Math Monday
The Math Monday undergraduate lecture series is the flagship event of MUSA. It is a series of talks, every Monday at 5 PM in Evans 1015, given by professors and other academics about mathematical research and special topics.
You can find older Math Mondays at the archive page.
Upcoming Math Monday: A basis of the alternating diagonal coinvariants
Yuhan Jiang
December 1
Abstract: We construct an explicit vector space basis in terms of bivariate Vandermonde determinants for the alternating component of the diagonal coinvariant ring \(DR_n\), answering a question of Stump. As a Corollary, we recover the combinatorial formula of the \(q,t\)-Catalan numbers. Moreover, we construct a decomposition of an \(m\)-Dyck path into an \(m\)-tuple of Dyck paths such that the area sequence and bounce sequence of the \(m\)-Dyck path are entrywise the sum of the area sequences and bounce sequences of the Dyck paths in the tuple.
Fall 2025
An Introduction to ART with X-ray Tomography
Ryan Hass
November 17
Abstract: X-ray tomography is used to recover a function from its line integrals and is directly related to medical imaging. An alternative to popular filtered-backprojection methods, the Algebraic Reconstruction Technique (ART) provides a robust, iterative reconstruction algorithm that is based on Kaczmarz’s method for solving linear systems. We will begin by reviewing the X-ray transform, describing a few scanning geometries used in Computed Tomography, and then have a small diversion on computing the X-ray transform of the unit square in \(\mathbf{R}^2\).
Hex
Felix Weilacher
November 10
Abstract: Hex is an abstract board game invented independently in the 1940’s by Piet Hein and John Nash. In it two players take turns placing stones in a hexagonal grid attempting to form a path between two opposite sides of a quadrilateral. The main mathematical theorem of interest about this game is that there are no ties; given a fully colored game board, exactly one player has a winning path. Much of the appeal of Hex lies not in this theorem itself but in the surprising variety of seemingly unrelated fields in which it shows up. We will prove the Hex theorem and highlight a connection to topology due to David Gale. We then discuss some variants of the game in other topological spaces and their implications.
The strange persistence of Dynkin diagrams
Vera Serganova
November 3
Abstract: I will talk about special graphs called Dynkin diagrams which appear in different areas of mathematics. We will begin with quivers and their representation theory, where Dynkin graphs correspond to finite type quivers. Then we will discuss roots systems and Coxeter groups. If time permits I will talk also about Kleinian singularities and Dynkin graphs.
Complex and p-adic uniformization
Avi Zeff
October 27
Abstract: A frequent theme in number theory is that "global" information about a number field, such as the rational numbers, can be studied in terms of its associated "local" fields, such as the p-adic numbers or the real or complex numbers. One way in which this principle manifests is by various kinds of "uniformization"; we will study some examples. An interesting feature here is that we often need to pick a distinguished "place" to use as our viewpoint, and we will try to understand how this choice affects the result.
Fermat's Last Theorem
Kenneth Ribet
October 20
Abstract: I was led to speak about Fermat's Last Theorem because a bunch of my Math 110 students asked me to explain my role in the proof of the theorem, which was completed in 1994. My talk will give an overview of the subject and its history but will contain even less hard mathematics than the previous Math Monday talks that I've given over the years. I hope that it will be fun for all of us!
Phase transitions
Vilas Winstein
October 13
Abstract: I’ll briefly discuss my experience with the Fulbright program at Budapest Semester in Mathematics/Rényi Institute and how I found my direction in research through making expository math videos. I’ll then give an introduction to phase transitions including a demo of an interactive simulation, and finish by talking about some of my current research on fluctuations in random graph models inspired by statistical mechanics
Integral formulas for the divergence operator and its generalizations
Sung-Jin Oh
October 6
Abstract: We will start with the elementary question: what does the divergence tell you about a vector field? In the first part of the talk, I will give an answer to this question by writing down formulas for "good" vector fields with a given divergence. The inspiration will come from the idea of duality in linear algebra, combined with the fundamental theorem of calculus on curves. Next, I will describe how this procedure generalizes to a large class of important under-determined differential operators, such as the linearizations of the scalar curvature in Riemannian geometry and the compatibility equations for initial data sets in general relativity.
Black hole thermodynamics
Ryan Unger
September 29
Abstract: One of the central principles of modern theoretical physics is the belief that black holes are thermodynamic objects. This principle was first formulated by Bardeen, Carter, and Hawking in 1973 as the “four laws of black hole thermodynamics,” which are precise mathematical statements within the framework of Einstein’s theory of general relativity. In this talk, I will introduce the basic concepts of general relativity, the theory of black holes, and the laws of black hole thermodynamics. I will then explain how recent advances in mathematics have led to the fall of the third “law” of black hole thermodynamics.
Irrationality and transcendence of numbers
Yunqing Tang
September 22
Abstract: In this talk, I will discuss some classical and recent results on irrationality of interesting numbers arising from number theory and geometry, including the recent joint work of Calegari, Dimitrov and myself on certain products of log values. I will also explain some famous conjectures on irrationality and transcendence of numbers coming from geometry. No prerequisite knowledge on number theory is needed for this talk.
