2024-2025

In this talk, we will discuss the crucial role that the algebraic properties of complex Bloch and Fermi varieties play in the study of periodic graph operators. I will begin with an introduction to the basics of periodic graph operators, followed by a discussion of recent discoveries, particularly focusing on the irreducibility of Bloch and Fermi varieties.
Next, I will demonstrate how these algebraic properties, combined with techniques from analysis and combinatorics, can be applied to address spectral and inverse spectral problems arising from periodic operators.

The variable Haar multipliers are analogous to pseudo-differential operators where the trigonometric functions have been replaced by the basis of Haar functions indexed on the dyadic intervals. In the classical case the symbol of the operator depends on space and frequency variables, in the Haar case, the symbols depend on the space and the dyadic frequency variables. We study a particular class of these operators that we call signed t-Haar multipliers, the symbol involves a weight w, a choice of signs \sigma, and the real parameter t.
We will discuss unweighted and weighted inequalities for signed t-Haar multipliers. We seek necessary and sufficient conditions on triples of weights (w,u,v) so that the operators are uniformly (on the choice of signs ) bounded from L^2(u) into L^2(v). We recover known necessary and sufficient conditions of Nazarov, Treil, and Volberg for two weight inequalities for the martingale transform (the case $w=1$ ), and the known necessary and sufficient conditions of Katz and Pereyra for the unweighted case ($u=v=1$).

This part is joint work with Daewon Chung, Jean Moraes, Weijan Huang, and Brett Wick.

This talk concerns the Benjamin-Ono (BO) equation of internal wave theory, and properties of the solution of the Cauchy initial-value problem in the situation that the initial data is fixed but the coefficient of the nonlocal dispersive term in the equation is allowed to tend to zero (i.e., the zero-dispersion limit). It is well-known that existence of a limit requires the weak topology because high-frequency oscillations appear even though they are not present in the initial data. Physically, this phenomenon corresponds to the generation of a dispersive shock wave. In the setting of the Korteweg-de Vries (KdV) equation, it has been shown that dispersive shock waves exhibit a universal form independent of initial data near the two edges of the dispersive shock wave, and also near the gradient catastrophe point for the inviscid Burgers equation from which the shock wave forms. In this talk, we will present corresponding universality results for the BO equation. These have quite a different character than in the KdV case; while for KdV one has universal wave profiles expressed in terms of solutions of Painlevé-type equations, for BO one instead has expressions in terms of classical Airy functions and Pearcey integrals. These results are proved for general rational initial data using a new approach based on an explicit formula for the solution of the Cauchy problem for BO. This is joint work with Elliot Blackstone, Louise Gassot, Patrick Gérard, and Matthew Mitchell.

We introduce a new finer limit point/limit circle classification for Sturm-Liouville equations by defining the regularization index. The results rely on the integrability of the product of the principal and nonprincipal solutions near singular endpoints and constructing a spectral parameter power series (i.e., a Taylor series in the spectral parameter z) for solutions of the Sturm-Liouville problem. The regularization index at the singular endpoint is then defined by comparing the growth in x of the coefficients of the power series. Implications include classifying when these series are well-behaved asymptotic series (in a precise sense), quantifying how far certain limit point endpoints are away from being Darboux transformed to a limit circle endpoint, and extending Weyl eigenvalue asymptotics to singular problems. These results will be motivated by multiple examples.

This talk is based on joint work with Mateusz Piorkowski.

I will explain how several exactly solvable polymer models are related to signed biorthogonal measures. These measures reduce to classical orthogonal polynomial ensembles in the zero-temperature limit. I will explain how asymptotic analysis of the signed biorthogonal measures provides insight in large deviations for the polymer partition functions and what the current limitations of this method are.
The talk will be based on joint works with Mattia Cafasso and Julian Mauersberger.

In this talk we will explore the dependence of the density of states for Schrödinger operators on the potential. The density of states characterizes the averaged spectral properties of a quantum system. Formally, it can be obtained as an infinite volume limit of the spectral density associated with finite-volume restrictions of a quantum system. Such a limit is known to exist for certain quantum mechanical models, most importantly for Schrödinger operators with periodic and random potentials.

Following ideas by J. Bourgain and A. Klein, we will consider the density of states outer measure (DOSoM) which is well defined for all Schrödinger operators. We will explicitly quantify the parameter dependence of the DOSoM by proving a modulus of continuity with respect to the potential (in $L^\infinity$-norm and weak topology). This result is obtained for all discrete Schrödinger operators on infinite graphs and captures the geometry of the graph at infinity. Applications of this result to random and ergodic operators will be presented.

This talk is based on joint work with Peter Hislop (University of Kentucky).

