2023-2024

Motivated by a recent study of Bessel operators in connection with a refinement of Hardy’s inequality involving 1/sin²(x) on the finite interval (0,π), we discuss domain properties of certain Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in L²((a,b);dx) associated with differential expressions of the form
and
ω_sa =-^d2⁄_dx²+^(s_a2-(1/4))⁄_(x-a)², s_a∈R, x∈(a,b),
τ_sa,sb=-^d2⁄_dx²+^(s_a2-(1/4))⁄_(x-a)²+^(s_b2-(1/4))⁄_(x-b)²+q(x), x∈(a,b),
s_a, s_b∈[0,∞), q∈L^∞((a,b);dx), q real-valued a.e. on (a,b),
where (a,b)⊂R is a bounded interval.
As an explicit illustration, we describe the bewildering variety of characterizations of the Friedrichs extension of the minimal operator corresponding to τ_sa,sb. Time permitting, we will illustrate the connection between the Sturm–Liouville problem τ_sa,sb,q=0u = zu and the confluent Heun differential equation.
This talk is based on joint work with Fritz Gesztesy and Michael M. H. Pang.

Rational approximations of functions offer a rich mathematical theory. Touching subjects such as orthogonal polynomials, potential theory and of course differential equations. In this talk we will discuss a specific type of rational approximant, factorial expansions. In recent work with O. Costin and R. Costin we have developed a theory of dyadic expansions which improve the domain and rate of convergence when compared to the classical methods found in the literature. These results provide a general method for producing rational approximations of Borel summable series with locally integrable branch points. Surprisingly, these expansions capture the asymptoticly important Stokes phenomena. Additionally, we find applications in operator theory on Hilbert spaces providing new representations for (bounded and unbounded) positive and self-adjoint operators in terms of the semigroups and unitary groups they generate. Finally, as an example of an important application we discuss representing the tritronquée solutions of Painlevé’s first equation.

In this talk we will look at some non-compact “optimal” Sobolev embeddings
and try to study their behavior (using different tools like entropy numbers, s-numbers and concept of strict singularity).

In this talk, I will present our recent result on the spherically symmetric Coulomb waves. We study the evolution operator of $H=-\Delta+q |x|^{-1}$, with $q>0$. By means of a distorted Fourier transform adapted to $H$, we compute the evolution kernel explicitly. A detailed analysis of this kernel shows that $e^{i t H}$ obeys an $L^1 \to L^\infty$ dispersive estimate with the natural decay rate $t^{-\frac{3}{2}}$. This is a joint work with Adam Black, Bruno Vergara and Jiahua Zhou

Doubly periodic tiling models have gained considerable attention in recent years due to their deep connections with random matrices, combinatorics, algebraic geometry, orthogonal polynomials and various other fields of mathematics. In this talk I will report on recent developments in the study of the doubly periodic Aztec diamond, and present some new results obtained in a collaboration with Arno Kuijlaars concerning the arctic curve. Our methods are based on the analysis of certain matrix-valued orthogonal polynomials introduced by Duift and Kuijlaars in 2021.

The KdV equation is a nonlinear wave equation used, among others, for modeling shallow water waves. In this talk I will focus on the Riemann-Hilbert analysis of the initial value problem for the KdV equation with steplike initial data. It turns out that in the transition region solutions converge to a modulated elliptic wave, i.e., a genus 1 finite-gap solution, also known as an Its–Matveev solution. The emphasis will be on the peculiarities that arise when analyzing this problem using the Deift-Zhou nonlinear steepest descent method for oscillatory Riemann-Hilbert problems. Part of this work has been a collaboration with Iryna Egorova and Gerald Teschl.

The study of stability and instability in nearly integrable Hamiltonian systems
has been considered ever since Poincare as one of the most important
problem in dynamical systems. The problem has interesting applications, e.g.,
to astrodynamics, in designing low-energy spacecraft orbits that follow prescribed paths.

In this work, we apply some tools from control theory to integrable Hamiltonian systems that are subject to small, generic perturbations.
We construct a family of vector fields that are induced by the perturbation, which are associated to orbits asymptotic to a certain invariant manifold.
We provide conditions on the Lie algebra of these vector fields that allow one to obtain orbits with prescribed itineraries.

This is based on joint work with Rafael de la Llave and Tere M-Seara.

We prove the Kato-Ponce inequality (normed fractional Leibniz rule) for multiple factors in the setting of \emph{multiple weights} ($A_{\vec P}$ weights). This improves existing results to the product of $m$ factors and extends the class of known weights for which the inequality holds.

