Speaker: Alec Helm

Abstract: A planar graph is a graph which can be represented in the plane such that vertices are distinct points, edges are continuous arcs between their incident vertices, and edges intersect vertices and other edges only at their endpoints. In this talk we will review some fundamental facts about this well-studied graph family, and survey some of the famous results about them. After carefully building this framework, we will expose how many of these results, as stated in numerous papers on the subject, are in fact false. We will launch our assault first against the more elementary so-called facts (such as the claim that maximal planar graphs are 3-connected), before turning our attention to the esteemed results of Kuratowski, Whitney, and 4-Color. Some of these will prove to be true, some utterly false, and some true with minor addendum.

Speaker: Benjamin Dial

Abstract: Gödel's completeness theorem is one of the foundational theorems of mathematical logic. This says that a statement is a semantic consequence of a theory (i.e. in every model of the theory, the statement is true) if and only if it is a syntactic consequence of the theory (i.e. the statement can be proven from the theory). This tells us that our choice of proof system is "complete", and we do not need to add any extra logical rules to it. In this talk, we will first get acquainted with the necessary definitions and then sketch a proof of Henkin's model existence theorem (every consistent theory has a model). Finally, we will see Gödel's completeness theorem and the compactness theorem (a theory is consistent if and only if every finite subtheory is consistent) as consequences of this.

Speaker: Jasdeep Singh

Abstract: One key limitation of numerically modeling differential equations is the high computation cost for accurate models. Because of the need for quick predictive models for real-time applications, there is a class of methods called Reduced Order Modeling (ROM), where we reduce the size of the model while retaining as much accuracy as possible. We will introduce classical approaches such as Proper Orthogonal Decomposition (POD) and Galerkin Projection, as well as non-intrusive methods. Another important characteristic of many dynamical models is energy-preservation, which leads to a class of methods focusing on Structure Preserving ROM. The goal of this talk is to familiarize the audience with ROM as well as further areas of research.

Speaker: Victoria Chebotaeva

Abstract: It’s academic job season, and the process can feel overwhelming. In this talk, I’ll break down the essentials: what makes a teaching or diversity statement stand out, what to include (and avoid) in your CV and cover letter, and a few lessons I learned along the way. Plus, I’ll share some tips for handling interviews. If you’re currently on the job hunt or planning to be in the future, I have some answers for you! 

Speaker: William Linz

Abstract:  The Shannon capacity of a graph is a graph invariant with origins in communications theory which is notoriously hard to compute or even approximate. The Lovasz number is an upper bound on the Shannon capacity which by contrast can be efficiently computed and sometimes gives the exact value of the Shannon capacity. In this talk, I will survey the history and basic results about the Shannon capacity and Lovasz number, leading up to a recent result of mine which gives the largest known gap between the Shannon capacity and the Lovasz number.

Speaker: Aditya Iyer

Abstract: The Riemann Hypothesis is a famous conjecture that describes the behavior of nontrivial zeroes of the Riemann zeta function. It is a fairly well known fact that the Riemann Hypothesis is not an easy problem — it is considered by many to be the hardest way to make a million dollars. Many of the brightest mathematical minds in history have tried and failed. There is, however, a plethora of statements equivalent to the Riemann hypothesis. Some of these seem to align strongly with our intuition of how the integers must behave, while others seem unexpected. This talk will be an exposition describing some of these statements -- why they are equivalences, and why we care.

Speaker: Matthew Booth

Abstract: One of the crown jewels in abstract algebra is the Fundamental Theorem of Galois Theory, which gives a one-to-one correspondence between intermediate fields of a Galois extension and subgroups of an associated group of automorphisms (the Galois group). This result is familiar to most students after a first-year graduate course in algebra. However, a key assumption in this correspondence is that the field extension in question be of finite degree. Infinite algebraic extensions do not obey the correspondence as “nicely” as their finite counterparts. The goal of this introductory/expository talk, after a quick review of the fundamentals and finite extensions, will be to develop the necessary theory to extend (no pun intended) the fundamental theorem to infinite extensions. In particular, we shall aim to understand the Krull topology and see that infinite algebraic extensions can arise naturally.

Speaker: George Brooks

Abstract: The spread of a graph $G$ is the difference $\lambda_1 - \lambda_n$ between the largest and smallest eigenvalues of its adjacency matrix. Breen, Riasanovsky, Tait and Urschel recently determined the graph on $n$ vertices with maximum spread for sufficiently large $n$. In this paper, we study a related question of maximizing the difference $\lambda_{i+1} - \lambda_{n-j}$ for a given pair $(i, j)$ over all graphs on $n$ vertices. We give upper bounds for all pairs $(i, j)$, exhibit an infinite family of pairs where the bound is tight, and show that for the pair $(1, 0)$, the extremal example is unique. These results contribute to a line of inquiry pioneered by Nikiforov aiming to maximize different linear combinations of eigenvalues over all graphs on $n$ vertices. Based on joint work with William Linz and Linyuan Lu.