Infinite Games
Antonio Montalbán
September 15
Abstract: Infinite two-player games have been a very useful tool to prove many results in logic and other areas. What makes them fascinating is the beautiful regularity properties one can derive from them, creating tension with unintuitive consequences of the axiom of choice. In the first half of the talk, we will describe these games and talk about their connection to the axiom of choice. In the second part of the talk, we introduce the necessary background to understand the answer-given by the author and Richard Shore-to the following question: How much determinacy of games can be proved without using uncountable objects?
From numerical semigroups to Riemann surfaces
David Eisenbud
September 8
Abstract: A numerical semigroup is a subset of the non-negative integers closed under addition. Despite the simplicity of the definition, there are many open problems both combinatorial and algebraic. I'll explain a few, and finish with an unsolved problem, posed by Hurwitz in 1892, from my current research.
Spring 2025
The Poincare-Birkhoff fixed point theorem
Rohil Prasad
April 28
Abstract: I'll discuss a classical theorem by Poincare and Birkhoff concerning maps from the annulus (a region between two concentric circles in the plane) to itself. If the map is an area-preserving homeomorphism and twists the two boundary circles in different directions, then it has a fixed point. I'll sketch the proof, which is very hands-on and topological.
Prerequisites: Some knowledge of topology is useful but not required.
When and why do efficient algorithms exist (for constraint satisfaction and beyond)?
Venkatesan Guruswami
April 21
Abstract: Some computational problems can be solved quickly; others seem hopelessly intractable. It would be simplistic to hope for a universal theory explaining the underpinnings of easiness and hardness, yet in the realm of constraint satisfaction problems (CSPs), there's a fascinating answer involving operations called polymorphisms. This talk will discuss CSPs and the polymorphic approach. We'll also explore extensions of the polymorphic approach beyond CSPs, to include problems such as approximate graph coloring, a “promise CSP."
Prerequisites: mathematical maturity and a basic familiarity with algorithms.
Finite topological spaces know everything
John Lott
April 14
Abstract: There's a notion of a topology on a set, including finite sets. I'll describe a surprising result saying that for a sphere of any dimension (or more generally any finite simplicial complex), there's a finite topological space with the same topological invariants.
Prerequisites: Math 104
From surfaces to algebras, and beyond
Jasper van de Kreeke
March 31
Abstract: In this talk, we will explore a connection between geometry and higher algebra through the study of surfaces with punctures. By considering systems of arcs on these surfaces, we can construct a specific type of algebra called a "gentle algebra." This idea plays a key role in mirror symmetry, a fundamental concept in modern mathematics. On the technical side, we will discuss categories and potentially how to upgrade them to \(A_\infty\)-categories, providing a powerful framework to capture surface topology in an algebraic way. We will provide an intuitive explanation of the construction, as well as some key results and insights.
Prerequisites: Math 110, 113, some topology
A different kind of fixed point theorem
Patrick Lutz
March 17
Abstract: Fixed point theorems are ubiquitous in mathematics. Often, to construct a fixed point of a function \(f\), one iterates \(f\) and takes a limit. However, there is a family of fixed point theorems in mathematical logic which are proved using self-reference rather than iteration. We will discuss how this sort of fixed point theorem is proved and how to view Russell's paradox as a fixed point theorem of this sort.
Prerequisites: Math 55
Quantum Hamiltonian Descent Algorithms for Nonlinear Optimization
Jiaqi Leng
March 10
Abstract: Nonlinear optimization is a vibrant field of research with wide-ranging applications in engineering and science. However, classical algorithms often struggle with local minima, limiting their effectiveness in tackling nonconvex problems. In this talk, we explore how quantum dynamics can be exploited to develop novel quantum optimization algorithms. Specifically, we introduce Quantum Hamiltonian Descent (QHD), present an open-source implementation of the algorithm (named QHDOPT) and demonstrate its real-world applications.
Crystals!
Nicolle González
March 3
Abstract: A basic question in linear algebra is finding the coefficients of a vector in a given basis. A fundamental goal in combinatorics is finding formulas for these coefficients when our vector spaces are certain polynomial rings. Questions like these can be answered using certain graphs, known as “crystals.” These crystals contain more information than the original polynomials, which can be seen as "shadows" of the crystals. In this talk we will introduce crystals from scratch and with lots of examples.
A different kind of fixed point theorem (cancelled)
Patrick Lutz
February 24
Abstract: Fixed point theorems are ubiquitous in mathematics. Often, to construct a fixed point of a function \(f\), one iterates \(f\) and takes a limit. However, there is a family of fixed point theorems in mathematical logic which are proved in a very different way, using self-reference rather than iteration. We will discuss how this sort of fixed point theorem is proved and applied and consider a few specific examples. Among other things, we will see how to view Russell's paradox as a fixed point theorem of this sort.
Geometry over the complex numbers and finite fields
Noah Olander
February 10
Abstract: We will compare complex solutions of polynomial equations with solutions in finite fields, and use this to reduce questions about complex geometry to questions about finite sets.
When are two objects equal?
Martin Olsson
February 3
Abstract: One of the key ideas in modern mathematics is the notion of category. In this talk I will discuss some examples that motivate this notion focusing on the distinction between equality vs isomorphism. The key point is that a perspective focused on maps between objects, rather than the objects themselves, can sometimes reveal symmetries and structures we would otherwise not see.
Newton's Principia
Richard Borcherds
January 27
Abstract: Newton's Principia is one of the most famous math books ever written, but few mathematicians have read it. This talk will discuss some of its mathematical results.
You can find older Math Mondays at the archive page.