Spring 2025
January 27: Sergei Treil, Brown University
Title: The inverse spectral problem for positive Hankel operators
Abstract: Hankel operators are bounded operators in \ell^2 =\ell^2(\mathbb Z_+) with matrix of form \{\gamma_{j+k}\}_{j,k\ge0} . They can be realized as the integral operators in L^2(\mathbb R_+),
\displaystyle \Gamma f(s) =\int_0^\infty h(s+t) f(t) dt.
We consider positive semidefinite Hankel operators with simple spectrum and solve the inverse spectral problemfor such operators, i.e. reconstruct coefficients \gamma_k or the kernel h from the appropriately chosen spectral measure (and some other spectral invariants). It turns out that solutions of the spectral problem are drastically different for discrete and continuous realizations: to reconstruct the kernel h one only needs the natural (for the problem) spectral measure, for the coefficients \gamma_k one needs extra spectral invariant.
The talk is based on a joint work with A. Pusnitski.
February 3: Constanze Liaw, University of Delaware
Title: An Overview of Aleksandrov-Clark Theory and some Generalizations
Abstract: We will begin by recalling the origination of Aleksandrov-Clark Theory: First note that Beurling’s Theorem says that any non-trivial shift-invariant subspace of the Hardy space $H^2(\mathbb{D})$ is of the form $\theta H^2(\mathbb{D})$ for an inner function $\theta$. Now, for a fixed inner $\theta$, we form the model space, that is, the orthogonal complement of the corresponding shift-invariant subspace in the Hardy space. Consider the compressed shift, which is the application of the shift to functions from the model space followed by the projection to the model space. Clark observed that all rank-one perturbations of the compressed shift that are also unitary have a particular, simple form. Following this discovery, a rich theory was developed connecting the spectral properties of those unitary rank one perturbations with properties of functions from the model space, more precisely, with their non-tangential boundary values. Some intriguing perturbation results were obtained via complex function theory.
Throughout we will allude to some generalizations such as non-inner $\theta$, finite rank perturbations, functions of several variables, etc., and we will also focus on some ongoing work on the bidisk.
March 17 : Dmitriy Bilyk, University of Minnesota
Title: Spherical cap discrepancy, sums of distances, and positive definite functions
Abstract: Spherical cap discrepancy measures the extent of equidistribution of a discrete set of $N$ points of the sphere and, due to the so-called Stolarsky invariance principle, is closely related to the sum of distances between points. We shall present a new proof of Beck's classical result stating that this discrepancy is always at least of the order $N^{-1/2 - 1/2d}$. This proof is completely elementary in nature and, unlike the other proofs, avoids using Fourier analysis or spherical harmonics/Gegenbauer polynomials. The argument is also flexible enough to provide inequalities for discrepancy in terms of various other geometric quantities, estimates for discrete Riesz energies, almost sharp constants in the discrepancy bounds, as well as new bounds for discrepancy of lines and general sets of Hausdorff dimension between $0$ and $d$ (rather than just point sets). Along the way we shall discuss some related topics that come up: positive definite functions, Welch bounds, frame energy, spherical designs, and energy minimization on the sphere. The talk, which is based on joint work with Johann Brauchart (TU Graz), will assume no prior knowledge of the subject.
April 9 (Wednesday): Nathan Wagner, Brown University
Title: Boundedness and compactness of Bergman projection commutators in two-weight setting
Abstract: The Bergman projection is a fundamental operator in complex analysis with connections to singular integral theory, and it is of interest to study the commutator operator of the Bergman projection with multiplication by a measurable function b. In particular, we study the boundedness and compactness of the Bergman projection commutators in two weighted settings via weighted BMO (bounded mean oscillation) and VMO (vanishing mean oscillation) spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. Complete characterizations of two weight boundedness and compactness are obtained in the analytic case, which parallel results of S. Bloom for the Hilbert transform. Our work initiates a study of the commutators acting on complex function spaces with different symbols. In this talk, we will discuss our main results, as well as the principal ideas of the proofs. This talk is based on joint work with Bingyang Hu and Ji Li.
April 14: Lukas Bundrock, University of Alabama
Title: Spectral Geometry of the Robin and Steklov Eigenvalue Problem
in Exterior Domains
Abstract: The spectrum of the Laplace operator under Robin boundary
conditions on bounded domains is a well-explored topic, with
implications for various physical phenomena. For instance, in
analyzing the long-term behavior of Brownian motion with particle
creation at the boundary, the principal eigenvalue provides
information about the expected number of particles within the domain.
However, when we consider an exterior domain, that is the complement
of a compact set, many open questions arise. This talk explores the
Robin and Steklov eigenvalue problems in exterior domains, with a
focus on how the geometry of the domain affects the behavior of the
eigenvalues, with particular attention to differences unique to the exterior setting.
April 21: Jose Madrid Padilla, Virginia Tech
Title: Analysis on discrete cubes and applications
Abstract: In this talk we will discuss a collection of inequalities for convolutions of real valued functions on the discrete cubes, motivated by combinatorial applications.
April 28: Brandon Sweeting, Washington University in St. Louis
Title: Two-Weight Multiplier Weak-Type Inequalities
Abstract: We discuss a class of weighted weak-type inequalities first studied by Muckenhoupt and Wheeden. In this formulation, the weight for the target space appears as a multiplier rather than as a measure, leading to fundamentally different behavior. Notably, Muckenhoupt and Wheeden showed that the class of weights characterizing such inequalities for the maximal operator in the one-weight case is strictly larger than the $A_p$ class.
In joint work with David Cruz-Uribe and Kabe Moen, we extend these inequalities to the two-weight setting for both the Hardy–Littlewood maximal operator and singular integrals. For the maximal operator, we establish a necessary and sufficient Sawyer-type testing condition, and for singular integrals, we provide sufficient Pérez-type bump conditions.