Speaker: Albert Luan

Abstract:  In this talk, we will look at the one-dimensional pressure-less Euler equations, a fundamental system in fluid dynamics that describes the flow of inviscid fluids. First I will talk about the intuitions behind these equations, and then introduce the method of characteristics, a powerful technique that allows us to transform these PDEs into ODEs. Through this approach, we will have better understanding about key behaviors of solutions, including phenomena such as finite-time blow-up and existence of classical solutions. I will also talk a bit about further questions related to this system, and if time permits, how does harmonic analysis help us in solving PDEs. 

Speaker: Dinesh Limbu

Abstract: "Yoneda lemma is a fundamental result in category theory. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is a vast generalization of Cayley’s theorem from group theory. In this talk, we will go over the basic ideas of Category theory and proof of Yoneda lemma. We will also go over several examples to understand the importance of this lemma. For this talk, no prerequisites are assumed."

Speaker: Viktor Stein

Abstract: Regularized optimal transport (OT) has received much attention in recent years starting from Cuturi's paper with Kullback-Leibler (KL) divergence regularized OT. In this paper, we propose to regularize the OT problem using the family of alpha-Rényi divergences for alpha in (0,1). Rényi divergences are neither f-divergences nor Bregman distances, but they recover the KL divergence in the limit alpha to 1. The advantage of introducing the additional parameter alpha is that for alpha to 0 we obtain convergence to the unregularized OT problem. For the KL regularized OT problem, this was achieved by letting the regularization parameter tend to zero, which causes numerical instabilities. We present two different ways to obtain premetrics on probability measures, namely by Rényi divergence constraints and by penalization. The latter premetric interpolates between the unregularized and KL regularized OT problem with weak convergence of the minimizer, generalizing the interpolating property of KL regularized OT. We use a nested mirror descent algorithm for solving the primal formulation. Both on real and synthetic data sets Rényi regularized OT plans outperform their KL and Tsallis counterparts in terms of being closer to the unregularized transport plans and recovering the ground truth in
inference tasks better.

Speaker: Pankaj Singh

Abstract: In this introductory talk, we will cover the basic ideas of linear algebraic groups. We will discuss the Jordan Decomposition Theorem, which helps us understand the structure of these groups, and look at the associated Lie algebras to see how they relate to each other. We will also introduce reductive groups and explore root systems and their data, which are important for classifying these groups. If time allows, we will mention Chevalley’s key result about simple modules. In future sessions, we may discuss Good filtrations and Weyl filtrations of G-modules for a given reductive group G, as well as Donkin’s conjecture, which is still an open problem. This talk aims to provide a foundation for understanding linear algebraic groups and prepare us for more advanced topics later on.

Speaker: Jonathan Smith

Abstract: Pick a field k and suppose X is an algebraic object defined over the separable closure of k. Our broad goal is to answer the question: “What is the smallest extension of k over which X is defined?” As stated, this question is naive, as there are numerous scenarios in which we know that a smallest extension does not exist. However, it is simple to show that if X is defined over an intermediate field L, then L contains what is known as the field of moduli of X, so the field of moduli is a natural candidate for a minimal field of definition. We will introduce the field of moduli, discuss some sufficient conditions for an algebraic object to be defined over its field of moduli, appreciate the generality of the original question, and sample many classes of algebraic objects that can or cannot be defined over the field of moduli.

Speaker: George Brooks

Abstract: For a fixed integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin proved that the maximum $\lambda_1(G)$ among all connected outerplanar graphs on $n$ vertices is achieved by the fan graph $K_1\vee P_{n-1}$. In this talk, we consider a similar problem of determining the maximum $\lambda_2$ among all connected outerplanar graphs on $n$ vertices. For $n$ even and sufficiently large, we prove that the maximum $\lambda_2$ is uniquely achieved by the graph $(K_1\vee P_{n/2-1})\!\!-\!\!(K_1\vee P_{n/2-1})$, which is obtained by connecting two disjoint copies of $(K_1\vee P_{n/2-1})$ through a new edge at their ends. When $n$ is odd and sufficiently large, the extremal graphs are not unique. The extremal graphs are those graphs $G$ that contains a cut vertex $u$ such that $G\setminus \{u\}$ is isomorphic to $2(K_1\vee P_{n/2-1})$. We also determine the maximum $\lambda_2$ among all 2-connected outerplanar graphs and  asymptotically determine the maximum of $\lambda_k(G)$ among all connected outerplanar graphs for general $k$.

