Winter 2025
March 10, 2025, 3:00–4:00pm, East Hall 3096
Speaker: Brian Hall (University of Notre Dame)
Title: Roots of polynomials under differential flows, with connections to random matrix theory
Abstract: I will discuss how the roots of polynomials evolve under differential flows, focusing on two main examples: heat flow and repeated differentiation. In the case of real roots, the situation is well understood and the evolution of the roots is described by constructions from random matrix theory. In the case of complex roots, I will present a general PDE-based conjecture, together with a random matrix interpretation. Finally, I will describe recent rigorous results for random polynomials.
The talk will be self-contained and have lots of pictures and animations.
March 17, 2025, 4:00–5:00pm, zoom
Speaker: Samir Donmazov (University of Kentucky)
Title: Long-Time Asymptotics of Solutions to the Kadomtsev-Petviashvili I Equation
Abstract: We establish the long-time asymptotics for solutions of the Kadomtsev-Petviashvili I (KP I) equation $$ (u_t+6uu_x+u_{xxx})_x = 3 u_{yy} $$ with small initial data, using the inverse scattering transform (IST) formalism developed by Zhou. Within this framework, the inverse problem for the KP I equation is formulated as a nonlocal Riemann-Hilbert problem (RHP) in two dimensions. As part of the asymptotic analysis, we also determine the long-time behavior of the solution to the nonlocal RHP, along with its $x$-derivative.
March 31, 2025, 4:00–5:00pm, East Hall B745
Speaker: Alexander Moll (Reed College)
Title: Fractional Brownian motions and Kerov's CLT
Abstract: Nonlinear functionals of Gaussian fields are ubiquitous in probability theory and PDEs. We introduce a family of random curves in the plane which encode the random values of certain nonlinear functionals of fractional Brownian motions on a circle with Hurst index s - 1/2. For a special choice of Cameron-Martin shift, the low variance limit of the fractional Brownian motion induces a LLN and CLT for the associated random curves that is nearly identical to the global behavior of Plancherel measures on large Young diagrams. The limit shape is independent of s and is that of Vershik-Kerov-Logan-Shepp. The global Gaussian fluctuations depend on s and coincide with the process in Kerov's CLT for s = -1/2. We give a dynamical explanation of this relationship using results of Eliashberg and Dubrovin. This is work in progress with Robert Chang (Rhodes College).
Fall 2024
October 21st 4-5pm EDT - Cade Ballew (University of Washington)
Title: Numerical solutions of Riemann--Hilbert problems on disjoint intervals
Abstract: We present a general approach to numerically compute the solutions of Riemann--Hilbert problems with jump conditions supported on disjoint intervals. Applied to the Fokas--Its--Kitaev Riemann--Hilbert problem, this enables the computation of orthogonal polynomials on multiple intervals, requiring only O(N) arithmetic operations to compute the first N recurrence coefficients. Such recurrence coefficients describe the flow of a semi-infinite Toda lattice, and expansions in these orthogonal polynomials yield a novel iterative method for solving indefinite linear systems and computing matrix functions, further yielding a fast algorithm for computing the solutions of Sylvester matrix equations. Other Riemann--Hilbert problems of this form yield finite-genus and soliton gas solutions of the Korteweg--de Vries equation. In particular, we compute large-genus solutions to simulate dispersive quantization and evaluate soliton gas solutions before asymptotic estimates are valid.
October 28th 4-5pm EDT - Nathan Hayford (KTH Royal Institute of Technology)
Title: Critical phenomena in the 2-matrix model: the Ising model coupled to gravity
Abstract: The unitary invariant ensembles of random matrix theory (collectively referred to as the `1-matrix model’) are one of the central objects of study in the theory of random matrices. Upon tuning the parameters of this model, one can realize certain `higher-order' eigenvalue correlations at the endpoints of the density of eigenvalues. In the 1-matrix model, these higher-order correlation kernels can be written in terms of special solutions of the Painlevé I and II hierarchies. We refer to such special tunings as critical phenomena.
