For the semester of Spring 2026, Alec Helm and I are organizing the Graduate Colloquium at the University of South Carolina. Below, you can find a schedule for the talks this semester.
Title: How to Define a Graph Property
Speaker: Alec Helm
Date, Time, and Location: Thursday, January 29th, 12-1pm, LC 440
Abstract: Greetings, sailors! This Thursday at 1200 we will be departing on this year's inaugural voyage of the good ship Colloquium. Our cargo will be sourced from the class of graph properties, each of which is itself a class of graphs. Before we set sail, our port captain will show us how to equip each of our graph properties with several partial orders in accordance with maritime law. The ship's manifest reports that our freight is finite and downward closed with respect to each partial order. Once this is cleared through customs, we set out for Port Robertson-Seymour where we will obtain a well-quasi-order for all our cargo. Be warned however, that after this stop we will dim the lights and covertly drop anchor in the Topological Bay to acquire some contraband. We will do our best to fabricate something close to a well-quasi-order for this new haul, but our forgery may not withstand scrutiny. Time permitting, a daunting journey will be made into the open seas, where traders deal in infinite properties, with infinite antichains and descending chains.
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Title: No MIS missed
Speaker: Isaiah Hollars
Date, Time, and Location: Thursday, February 5th, 12-1pm, LC440
Abstract: Let $\tau(G)$ denote the size of a smallest set of vertices which shares some vertex with every maximal independent set (MIS) in $G$. In 1992, Erdős, Gallai, and Tuza conjectured that if every MIS contains at least $cn$ vertices, then $\tau(G)=o(n)$. The problem remains open. In this talk, we discuss the partial progress made on a related conjecture and possible strategies for attacking this conjecture. Along the way, we look at several interesting graphs including Alon's arrow graph, random split graphs, and stochastic block model graphs. Many of our arguments and examples make basic use of probabilistic methods.
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Title: On character sums of all shapes and sizes
Speaker: Aditya Iyer
Date, Time, and Location: Thursday, February 12th, 12-1pm, LC440
Abstract: The story I'm about to tell you is one of conflict and harmony, structure and chaos, and the origins of analytic number theory. The main characters (pun intended) of our story are multiplicative group characters — homomorphisms from multiplicative abelian groups (usually groups of the form (R/p)^\times for some suitable ring R, usually one of interest to number theorists, and p a prime ideal in the ring). These maps come from sets with lots of underlying structure but seem to enjoy fighting among themselves to the point where sums of characters don't grow too large in size, implying an inherent randomness. We'll discuss short character sums (and what "short" really means here) and why they're fun.
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Title: Bounding Circuit Size
Speaker: Henry Simmons
Date, Time, and Location: Thursday, February 19th, 12-1pm, LC440
Abstract: Boolean circuits are a model for defining sets, Boolean functions, and computation. Minimizing the number of vertices (logic gates) in a circuit is a natural measure of its complexity. We can define the complexity class P/Poly as the class of binary strings which are decidable by a polynomial sized circuit. This complexity class is different from P and NP but closely related. In many ways circuits are easier to work with than Turing Machines; for example, the 3-SAT problem is defined with Boolean circuits and can be widely used to show other problems are NP-Complete. Today we will talk about how to find a small circuit for a given Boolean function. Then, we will discuss how bounding the size of circuits could help us decide P vs. NP.
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Title: Edge Distributions in Gallai Colorings of Complete Graphs
Speaker: Luke Hawranick
Date, Time, and Location: Thursday, February 26th, 12-1pm, LC440
Abstract: A Gallai coloring of a graph is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1 \ge e_2 \ge \ldots \ge e_k$ with $\sum_{i=1}^k e_i = \binom{n}{2}$, for some $n$, does there exist a Gallai $k$-edge-coloring of $K_n$ with $e_i$ edges in color $i$? In $2019$, Gy\'arf\'as, P\'alv\"olgyi, Patk\'os, and Wales showed that for any fixed $k$ and sufficiently large $n$, every such edge distribution is achievable. They also established bounds on the smallest $n=g(k)$ where this is the case: $2k-2 \le g(k) \le 8k^2+1$. In this talk, I'll discuss these results and describe strategies to tighten the existing bounds on $g(k)$.
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Title: Les Liaisons Dangereuses
Speaker: AJ Greene
Date, Time, and Location: Thursday, March 5th, 12-1pm, LC440
Abstract: Liaison or linkage theory studies invariant properties shared by related ideals. This area is common ground for those interested in Algebraic Geometry and Commutative Algebra; though, the language of the latter will be used throughout the talk.
