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.

Slides from some recent Math Mondays can be found here.

You can find older Math Mondays at the archive page.


Upcoming Math Monday: 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.


Spring 2025

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.


Fall 2024

Chromatic symmetric functions

Sylvie Corteel
December 9

Abstract: The chromatic symmetric function is a symmetric function invariant of graphs studied in algebraic graph theory, a branch of mathematics. It is the weight generating function for proper graph colorings, and was originally introduced by Richard Stanley as a generalization of the chromatic polynomial of a graph. R. Stanley conjectures that the chromatic symmetric function of a tree uniquely determines the tree. Stanley and Stembridge conjectured in 1993 that the chromatic symmetric function of unit interval graphs are e-positive. This conjecture was proven in October 2024 by T. Hikita.


How not to prove the Riemann hypothesis, or, The zeta function as a scattering matrix

Maciej Zworski
December 2

Abstract: I will explain definitions and basic properties of two seemingly unrelated objects: the Riemann zeta function of number theory and the scattering matrix of quantum mechanics. I will then show that in fact they are related when one considers analysis on the modular surface.


Singularity formation in fluid dynamics

Federico Pasqualotto
November 15

Abstract: In physical models of fluids, a singularity occurs when a quantity of interest (velocity, pressure, vorticity…) becomes infinite at some finite time, starting from a "nice" initial configuration. In this talk, we will first introduce some classical models of compressible and incompressible fluids. We will then describe several mechanisms by which fluids can form singularities in finite time, allowing us to discuss the singularity formation problem for the incompressible three-dimensional Euler equations and the Navier—Stokes equations. We will finally touch upon computer-assisted methods and their applications to singularity formation in fluids. We will only assume familiarity with multi-variable calculus and some concepts of real analysis.
Prerequisites: Multi-variable calculus is enough. Intro to real analysis is preferable.


Interactive proofs

John Wright
Novermber 4

Abstract: What is a proof? In principle, we think of a proof as a sequence of axioms and inference rules that allow us to derive a mathematical statement. In practice, it's something that typically appears in a textbook or a math journal. In this talk, I'll discuss a different philosophy coming from theoretical computer science, in which a proof can be anything which convinces you. This opens the door to proofs having a variety of different attributes, including being interactive, zero-knowledge, and even quantum.


Symplectic embeddings and infinite staircases

Nicki Magill
October 28

Abstract: When mailing an object, a question arises: Can I fit this object into this particular box? The answer depends on which aspects of the object we want to preserve when placing it in the box. For example, different constraints apply when shipping a foam ball compared to a wooden one. For some objects, we might only care about preserving their volume, while others may be more rigid. In this talk, we will focus on preserving the symplectic structure, which can be quite flexible at times and rigid at others. Determining whether one object can fit into another while preserving the symplectic structure is often difficult to answer. Our focus will be on embedding four-dimensional ellipsoids. We will explore the intricate structure that determines when these symplectic ellipsoids can be embedded into other four dimensional objects.
Prerequisites: None!


L-functions and integrals

Danielle Wang
October 21

Abstract: We discuss how the Riemann zeta function, Dirichlet \(L\)-functions, and \(L\)-functions of modular forms are defined and analyzed by expressing them as Mellin transforms (a type of integral). This kind of relationship between \(L\)-functions and integrals has generalizations that are currently of interest in the relative Langlands program.
Prerequisites: I think really only things from calculus, like integrals on \(\mathbb R\) (except the functions are complex-valued), the idea of series/integrals converging, Taylor series. Some complex analysis feels like it would be nice, but shouldn't be necessary (at least in my mind). I will define holomorphic function, we won't do any contour integrals.


Finding and creating periodic orbits

Michael Hutchings
October 14

Abstract: A dynamical system can be described by a vector field. A periodic orbit is a trajectory of the vector field which starts and ends at the same point. This corresponds to repeating behavior in the dynamical system, such as a planet orbiting around a star. In general there are many open questions regarding when periodic orbits exist and what properties they have. This talk will introduce some recent results about existence of periodic orbits of certain vector fields in three dimensions.
Prerequisites: The beginning of the talk should be understandable to anyone who knows what a vector field is, e.g. from Math 53, while later in the talk we will touch on some more sophisticated notions, but without getting into technical details.