The 2-matrix model is an extension of the 1-matrix model, which came to be well-studied in part because it admits a much richer class of critical phenomena. In this talk, I will survey some of the conjectures regarding these critical phenomena from the physics literature, and discuss some of the physical implications of these results. In particular, I will discuss some forthcoming work (joint with Maurice Duits and Seung-Yeop Lee) regarding a special critical phenomenon in the 2-matrix model, which has implications in the theory of the Ising minimal model coupled to topological gravity.
November 4th 4-5pm EDT - Patrick Sprenger (University of California, Merced)
Title: The Riemann problem for a discrete conservation law
Abstract: The hydrodynamics of a system of conservation laws can often be understood by studying solutions to a Riemann problem, i.e., the evolution of step initial data in a hyperbolic system. When dispersion compensates for the large gradients induced by this initial data, wave-breaking is often resolved by a dispersive shock wave (DSW). An active area of research is the investigation of quantitative features of DSWs in continuum models arising in physical applied mathematics. This talk leverages modern techniques to investigate solutions of the Riemann problem for a semi-discrete conservation law. The semi-discrete model is obtained by applying a first-order centered difference scheme to the spatial derivative of the Hopf equation. Solutions to the Riemann problem reveal a surprisingly elaborate set of solutions to this example system. In addition to discrete analogs of well-known dispersive hydrodynamic solutions—rarefaction waves (RWs) and DSWs—additional unsteady solution families and finite-time blow-up are observed and characterized. We will also compare the dynamics of the Riemann problem to an integrable discretization of the spatial derivative.
November 18th 4-5pm EST - Seung-Yeop Lee (University of South Florida)
Title: Planar orthogonal polynomials with non-Hele-Shaw type polynomial potentials
Abstract: Planar orthogonal polynomials in the double scaling limit have been much studied for their connection to Coulomb gas system in two dimensions. Most exact results have been known either for radially symmetric potential or for so-called Hele-Shaw potential, where the limiting density of the Coulomb gas is uniform over its support. When the potential is not Hele-Shaw type nor radially symmetric, we expect to observe a new type of singular behaviors. Unfortunately, in such cases, there is no known multiple orthogonality that we can use for asymptotic analysis of the planar polynomials.
In this talk, we will propose a matrix Riemann-Hilbert problem for some polynomial potential that is not radially symmetric and not Hele-Shaw type. More explicitly we will consider the case when the Laplacian of the potential is |z|^2. This work is a preliminary report of the work by Abril Arenas and by Seong-Mi Seo.
November 25th 4-5pm EST - Michel Alexis (Hausdorff Center for Mathematics)
Title: How to represent a function in a quantum computer?
Abstract: Quantum Signal Processing (QSP) is an algorithmic process by which one represents a signal $f:[0,1] \to (-1,1)$ as the upper left entry of a product of SU(2) matrices parametrized by the input variable $x \in [0,1]$ and some "phase factors'' $\{\psi_k\}_{k \geq 0}$ depending on $f$. We show that, after a change of variables, QSP is actually the SU(2)-valued nonlinear Fourier transform, and the phase factors correspond to the nonlinear Fourier coefficients. By exploiting a nonlinear Plancherel identity and using some basic spectral theory, we show that QSP can be done for any signal f satisfying the log integrability condition
\int\limits_{0} ^1 \log (1-f(x)^2) \frac{dx}{\sqrt{1-x^2}} > - \infty .
December 2nd 4-5pm EST - Ahmad Barhoumi (KTH Royal Institute of Technology)
Title: Non-Hermitian orthogonality in the q^(Volume) tiling model
Abstract: Consider a hexagon overlayed on a regular triangular grid, where the equilateral triangles have sidelength 1. A "lozenge" is a pair of adjacent triangles, and there are three types of lozenges that can be built on this grid. Using these lozenges, one can tile the entire hexagon. Lozenge tilings of a regular hexagon are in bijection with boxed plane partitions (i.e. stacks of boxes in the back of a cubic room) and can therefore be assigned a volume; a fact that is best illustrated by staring at a picture of one such tiling. The q^(Volume) tiling model is a measure on the space of tilings of the hexagon which assigns to each tiling a probability proportional to q^(Volume), where q is a real parameter. In this talk, I will recall the model and basic result about it and propose an approach to studying its statistical properties as the size of the hexagon grows by analyzing a related family of non-Hermitian orthogonal polynomials. The talk is based on ongoing joint work with Maurice Duits.