Speaker: Swati

Abstract:  Modular forms have played significant roles in some of the most celebrated results in number theory in the last thirty years. For example, they featured prominently in the Wiles’ celebrated proof of Fermat’s Last Theorem in 1995 and more recently, they were crucial to Viazovska’s solution to the “spherepacking problem” in 8 and 24 dimensions which earned her a Fields medal in 2022. In this talk, the goal is to discuss the arithmetic properties of a map called the “Shimura Correspondence,” which is the fundamental link between modular forms of integral and half-integral weights. I’ll wrap up by discussing the work done in collaboration with my advisor in the direction of obtaining explicit formulas for the Shimura correspondence on forms with eta-multiplier.

Speaker: Shreya Sharma

Abstract: Minimal Model Program, MMP in short, has been one of the major discoveries in algebraic geometry in the past few decades. This talk will introduce audiences to the basic ideas of the MMP. We will see how the MMP for 3-folds arises as a generalization of the classification theory of surfaces. Time allowing, we will also see a few examples of the minimal models.

(Halloween Special)

Speaker: Alec Helm

Abstract: Tanglegrams are a combinatorial object which arise in the analysis of phylogenetics and clustering. It is often of interest to determine how nicely a given tanglegram can be drawn, which typically amounts to determining the required number of edge crossings in a drawing. In general, it is a very hard problem to determine the crossing number of a tanglegram, but planarity can be determined relatively easily and there exists a simple characterization of non-planar tanglegrams through critical substructures. In this talk I will provide background on crossing-critical graphs to motivate the known characterization of crossing critical subtanglegrams. Then, I will share some progress and hopes towards extending this result to 2-crossing critical tanglegrams.

Speaker: Bailey Heath

Abstract: Algebraic tori over a field k are special examples of affine group schemes over k, such as the multiplicative group of the field or the unit circle.  Any algebraic torus can be embedded into the group of n x n invertible matrices with entries in k for some n, and the smallest such n is called the representation dimension of that torus.  In this work, I am interested in finding the smallest possible upper bound on the representation dimension of all algebraic tori of a given dimension d.  After providing some background, I will discuss how we can rephrase this question in terms of finite groups of invertible integral matrices.  Then, I will share some progress that I have made on this question, including exact answers for certain values of d.

Speaker: Victoria Chebotaeva

Abstract: We examine the effects of cross-diffusion dynamics in epidemiological models. Using reaction-diffusion dynamics to model the spread of infectious diseases, we focus on situations in which the movement of individuals is affected by the concentration of individuals of other categories. In particular, we present a model where susceptible individuals move away from large concentrations of infected and infectious individuals.

Our results show that accounting for this cross-diffusion dynamics leads to a noticeable effect on epidemic dynamics. It is noteworthy that this leads to a delay in the onset of epidemics and an increase in the total number of people infected. This new representation improves the spatiotemporal accuracy of the SEIR Erlang model, allowing us to explore how spatial mobility driven by social behavior influences the disease trajectory.

One of the key findings of our study is the effectiveness of adapted control measures. By implementing strategies such as targeted testing, contact tracing, and isolation of infected people, we demonstrate that we can effectively contain the spread of infectious diseases. Moreover, these measures allow achieving such a result, while minimizing the negative impact on society and the economy.

Speaker: Chase Fleming

Abstract: Generators for Tychonoff spaces give a way to capture the topology on a space through real valued continuous functions. Every Tychonoff space $X$ has a generator, namely$$C_p(X)$. However, finding non-trivial generators is an interesting task. We show that, with a small restriction, any tree of arbitrary height has a discrete $(0,1)$-generator. 

Speaker: Jonah Klein

Abstract: A covering system is a finite set of arithmetic progressions with the property that each integer belongs to at least one of them. Given a covering system C, we will look at various ways to construct other covering systems that have the same moduli as C. Let η(C) be the set of covering systems with the same moduli as C. We will look at some divisibility conditions on |η(C)|, and how to count |η(C)| for a few examples.

Speaker: Scotty Groth

Abstract: A  Sierpiński  number is a positive odd integer k such that k*2ⁿ + 1 is composite for all n ∈ Z⁺. Fix an integer A with 2 ≤ A. We show there exists a positive odd integer k such that k · aⁿ + 1 is composite for all integers a ∈ [2, A] and all n ∈ Z⁺. This is joint work with Michael Filaseta and Thomas Luckner.