Ideals related, or ‘linked,’ to complete intersections are a main focus
in linkage theory because of the invariant qualities held by complete intersections. These ideals are called licci; rather, they are in the linkage
class of a complete intersection.
Another family of ideals of arguably almost as much interest are almost complete intersections. In this talk, we will cover appropriate definitions from commutative algebra and what it means for two ideals to be linked. We will then give proof that all monomial almost complete intersections are licci and in fact directly linked to a complete intersection.
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Title: Symbolic Methods in Combinatorics
Speaker: Summer Southwood
Date, Time, and Location: Thursday, March 19th, 12-1pm, LC440
Abstract: Often times when counting objects, it can be useful to encode information in power series known as generating functions. In this graduate colloquium, we will discuss how symbolic methods in combinatorics can help provide a general framework for working with and constructing generating functions. To do so, we will use objects known as combinatorial classes, which are defined as finite sets with a size function. We then define operations on these classes from which we will be able to construct more classes and derive information about the generating function of a class. We then apply these results to various objects such as graphs and trees, giving a "surprising" result for increasing trees. (author's opinion).
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Title: Really Big Sets: A survey of the world's alphabets
Speaker: Benji Dial
Date, Time, and Location: Thursday, March 26th, 12-1pm, LC440
Abstract: On November 29th, 1873, Georg Cantor would send Richard Dedekind a letter asking whether there is a bijection between the set of positive real numbers and the set of positive integers. Dedekind would reply that he did not know, adding that the question "did not deserve too much effort because it has no particular practical interest." In 1874, Cantor would publish a proof that there is no such bijection, opening the rich world of infinite cardinality and enabling untold amounts of illegal thought.
Our goal today will be to survey a couple alphabets from around the world. We will do this by constructing larger and larger ordinals and cardinals, and then seeing what set theorists have decided to name them. Today's talk will include: Arabic numerals, Greek letters, Hebrew letters, ordinals, cardinals, transfinite induction / recursion, ordinal arithmetic, and cardinal arithmetic. Today's talk will not include: Cyrillic letters, any non-alphabetic scripts, a definition of the word "set," or any particular practical interest.
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Title: TBA
Speaker: Sebas de Vega Potts
Date, Time, and Location: Thursday, April 2nd, 12-1pm, LC440
Abstract: TBA
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Title: TBA
Speaker: Joshua Lowrance
Date, Time, and Location: Thursday, April 9th, 12-1pm, LC440
Abstract: TBA
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Title: TBA
Speaker: Declan Liming
Date, Time, and Location: Thursday, April 16th, 12-1pm, LC440
Abstract: TBA
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Title: TBA
Speaker: Matthew Booth
Date, Time, and Location: Thursday, April 23rd, 12-1pm, LC440
Abstract: TBA
For the semester of Fall 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 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.
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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.
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Title: Hedetniemi's homomorphisms
Speaker: Isaiah Hollars
Date, Time, and Location: Thursday, September 18th, 1-2pm, LC440
Abstract: Hedetniemi conjectured in 1966 that $\chi(G\times H)= \min\{\chi(G), \chi(H)\}$ for all graphs $G$ and $H$. The conjecture was recently disproved by Shitov in 2019. We will begin with an introduction to graph homomorphisms, graph products, and the exponential graph. After proving some preliminary results about these objects, we'll be able to give a proof sketch for Shitov's counterexample. We will conclude by discussing related open problems.
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Title: On the damped Euler-Monge-Ampere equations with radial symmetry: critical thresholds and large time behavior.
Speaker: Albert Luan
Date, Time, and Location: Thursday, September 25th, 1-2pm, LC440
Abstract: We study the global well-posedness of the pressureless Euler–Monge–Ampère (EMA) system with linear damping in multiple dimensions, under radially symmetric initial data. We compare this system with the Euler–Poisson equations and examine the critical threshold phenomenon: subcritical initial data lead to global regularity, while supercritical data cause finite-time singularity formation. In particular, we construct an explicit and sharp critical threshold, recovering several earlier results as special cases under corresponding assumptions.
