For the semester of Fall 2025, Alec Helm and I are responsible for organizing the Graduate Colloquium at the University of South Carolina. Below, you can find a schedule for the talks this semester.
Title: Functional Countability and the Kunen Line
Speaker: Chase Fleming
Date, Time, and Location: Thursday, September 4th, 1-2pm, LC 440
Abstract: We answer a question posed by Tkachuk in $2021$. A topological space is called functionally countable if, for any real valued continuous function $f:X \to \mathbb{R}$, the image, $f(X)$ is at most countably infinite. We will show that, by assuming the Continuum Hypothesis, that there exists a hereditarily separable, non-Lindelof, regular, first countable, locally compact, locally countable space $X$ of cardinality $\omega_1$ such that $(X+1)^2\setminus \Delta$ is functionally countable. We will cover the construction of the Kunen line and give a direct proof that the space $\omega_{1}^{2}\setminus\Delta$ is not functionally countably, but the square of Alexandroff one-point compactification of a space of size $\omega_1$ minus the diagonal is indeed functionally countable.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Fröberg’s Conjecture
Speaker: Dinesh Limbu
Date, Time, and Location: Thursday, September 11th, 1-2pm, LC440
Abstract: Fröberg’s Conjecture is a fundamental open problem in commutative algebra that predicts the behavior of Hilbert functions of algebras generated by generic homogeneous polynomials. Despite its simple statement, the conjecture remains unresolved in general, with only special cases fully understood. In this talk, I will provide an overview of the conjecture, discuss known results and examples. I will also touch on techniques used to study the problem and outline some open questions.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: TBD
Speaker: Isaiah Hollars
Date, Time, and Location: Thursday, September 18th, 1-2pm, LC440
Abstract: TBD
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: TBD
Speaker: Albert Luan
Date, Time, and Location: Thursday, September 25th, 1-2pm, LC440
Abstract: TBD
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: TBD
Speaker: Jesse Singh
Date, Time, and Location: Thursday, October 2nd, 1-2pm, LC440
Abstract: TBD
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: TBD
Speaker: TBD
Date, Time, and Location: Thursday, October 16th, 1-2pm, LC440
Abstract: TBD
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: TBD
Speaker: TBD
Date, Time, and Location: Thursday, October 23rd, 1-2pm, LC440
Abstract: TBD
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: TBD
Speaker: AJ Greene
Date, Time, and Location: Thursday, October 30th, 1-2pm, LC440
Abstract: TBD
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: TBD
Speaker: Sebastian de Vega Potts
Date, Time, and Location: Thursday, November 6th, 1-2pm, LC440
Abstract: TBD
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: TBD
Speaker: Matthew Booth
Date, Time, and Location: Thursday, November 13th, 1-2pm, LC440
Abstract: TBD
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
For the semester of Spring 2025, Alec Helm and I were responsible for organizing the Graduate Colloquium at the University of South Carolina. Below, you can find an archive of the talks.
Title: On a paper of Serre on a theorem of Jordan
Speaker: Alexandros Kalogirou
Date, Time and Location: January 29th, 2025, 1-2pm, LC 440
Abstract: Using elements of representation theory, we present a proof of a theorem of Jordan for group actions. We relate this to the study of roots of polynomial equations modulo primes. We present the theory needed to decide the average number of solutions, and with the use of examples we motivate the question of finding explicit formulas for the same number. That is a natural setting in which to discuss some basics of class field theory and related topics.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: On a paper of Serre on a theorem of Jordan (Continued)
Speaker: Alexandros Kalogirou
Date, Time and Location: February 5th, 2025, 1-2pm, LC 440
Abstract: Using elements of representation theory, we present a proof of a theorem of Jordan for group actions. We relate this to the study of roots of polynomial equations modulo primes. We present the theory needed to decide the average number of solutions, and with the use of examples we motivate the question of finding explicit formulas for the same number. That is a natural setting in which to discuss some basics of class field theory and related topics.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Something about Cohen-Macaulay...
Speaker: Matthew Booth
Date, Time and Location: February 12th, 2025, 1-2pm, LC 440
Abstract: Commutative algebra is a delight. Problems from all over mathematics can both motivate and find motivation from the theory of rings. It is no wonder that a standard graduate abstract algebra sequence will get students acquainted with \textit{integral domains} and some of its special sub-variants (e.g. UFDs, PIDs, fields). And yet, the banishment of zero-divisors does limit our scope significantly, and indeed something as simple as taking a quotient shows how fragile the integral domain property is. (Just mod the ring $\mathbb{Z}$ by the ideal generated by your favorite composite number!)
