Spring 2026

Mathematics Colloquium

Location and Time: Minard 310 at 3:00 PM (Refreshments at 2:30 in Minard 404)

*Special Colloquia or Tri-College Colloquia venues and times may vary, please consult the individual listing.

Tuesday, February 10

Pranav Arrepu, UC Santa Barbara

Recovery of Coefficients in Nonlinear Schrödinger Equations by Carleman Estimates

Abstract: We consider a class of dynamical Schrödinger equations with locally analytic nonlinear terms on a smooth bounded domain and take up the inverse problem of recovering the linear potential and nonlinear interaction coefficients. Assuming the coefficients are known in a given neighborhood of the boundary, we establish unique and stable determination by measurements of the Neumann data on an arbitrary part of the boundary. Our proof relies on a transformation to an associated linear parabolic equation along with a direct application of Carleman estimates for the linear parabolic and Schrödinger equations. This is joint work Hanming Zhou.

Tuesday, February 24

Bridget Tenner, DePaul University

Majority relations: how do ranked ballots shake out?

Abstract: Suppose you have an election in which each voter ranks the full slate of candidates. If we want to draw an aggregated conclusion from all of the ballots cast, what is the "winning" candidate ranking? We will study this question on so-called Condorcet domains of tiling type, which can be defined in terms of rhombic tilings of certain polygons (equivalently, in terms of reduced decompositions of permutations). We can then use heaps and poset theory to show important properties of the majority relation in these domains. We will demonstrate these results by computing the majority relation explicitly for several important classes.

This talk is based on joint work with Vic Reiner.

Tuesday, March 3

Azer Akhmedov, NDSU

Tiles of Groups and Spaces

Abstract: We will discuss various tiling problems in the Euclidean plane, the hyperbolic plane, and in groups. It is not known which finite subsets of the group ℤ (the integers) tile ℤ. We will review some recent results in this direction, including work related to the Fuglede conjecture, which was disproved by Tao in dimension ≥ 5, later by Matolcsi in dimension 4, and by Farkas–Matolcsi–Móra in dimension 3. We will also review results on aperiodic tilings of the hyperbolic plane (by Margulis and Mozes), as well as more recent results on aperiodic tilings of the lattices of Euclidean spaces and in the Euclidean plane (by Greenfeld–Tao and Goodman–Straus–Kaplan–Myers–Smith).

It is unknown whether, for every countable group G and every finite subset K of G, the set K is contained in a tile of G. We prove this for hyperbolic groups. This result was later generalized by MacManus and Mineh to acylindrically hyperbolic groups, which include interesting examples of groups that are not hyperbolic.

Thursday, March 19

Reynold Fregoli, U of Michigan

Orbits, Chaos, and Rational Numbers

Abstract: It is a well-known fact that rational numbers are dense in the real line, but attempting to quantify how well they approximate a given real number can lead to deep and subtle questions. Mathematicians have studied approximation by rational numbers for a long time, making impressive progress, yet many intriguing problems remain open today.

A modern approach to these questions is based on ideas from dynamical systems — an area of mathematics originally inspired by attempts to understand the motion of planets. In this talk, I will provide a gentle introduction to this connection and present a recent result, joint with Prasuna Bandi and Dmitry Kleinbock, that illustrates how dynamical methods can shed new light on classical problems about rational numbers.

Tuesday, March 24

Hannah Hoganson, U of Maryland

Groups of Proper Homotopy Equivalences of Graphs

Abstract: Mapping class groups are classical objects of study in geometric group theory that record algebraic information about the symmetries of a geometric object. We will start with an overview of some basic ideas in geometric group theory, before focusing in on the mapping class group of a locally finite, infinite graph. These are large topological groups, which are not finitely generated, so classical tools and techniques don't apply. Instead, we turn to descriptive set theoretic techniques of Rosendal to study their large-scale geometry. This talk includes joint work with George Domat and Sanghoon Kwak.

Thursday, March 26

Jacob Shapiro, U of Dayton

Local energy decay for the acoustic wave equation in low regularity

Abstract: The wave equation is a partial differential equation describing how an initial disturbance propagates through space and time. The wavespeed is a coefficient in the equation that determines how quickly the wave travels at each point in space.

In three spatial dimensions, the wave equation models acoustic waves propagating through the Earth's subsurface. In this setting, the wavespeed represents how the propagation speed varies across different geological layers.

In this talk, we study the relationship between the regularity of the wavespeed and the rate of local energy decay for solutions of the wave equation. When the wavespeed is smooth, a classical result of Burq shows that the decay rate is logarithmic. When the wavespeed has discontinuities, I showed energy still escapes a localized region, but at a sublogarithmic rate. It is an open problem to determine whether this slower decay is optimal. This question is motivated by the geophysical setting described above, where the wavespeed may be discontinuous across material interfaces.

Using Fourier transform methods, we relate decay estimates to high-frequency resolvent bounds for associated Schrödinger operators. The key tool in proving these bounds is a Carleman estimate. I will present one case of such an estimate that is derived using separation of variables.

Thursday, April 2

Ashley Zhang, Vanderbilt U.

Spectral Theory for Canonical Systems and the Nonlinear Fourier Transform

Abstract: Many fundamental equations in mathematics and physics, including the Schrödinger and Dirac equations, belong to a broad family known as canonical Hamiltonian systems. The central goal of spectral theory is to understand the relationship between the coefficients of these systems and their solutions. Despite the long history of these problems, constructive and algorithmic solutions have remained elusive in much of the theory.

In this talk, I will present recent progress on inverse and direct spectral problems for canonical systems using tools from complex function theory, in particular truncated Toeplitz operators. Building on the algorithm developed by Makarov and Poltoratski, I will describe convergence results for inverse spectral problems and a parallel algorithm for the direct problem.

I will then turn to the nonlinear Fourier transform (NLFT), which arises naturally from scattering problems and is closely related to spectral theory. The spectral methods developed earlier provide a natural and powerful framework for understanding the NLFT: they yield an equivalent definition of the transform, convergence results connecting discrete and continuous versions of the theory, and an extension of the theory beyond the classical L^p and l^p decay assumptions.

Concrete examples will be woven throughout the talk, and I will discuss directions for future work. This talk is partially based on joint work with Alexei Poltoratski.

Tuesday, April 7

*Special Tri-College Colloquium at Concordia
Talk: Integrated Science Center 301 (Refreshments: Integrated Science Center 362)

Jessica Striker, NDSU

Unraveling a web of mysteries

Abstract: Dynamical algebraic combinatorics studies actions on mathematical objects with strikingly nice counting formulas and algebraic significance. Often, the study of such dynamics provides insight into the structure of the objects, revealing hidden symmetries and connections. In this talk, I'll describe some recent work on mathematical objects called webs, in which we found a beautiful, visual explanation of nice dynamical behavior which led us to a solution of a 30-year old problem. This is joint work with Christian Gaetz, Oliver Pechenik, Stephan Pfannerer, and Joshua Swanson. I’ll also discuss some very recent work with Bridget Tenner that connects webs and permutation pattern avoidance.