and regularity of the viscoelastic Timoshenko beam model. We finally investigate the performance of the model, in comparison with the widely used Euler–Bernoulli and Timoshenko beam models, which shows the utility of the new model.

Wednesday 3/29/23 at 4:30 in LC422

Speaker: Alexandros Kalogirou

Title: Sieve methods, and a proof of Chen's theorem

Abstract: We give an overview of some sieve methods, like the Selberg sieve, and the weighted sieve, and how they apply in a proof of Chen's theorem.

Thursday 4/6/23 at 4:30 in TBA

Speaker: Bailey Heath

Title: Bounding the Representation Dimensions of Algebraic Tori

Abstract: Let k denote a field. Algebraic tori are special functors from the category of finitely generated k-algebras to the category of groups, such as the multiplicative group and the circle group. Any algebraic torus can be embedded into the group of n x n invertible matrices for some n, and the representation dimension of an algebraic torus is the minimal such n for that torus. In this work, we seek the smallest possible upper bound on the representation dimension of all algebraic tori of a given dimension. To help us do this, we will (briefly) discuss symmetric ranks of G-lattices, finite groups of integral matrices, root systems and their Weyl groups, and quadratic forms and their theta series. This talk will be aim to be accessible to all graduate students!

Wednesday 4/12/23 at 4:30 in LC422

Speaker: Grant Fickes

Title: Zero Loci of Graph and Hypergraph Nullvectors

Abstract: The adjacency nullity of graphs’ and hypergraphs’ adjacency matrices is something of a mystery, though there are nice results for some narrow classes of graphs such as trees. There is, however, rich structure in their nullspaces (and, for hypergraphs, their nullvarieties), visible by partitioning nullvectors according to their zero loci: vertex sets which are indices of their zero coordinates. The zero loci of nullvectors are closely related to sets which are "stalled" under the skew zero forcing (SZF) definition. In this talk, we examine the relationship between these two set systems for graphs and extend the ideas to linear hypertrees.

Wednesday 4/19/23 at 3:00 in LC205

Speaker: Swati

Title: Theory of half-integer weight modular forms and Shimura Lifts

Abstract: Modular forms lie at the center of a terrific amount of research activity. Two prominent examples of their importance come from their significant role in Wiles’ proof of Fermat’s Last Theorem and in Borcherds’ proof of the Monstrous Moonshine Conjecture in Lie Theory. In this talk, we will recall basic definitions and essential properties of integer and half-integer weight modular forms, the theory of Hecke operators, and the notion of oldforms and newforms. We also discuss the Shimura Correspondence and Waldspurger's Theorem. We will conclude the talk by sharing the recent progress on a problem concerning explicit formulas for Shimura images of a certain family of integer and half-integer weight modular forms.

Thursday 4/20/23 at 4:15 in LC422

Speaker: Felix Bartel

Title: The power of least squares in the worst-case and learning setting

Abstract: Least squares approximation is a time-tested method for function approximation based on samples. It is a natural to compare it to approximations based on arbitrary linear functionals as a benchmark. In this talk we will present recent results from the information-based complexity community showing the optimality of the least squares algorithm. We will consider the worst-case setting: drawing points which are good for a class of functions and the learning setting: we approximate an individual function based on possibly noisy samples. We support our findings with numerical experiments.