In his classic text, von Neumann emphasized that quantum mechanics contains “two fundamentally different types” of state changes: unitary evolution and measurement. Although Born’s rule provides an unambiguous procedure for computing the probability distribution for the outcomes of a measurement, the state change induced by a measurement, sometimes referred to as “the collapse of the wave function,” is less clearly defined. In the 1960’s and 1970’s, careful physical analyses led to the formulation of the notion of “generalized” measurement, in which an ancillary system is measured and then discarded.

In this talk, a general framework for describing “quantum trajectories,” i.e., the evolution of the state of a system under repeated generalized measurements, will be described. At a mathematical level, quantum trajectories are a particular type of Markov chains. Two classes of results will be discussed: 1) results on asymptotic purification of quantum trajectories, originally due to Kümmerer and Maassen, and 2) more recent work by various authors on large deviation principles for the results of measurements. In each case, recent work of the PI and coauthors has generalized these results to repeated random measurements subject to stationary and ergodic noise.

We investigate obstructions to the existence of frame vectors, in particular related to group representation and we provide examples of groups having faithful representations without frame vectors. As a result of these constructions we obtain a new characterization of amenability and a weaker form of Kazhdan’s property (T). This is joint work with Gabriel Picioroaga.

The study of Sobolev embedding into function spaces such as the Lebesgue spaces and Lorentz spaces plays a fundamental role in the field of PDE and approximation theory. In the classical setting, the Rellich–Kondrachov theorem provides condition on which the Sobolev embedding is compact. In the case where the embedding is non-compact, there are various levels to which one can measure the quality of non-compactness. In this talk, we discuss the compactness of Sobolev embedding into the variable Lorentz space where the behavior of non-compactness is concentrated around a single point. The quality of non-compactness is also discussed in this case. If time permits, we also discuss the measurement of non-compactness for various function spaces. This is a joint work with Jan Lang and Liding Yao.

I will discuss how to disprove a conjecture by Iwaniec from 1983 about Calderon-Zygmund estimates for $L^r $where $r$ is close to $p-1$.

The behavior of solutions of integrable nonlinear PDEs with periodic boundary conditions is receiving renewed interest in recent years, due in part to the connection with the emerging subject of soliton and breather gases. In this talk I will review some recent results concerning a one-dimensional periodic Dirac operator, which is asssociated to the spectral problem for the focusing nonlinear Schrodinger equation. These include: (i) An asymptotic characterization of the spectrum in the semiclassical limit; (ii) The characterization of a two-parameter class of exactly solvable elliptic potentials, (iii) The development of the inverse spectral problem via a Riemann-Hilbert problem approach, and (iv) The characterization of behavior of solutions in the semiclassical limit.

The Henon-Heiles system, initially introduced as a simplified model of galactic dynamics, has become a paradigmatic example in the study of nonlinear systems. Despite its simplicity, it exhibits remarkably rich dynamical behavior, including the interplay between regular and chaotic orbital dynamics, resonances, and stochastic regions in phase space, which have inspired extensive research in nonlinear dynamics.
In this work, we investigate the system’s solutions at small energy levels, deriving asymptotic constants of motion that remain valid over remarkably long timescales—far exceeding the range of validity of conventional perturbation techniques. Our approach leverages the system’s inherent two-scale dynamics, employing a novel analytical framework to uncover these long-lived invariants.
The derived formulas exhibit excellent agreement with numerical simulations, providing a deeper understanding of the system’s long-term behavior.

This is joint work with Ovidiu Costin and Kriti Sehgal.

The theory of Muckenhoupt A_p weights, 1≤p<∞, has been an important area of harmonic analysis for more than 50 years. As part of this, a rich structure theory for these weights hase been developed, including the reverse Hölder inequality, left-openness, Jones factorization, and Rubio de Francia extrapolation.
In the past 15 years this theory has been extended to the variable Lebesgue spaces, with the introduction of the A_{p(⋅)} weights by Cruz-Uribe, Fiorenza, and Neugebauer. Given an exponent function p(⋅), we say that a weight w is in A_{p(⋅)} if [w]_{A_{p(⋅)}}=\sup_Q |Q|^{-1}\|w\|_{L^{p(⋅)}(Q)}\|w^{-1}\|_{L^{p'(⋅)}(Q)} is finite.

These weights give sufficient (and in some cases necessary) conditions for maximal operators, singular integrals, and other operators of classical harmonic analysis to be bounded on the weighted variable Lebesgue space L^{p(⋅)}(w).
However, unlike in the constant exponent case, very little is known about the structural properties of A_{p(⋅)} weights, beyond a theory of extrapolation developed by Cruz-Uribe and Wang. In this talk I will discuss recent progress in this area, proving a reverse Hölder inequality for weights in A_{p(⋅)}. As an application, I will use it to prove left and right-openness of these weight classes, a result which we can also prove for matrix weights. If time permits I will also discuss other partial results on the structure of A_{p(⋅)} weights.
This research is joint with my PhD student, Michael Penrod.

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