Our aim in this talk is to give a flavor of the types of rings one encounters in commutative algebra at large, using examples as much as possible to give an idea of the general theory. Stepping beyond the safety of integral domains, one can examine classes of rings that arise in connection with algebraic geometry and algebraic topology. The most important classes include \textit{regular local rings}, \textit{Gorenstein rings}, and \textit{Cohen-Macaulay rings}. Toward at least defining these rings, we will encounter \textit{regular sequences} and see a little of the interplay between certain ring invariants like \textit{dimension} and \textit{depth}.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Sums of squares and large values of r_3(n)
Speaker: Jonah Klein
Date, Time and Location: February 19th, 2025, 1-2pm, LC 440
Abstract: The problem of determining which integers can be written as a sum of 2, 3, and 4 squares has a long and interesting history. In the first part of this talk, we will go over this history, which includes the work of Fermat, Euler, Legendre, Lagrange, Jacobi, and Gauss, among others.
Let r_3(n) denote the number of ways of writing an integer n as a sum of 3 integer squares. In the second part of this talk, we will investigate the behavior of this function, and outline how, in joint work with Michael Filaseta and Cihan Sabuncu, we showed that this function exceeds its average value by a factor of \gg_m \log\log(n) infinitely often in any suitable arithmetic progression with modulus m.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Finding Small Clique-Transversals for $c$-Thick Graphs
Speaker: Isaiah Hollars
Date, Time and Location: February 26th, 2025, 1-2pm, LC 440
Abstract: In this talk, we explore an open problem posed by Paul Erd\H{o}s in 1991 on finding small clique-transversals. We start by discussing the problem and some important examples of $c$-thick graphs. In the remainder of the talk, we describe techniques for solving a particular case of the problem.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Critical Thresholds in Pressureless Euler-Poisson-Alignment System with Background States
Speaker: Albert Luan
Date, Time and Location: March 5th, 2025, 1-2pm, LC 440
Abstract: In this talk, I will introduce our recent work, joint with Dr. Changhui Tan and Qiyu Wu, on pressureless Euler equations with bounded nonlocal alignment interactions and non-vanishing, variable background states. We examine the critical threshold phenomenon in this system, and we demonstrate that subcritical initial data lead to global-in-time regularity, while supercritical initial data result in finite-time singularity formation. Our analysis accommodates variable alignment forces and background states, providing a unified framework to study these phenomena.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Vajda's Identity for Second-Order Linear Recurrence Relations
Speaker: Joseph Aulenbacher
Date, Time and Location: March 19th, 1-2pm, LC 440
Abstract: This talk is based on undergraduate research done with Dr. Marc Renault during Spring 2024. In this talk, we will make use of matrix methods to provide a proof for two known versions of Vajda's identity for generalized second-order linear recurrence relations. While previous proofs have used the Binet formula or induction, our matrix method sheds light on how these two generalizations are related and how they might arise naturally.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Injective Hull and Matlis duality
Speaker: Dinesh Limbu
Date, Time and Location: March 26th, 1-2pm, LC 440
Abstract: Matlis duality and injective hulls play a fundamental role in commutative algebra and homological algebra, particularly in the study of Noetherian Complete local rings. The injective hull of a module provides a minimal injective extension, which is crucial for understanding the structure of injective modules and their applications in module theory. Matlis duality, a powerful duality theorem, establishes a contravariant equivalence between Noetherian and Artinian modules via the Matlis dual functor, defined using the injective hull of the residue field. This talk will explore the construction and key properties of injective hulls, discuss their role in homological algebra, and illustrate how Matlis duality interchanges Noetherian and Artinian structures.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: A Concise Introduction to Hyperbolic Groups
Speaker: AJ Greene
Date, Time and Location: April 2nd, 1-2pm, LC 440
Abstract: Geometric Group Theory seeks to study finitely generated groups through the topological and geometric spaces on which they act. One may approach from either side by exploring the symmetries of a particular space—-such as the hyperbolic plane—-or building from a group a space on which it acts—-such as its Cayley Graph. In this talk we will discuss a slice of this field focusing on hyperbolic groups. We will go through some equivalent definitions on what makes a group/space hyperbolic. We will conclude with a statement of the Tits-Alternative and a proof of the Ping Pong lemma, both of whom appear frequently in the field.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Strong Generation and Dimensions of Derived Categories.
Speaker: Anirban Bhaduri
Date, Time and Location: April 9th, 1-2pm, LC 440
Abstract: Triangulated categories, and more specifically, Derived Categories have been a popular area of research in Algebraic Geometry and Commutative Algebra. Bounded Derived Categories of coherent sheaves of varieties and stacks is an ongoing area of research. In recent times, beyond commutative algebra, derived categories have been helpful in studying non-commutative algebra and have given rise to our understanding of what can be called non-commutative algebraic geometry. In this talk we discuss the notion of dimension and generation for derived categories. We also look at some interesting results related to dimension and strong generation in some well-known derived categories. Lastly, we discuss some examples arising from non-commutative algebra.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: What If Every Module Was...?