Let E ⊂ R^d be a domain such that 0 < |E| < ∞. In my talk I will
discuss the regularity of the associated characteristic function χ_E in
the framework of Besov and Lizorkin-Triebel spaces. The main goal
will be a description of if and only if – conditions given in terms of
the volume of the δ neighborhoods of the boundary of E for a certain
class of domains.

We establish a solution theory (i.e., unique existence) for state-dependent delay equations for arbitrary Lipschitz continuous pre-histories and suitably Lipschitz continuous right-hand sides on H^1. The solution theory is independent of previous ones and based on the contraction mapping principle yielding global in time well-posedness right away. Natural restrictions on the initial data or right-hand sides asked for by other approaches are not needed in the present setting. In particular, solution manifolds or techniques from the theory of dynamical systems are not required. The work is based on a joint work with Johanna Frohberg and can be found at arxiv:2308.04730.

We introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval [a,b] into the space of compact non-empty subsets of R^d. All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a set-valued function F, we obtain pointwise error estimates for sequences of integral operators.

We also provide a global approach for error bounds. A multifunction F is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in L^1 provides our global estimates.

The theory is illustrated by examples, including the Bernstein-Durrmeyer operator and the Kantorovich operator.

Joint work with Nira Dyn, Elza Farkhi and Alona Mokhov (Tel Aviv, Israel).

We begin the talk by some introductory notions in General Relativity. The aim of this talk is to discuss the question of determinism in classical General Relativity, most often called Strong Cosmic Censorship conjecture (SCC). Since this conjecture is notoriously difficult we restrict in Strong Cosmic Censorship near a Kerr de Sitter black hole spacetime. As we will see, quasinormal modes (QNMs), namely solutions of the linear wave equation that decay exponentially in time, play a very important role in the linear resolution of this conjecture near the Kerr de Sitter geometry. Time permitting, we will discuss recent work and work with progress with Rodica Costin.

When a multivariate, high-dimensional approximation problem is to be solved (mainly by interpolation, but also quasi-interpolation; currently a hot topic), radial basis functions and especially multiquadric interpolation are state-of the-art. They provide excellent accuracy, the asymptotic orders even increasing with dimension, and almost-positive definite interpolation matrices. In particular, the so-called curse of dimensionality turns into a blessing of dimensionality.

In this talk, we present a (to the best of our knowledge) previously unknown class of generalised multiquadric radial functions (by B. and Ortmann), and we study their distributional Fourier transforms that are relevant to existence of (quasi-)interpolants, polynomial reproduction, and asymptotic accuracy. We use them for quasi-interpolation. We then switch from almost-positive definiteness of other generalisations of radial basis functions connected to positive definite functions for interpolation on simplices and other regular domains with new concepts of distance functions (in collaboration with Yuan Xu).

I will present some of my recent joint work with Abdul-Rahman, Darras, and Stolz (https://arxiv.org/abs/2401.11508). I will begin with a quick crash course on discrete Schrödinger (Jacobi) operators with periodic potential and on Lieb-Robinson bounds. While periodic Schrödinger operators have purely ac spectrum and exhibit ballistic transport, I will show that if the potential is large enough, it is possible to make the velocity of this transport arbitrarily small. I will discuss the special case of period 2, where things can be computed explicitly and then talk about the case of general period p.

Since the publication of the seminal paper by Clarkson (1936), uniform convexity has been a concept historically associated to a norm. In this talk, a class of boundary value problems will be introduced (and solved) whose understanding requires the consideration of uniform convexity in a far more general setting. Open problems will be discussed.

Sobolev embeddings that are in a sense optimal, or nearly optimal, are typically noncompact. There are various quantities measuring “how bad” noncompactness of operators (e.g., of Sobolev embeddings) is, such as the ball measure of noncompactness or some so-called s-number. We will investigate some (nearly) optimal Sobolev embeddings from such a quantitative point of view.

Projection operators are fundamental algorithmic operators in Analysis and Optimization. It is well known that these operators are firmly nonexpansive; however, their composition is generally only averaged and no longer firmly nonexpansive. In this talk, we introduce the modulus of averagedness and provide an exact result for the composition of two linear projection operators. As a consequence, we deduce that the Ogura-Yamada bound for the modulus of the composition is sharp. Based on joint work with Theo Bendit and Walaa Moursi.

We establish improved Hardy inequalities for the magnetic p-Laplacian due to adding nontrivial magnetic fields. We also prove that for Aharonov-Bohm magnetic fields the sharp constant in the Hardy inequality becomes strictly larger than in the case of a magnetic-free p-Laplacian. We also post some remarks with open problems. This is based on a joint work with D. Krejčiřík, N. Lam and A. Laptev.