Kazhdan-Lusztig polynomials, combinatorial invariance, and machine learning

Christian Gaetz
October 7

Abstract: Kazhdan-Lusztig polynomials are an important family of polynomials with both algebrogeometric and representation-theoretic interpretations. I'll explain how to compute these polynomials and introduce the Combinatorial Invariance Conjecture (CIC), which asserts that they only depend on a certain directed graph. I'll then discuss work of Blundell, Buesing, Davies, Veličković, and Williamson, who used machine learning techniques to conjecture some recurrences which would imply the CIC. I'll end by mentioning some work of mine, joint with Grant Barkley, in which we use this approach to prove some new cases of the CIC.


The hardness of classification

Forte Shinko
September 30

Abstract: In mathematics, we usually talk about objects up to a notion of equivalence, such as groups up to isomorphism, manifolds up to homeomorphism, and so on. We typically use invariants to distinguish these objects; for instance, the groups \(\mathbb{Z}\) and \(\mathbb{Z}^2\) are not isomorphic, because "the size of the smallest generating set" is 1 in the former case, and 2 in the latter. In very nice situations, there are complete invariants which completely classify a given class of objects up to equivalence, such as for closed surfaces, which are homeomorphic iff they have the same genus. However, in many situations, there do not seem to exist such invariants, and we would like a formal way to say what it means for a nice classification to exist. We will use the language of descriptive set theory to formalize the notion of classification in terms of Borel equivalence relations, and then show that many natural classification problems have no complete invariants, or even much weaker versions of such.
Prerequisites: It will help to have some background in linear algebra and group theory.


Quantum advantage in scientific computation?

Lin Lin
September 23

Abstract: The advent of fully error-corrected quantum computers is anticipated to usher in a new era in computing, with Shor's algorithm poised to demonstrate practical quantum advantages in prime number factorization. However, cryptography problems are typically not categorized as scientific computing problems. This raises the question: which scientific computing challenges are likely to benefit from quantum computers? I will first explore some essential criteria and considerations to realize quantum advantages in these problems. I will then introduce some recent advancements in quantum algorithms related to scientific computing applications. At least the first half of the presentation is intended to be accessible to a broad audience.


Squaring the Circle

Andrew Marks
September 16

Abstract: In 1925 Tarski posed the problem of whether a solid two dimensional square can be partitioned into finitely many pieces which can be rearranged to form a disk of the same area. We will explain the history of this problem, and the ideas used in its 1990 solution by Laczkovich. We will emphasize tools for converting between solutions to combinatorial problems on finite vs infinite graphs, and also the study of effective bounds on the ergodic theorem. At the end of the talk, we will discuss some recent work joint with Spencer Unger giving better upper bounds on the number of pieces needed to square the circle. This uses results from diophantine approximation in number theory. Diophantine approximation studies how well irrational numbers can be approximated by rationals. A landmark result in this area is Roth's theorem on diophantine approximation of algebraic irrationals, for which Roth won the Fields medal in 1958. It is still an open problem if the constants in Roth's theorem can be computed effectively. We will explain how recent partial results on this problem (giving effective irrationality exponents for the cube root of 2) can be used to improve the upper bound on the number of pieces used to square the circle.
Prerequisites: It may be helpful (but is not required) to know the definition of a group action for the talk.


Sequences and Hidden Structure

Alexander Paulin
September 9

Abstract: A very simple way to generate a sequence of numbers is to fix a function and repeatedly apply it to zero. What do sequences constructed in this manner look like? What happens when we vary the function in a continuous way? In this lecture we will discover that such questions hold vast hidden depth, opening the door to some of the most beautiful mathematics ever discovered.


You can find older Math Mondays at the archive page.