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Title: The hardest race to a million dollars
Speaker: Jesse Singh
Date, Time, and Location: Thursday, October 2nd, 1-2pm, LC440
Abstract: In the 19th century, the works of Navier and Stokes were unified to describe the flow of a viscous, incompressible fluid. These were aptly named the Navier-Stokes equations, a PDE that remains challenging to analyze in the 3D setting. The Clay Millenium Prize offers a $1 million reward for breakthroughs in not just the existence of smooth solutions, but the problem of characterizing whether finite time singularities form given initial data with certain regularity. In this talk, we will outline the problem concerning regularity and finite time blowup as well as phenomena which are of interest in this problem. Then we will talk about advancements towards the conjecture, from Leray, Šverák, Tao, and others. Additionally, we will discuss the rise of computer assisted approaches to unravelling similar problems, and their attempts at the Millennium Problem. Maybe we’ll even resolve this elusive problem at the end of the talk.
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Title: Mean Graphs Drawn Nice
Speaker: Alec Helm
Date, Time, and Location: Thursday, October 16th, 1-2pm, LC440
Abstract: A graph drawing is an assignment of vertices to points and edges to continuous curves whose endpoints are the points assigned to its incident vertices. In general, drawings can be very messy objects: for example, edge curves can intersect in a continuum of points and even a single edge curve can fill a subspace of positive Lebesque measure. In this talk, we will discuss several 'niceness' conditions which are typically employed in the context of finite graphs and show that every finite graph has a 'nice' drawing which achieves its crossing number. Then, we will turn our attention to mean graphs, defined in the natural way. Many of these graphs admit nice crossing-optimal drawings, but some can be shown to not admit any nice drawings.
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Title: Did you know that there are infinitely many primes?
Speaker: Jonah Klein
Date, Time, and Location: Thursday, October 23rd, 1-2pm, LC440
Abstract: In this talk, we will show that there are infinitely many primes. Then, we will show that there are infinitely many primes. Afterwards, to complement the previous proofs, we will show that there are infinitely many primes. Finally, if time permits, we will show that there are infinitely many primes.
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Title: Exiting the Cave
Speaker: AJ Greene
Date, Time, and Location: Thursday, October 30th, 1-2pm, LC440
Abstract: Midway upon the journey through Graph and Set theory, I found myself with an equality dark. An equivalence---nay an answer---which led to more queries; it stokes an unquenchable flame casting shadows cavorting. Unshackled, I've emerged, explored, and returned no wiser than when I first left.
In this talk, I sing of a graph and a prime; a contrarian donning many masks. We will tease apart notation at the intersection of many branches of math in hopes to find deeper meaning and closing the longstanding gaps betwixt them. This will be less of a talk and closer to a group expedition.
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Title: Very Brief Introduction to Topological Vector Spaces
Speaker: Sebastian de Vega Potts
Date, Time, and Location: Thursday, November 6th, 1-2pm, LC440
Abstract: Thank you for accepting my invitation to this Analytical Dinner. Fear not, for the meal will be perfectly balanced for all appetites. We begin with an appetizer of Topological Vector Spaces. It may not be to everyone's taste, as it includes one ingredient not present in all recipes, but for the main entree we will explain the flavor of this ingredient. As a dessert, we will talk about seminorms and the Minkowski Functional.
Too cryptic? Worry not. I do not want any moles going around.
Oh? What's this? I have just been informed that there will be, in fact, a lot of moles in this talk. Be prepared or you might also get absorbed.
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Title: A Pleasure, Professor Lefschetz
Speaker: Matthew Booth
Date, Time, and Location: Thursday, November 13th, 1-2pm, LC440
Abstract: This is a talk with modest aims to answer a seemingly modest question: what is the \textit{Weak Lefschetz Property}, or \textit{WLP} for short? (Do not mistake me for an analyst, however...``Lipschitz'' and his ilk will have no place in this talk.) Our story will begin in the land of multivariable polynomial rings, which we will quotient by a homogeneous ideal. Each graded piece of this quotient is a finite-dimensional vector space, and what else is one to do but study maps between them? In particular, we will ask if a specific linear transformation has maximal rank when applied to any of the graded pieces. Our attention then turns to examining some tools that assist us with this answering this question along with surveying some examples and known results. (If time permits, I may even offer a few words about some of my recent work!) Above all, I hope this talk will serve as a gentle (as best as I can make it) introduction to the WLP.
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Title: A Brief Introduction to Quantum Physics.
Speaker: Ethan Mahan
Date, Time, and Location: Thursday, November 20th, 1-2pm, LC440
Abstract: In this talk, we give an accessible introduction to quantum physics. After a brief look at the history and motivations of the subject, we will explore some of the fundamental properties of the wavefunction, and conclude with some solution techniques for the Schrodinger equation for simple potentials.
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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.
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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.
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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}.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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!
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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.
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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.
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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.