### 2023–2024 Academic Year

**Organized by:**Bailey Heath (jbheath@email.sc.edu) and Pankaj Singh (pksingh@email.sc.edu).

At the Graduate Colloquium, students are provided with a unique opportunity to showcase their research, share their insights, and engage in lively discussions with peers and faculty members. The event serves as a catalyst for intellectual curiosity, encouraging graduate students to explore interdisciplinary connections and refine their research methodologies. It is a space where novel ideas take shape and where academic boundaries are pushed, leading to the emergence of groundbreaking research.

As active researchers, we come across many new ideas and theories which help us in understanding our field in a better way. The seminar presents a great platform to discuss these ideas with our peers and help each other to grow into a better mathematician/Researcher. It also offers an opportunity to bring new problems among the fellow graduate students and faculty members to create an active environment of collaboration.

**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*2^{n} + 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 · a^{n} + 1 is composite for all integers a ∈ [2, A] and all n ∈ Z^{+}. This is joint work with Michael Filaseta and Thomas Luckner.