Speaker: Matthew Booth
Date, Time and Location: April 16th, 1-2pm, LC 440
Abstract: Module theory is an incredibly rich source of mathematics. One sees in a standard graduate abstract algebra sequence that modules provide fertile ground to generalize numerous results about vector spaces (e.g. the “rank-nullity theorem”), abelian groups (e.g. the “fundamental theorem of finitely generated Abelian groups”), and ideal theory. This motivates the definition of various kinds of modules. Among the most important are the free, projective, and flat modules, as well as the “dual” to projectives (known as injective modules). Of course, modules do not exist in a vacuum; one first fixes a ring R and examines Abelian groups on which R acts, and it is the connection between R and modules over it that we wish to examine.
In particular, this introductory talk will attempt to address two questions that look in opposing directions. First, for a fixed ring R, if all modules over R are “nice” (e.g. all are free, projective, or flat), what can be said about the ring R? The notions of semisimple and von Neumann rings arise from this pursuit, the former having fundamentally important connections to representation theory. And second, if all the ideals of a given ring R are “nice” in one of the senses described above, what can be said about all modules over that ring? It is here that we will meet hereditary and semihereditary rings, which include as special cases the Dedekind domains and Pr¨ufer domains, respectively.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: The H-Cobordism Theorem & Smooth Structures
Speaker: Bobby Estrada
Date, Time and Location: April 23rd, 1-2pm, LC 440
Abstract: The h-Cobordism Theorem is a result in differential topology that provides a profound connection between the smooth structures of manifolds and their topological properties. It asserts that two smooth manifolds of the same dimension connected by an h-cobordism must be diffeomorphic, i.e. they have 'identical' smooth structures. The talk will focus on the key implications of this theorem, while explaining the significance of Reidemeister’s moves and the topological nature of cobordisms. We will also examine the case of dimension 4, where the theorem's result fails.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Monotonicity of Quantum Entropy
Speaker: Chase Fleming
Date, Time and Location: April 30th, 1-2pm, LC 440
Abstract: In the 1940’s, Shannon proved that the entropy of a statistically normalized sum of two independent, identically distributed random variables is larger than the entropy of one of the random variables. Naturally, in 1978, Lieb conjectured H(\hat{X}_{n+1}) ≥ H(\hat{X}_n) where \hat{X}_n = \sum_{i=1}^n X_i/ \sqrt{n} and {X_i}_{i=1}^\infty is a sequence of i.i.d. r.v.’s. This result was finally proven in 2004, and its quantum counterpart remained open until April, 2025. This talk will present the basics of quantum theory and quantum information theory qua study of entropy, and offer a summary of the known results in the field up until the April paper of Beigi and Mehrabi. We will see the results of the quantum central limit theorem, convergence of quantum relative entropy, and monotonicity of quantum entropy of the n-fold symmetric convolution of states.
For the semester of Fall 2024, Alec Helm and I were responsible for organizing the Graduate Colloquium at the University of South Carolina. Below, you can find an archive of the talks.
Title: Root system and root data of reductive Group
Speaker: Pankaj Singh
Date, Time and Location: September 5th, 2024, 1-2pm, LC 440
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.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Interpolating between Optimal Transport and KL regularized Optimal Transport using Rényi Divergences
Speaker: Viktor Stein
Date, Time and Location: September 12th, 2024, 1-2pm, LC 440
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.
This is joint work with Jonas Bresch (TU Berlin) and available at
(regularized) optimal transport, so that this talk should be
accessible to graduate students and interested undergrads of all
mathematical disciplines. The slides will be available before the talk
at viktorajstein.github.io.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Yoneda Lemma
Speaker: Dinesh Limbu
Date, Time and Location: September 19th, 2024, 1-2pm, LC 440
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.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: The 2nd PDE of humanity; a modern approach
Speaker: Albert Luan
Date, Time and Location: September 26th, 2024, 1-2pm, LC 440
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.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Maximum spectral gaps of graphs
Speaker: George Brooks
Date, Time and Location: October 3rd, 2024, 1-2pm, LC 440
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.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Galois Meets Infinity
Speaker: Matthew Booth
Date, Time and Location: October 10th, 2024, 1-2pm, LC 440
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.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Equivalences to the Riemann Hypothesis.
Speaker: Aditya Iyer
Date, Time and Location: October 24th, 2024, 1-2pm, LC 440
Abstract: The Riemann Hypothesis is a famous conjecture that describes the behaviour 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.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: The Shannon capacity and the Lovasz number
Speaker: William Linz
Date, Time and Location: October 31st, 2024, 1-2pm, LC 440
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.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: How to Get a Job in Academia, or At Least How Victoria Got One
Speaker: Victoria Chebotaeva
Date, Time and Location: November 7th, 2024, 1-2pm, LC 440
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!
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: An Introduction to Reduced Order Modeling for High-Fidelity Dynamical Systems
Speaker: Jesse Singh
Date, Time and Location: November 14th, 2024, 1-2pm, LC 440
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.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Gödel's Completeness Theorem for First-order Logic
Speaker: Benji Dial
Date, Time and Location: November 21st, 2024, 1-2pm, LC 440
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.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Title: Exposed: The Lies of Planar Graphs!
Speaker: Alec Helm
Date, Time and Location: December 5th, 1-2pm, LC 440
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.