VT Mathematics Colloquium

Fridays 4:00-5:00 PM Eastern Time

McBryde 455

February 7, 2025

Speaker: Paata Ivanisvili (UC Irvine)

Title: Discrete approximation theory

I will discuss polynomial approximation problems on the n-dimensional hypercube, focusing on quantitative estimates of the approximation error as n grows. This topic represents a burgeoning area in the analysis of Boolean functions—one that is far less understood than its classical counterpart on the real line. I will present several recent results together with its applications, as well as highlight ongoing challenges and open problems in the field.
February 14, 2025

Speaker: Luis Núñez-Betancourt (CIMAT)

Title: Singularities of Polynomials with Square-Free Support

A polynomial over a field is square-free supported if every variable appearing in it has degree at most one. Interest in such polynomials arises from matroid support polynomials, studied by Bath and Walther. In this talk, we will provide an overview of the study of singularities in prime characteristic via the Frobenius map. We will show that the algebraic variety defined by an irreducible polynomial with square-free support has mild singularities, answering a question posed by Bath, Mustață, and Walther. We will also discuss the consequences of these results, including their implications in characteristic zero.
February 28, 2025

Speaker: Jake Fillman (Texas A&M)

Title: Schrödinger operators with thin spectra

The Schrödinger equation is the foundational equation of nonrelativistic quantum mechanics. Studying the time-independent part leads to the eigenvalue equation of a Schrödinger operator, whose spectra contain a wealth of information about the underlying quantum system. We will survey some results about Schrödinger operators having exotic properties, such as spectra that can be "thin" Cantor sets.
March 21, 2025

Speaker: Phan Thành Nam (LMU Munich)

Title: Isoperimetric Inequalities and Binding Problems in Atomic Physics

I will discuss the connection between the classical isoperimetric inequality and the critical mass problem in nuclear fission, which is described by the liquid drop model. Techniques developed for this problem are also helpful in establishing certain universality results for large atoms, particularly in relation to the ionization problem in quantum physics: How many electrons can a nucleus bind?

March 28, 2025

Location: 113 McBryde

Speaker: Lillian Pierce (Duke)

Title: Superorthogonality

How do we check if two vectors are orthogonal? We compute their dot product, which by definition takes two vectors as inputs. How do we check if two functions are orthogonal? We compute their inner product, which by definition takes two functions as inputs. Why only two? What would it mean for 4 functions to be “orthogonal”? Or 8 functions? Or 7 functions? Let’s call this superorthogonality. What can we deduce about collections of functions that are superorthogonal? In this accessible talk, we will explore how accidental encounters with papers spanning 90 years led to a systematic investigation of these questions, and a way to see that previously “unrelated” theorems in harmonic analysis and number theory share a very interesting structure deep under their surface.
April 4, 2025

Speaker: Alexander Barg (U Maryland)

Title: Classical and quantum codes on Coxeter groups

Classical Reed-Muller codes form a well-known family of binary codes on the cubical complex, which also gives rise to a family of quantum codes. In the first part of the talk we extend the construction of the classical RM family to binary codes defined on an arbitrary finite Coxeter complex and establish basic properties of the new code class. In the second part we present a geometric description of a set of transversal logical operators of quantum RM codes. The talk assumes no background in coding theory or quantum computations, developing the constructions from the first principles. We will define classical and quantum codes and discuss the motivation for the construction of their logical operators. Based on joint works with Nolan Coble, Dominik Hangleiter, and Christopher Kang.
April 11, 2025

Speaker: Ken Ono (University of Virginia)

Title: Integer partitions detect the primes

This talk presents “partition theoretic” analogs of the classical work of Matiyasevich that resolved Hilbert’s Tenth Problem in the negative. The Diophantine equations we consider involve equations of MacMahon’s partition functions and their natural generalizations. Here we explicitly construct infinitely many Diophantine equations in partition functions whose solutions are precisely the prime numbers. To this end, we produce explicit additive bases of all graded weights of quasimodular forms, which is of independent interest with many further applications. This is joint work with Will Craig and Jan-Willem van Ittersum, and this paper was awarded Runner-Up recognition for the 2025 Cozzarelli Prize by the US National Academy of Sciences.
April 18, 2025

Speaker: Kreso Josic (University of Houston)

Title: Fast Decisions Reflect Initial Biases, While Slow Decisions Do Not

Drift-diffusion models are widely used to model how humans and other animals make decisions. Such models describe how the accumulation of uncertain evidence results in a choice. I will show how extending these models to social groups can give some interesting insights into collective decisions of rational agents. For instance, the order in which decisions are made can strongly impact their accuracy: In large groups the first agents to decide almost always hold the strongest initial bias and decide accordingly. Slow agents, conversely, decide as if they held no initial bias. When agents receive correlated evidence, decision accuracy depends on decision order in the absence of initial bias. Since these are rational agents using the same decision criterion, they are all equally confident in their decisions even when their accuracy differs dramatically. Although these are idealized models, our analysis offers general insights about the quality of decisions in groups.
April 25, 2025

No Colloquium

May 2, 2025

Speaker: Benjamin Peherstorfer (New York University)

Title: DICE: Discrete inverse continuity equation for marginal trajectory matching

The aim of this work is to learn models of population dynamics of physical systems that feature stochastic and mean-field effects and that depend on physics parameters. The learned models can act as surrogates of classical numerical models to efficiently predict the system behavior over the physics parameters. Building on the continuity equation, we use a variational problem to infer parameter- and time-dependent gradient fields that represent approximations of the population dynamics. The inferred gradient fields can then be used to rapidly generate sample trajectories that mimic the dynamics of the physical system on a population level over varying physics parameters. We show that a judicious discretization in time is critical for accurately estimating the training objective from sample data and for stabilizing the training process. We demonstrate on Vlasov-Poisson instabilities as well as on high-dimensional particle and chaotic systems that our approach accurately predicts population dynamics over a wide range of parameters and outperforms state-of-the-art diffusion-based and flow-based modeling that simply condition on time and physics parameters.