Analysis and Geometry Seminar
Center of Excellence in Complex and Hypercomplex Analysis
Schmid College of Science and Technology
Chapman University

"Reason is immortal, all else mortal." - Pythagoras

Fall 2022
Organized by Kamal Diki, Mario Stipčić, and Mihaela B. Vajiac, sponsored by CECHA.

The talks will be held in a hybrid format, on Zoom (meeting ID: 99396752824) and/or on campus at Keck Center of Science and Technology (no. 30 on the Campus map, at the intersection of Walnut Ave and Center St), room KC 156.

Schedule

• Tuesday, July 26, 2:00pm - 3:00pm, Keck 156
  Jussi Behrndt, Technische Universität Graz
  The generalized Birman-Schwinger principle

In this talk we discuss a generalized Birman-Schwinger principle in the non-self-adjoint context. In particular, we provide a detailed discussion of geometric and algebraic multiplicities of eigenvalues of the basic operator of interest (e.g. a Schrödinger operator) and the associated Birman-Schwinger operator, and additionally offer a careful study of the associated Jordan chains of generalized eigenvectors of both operators.

This talk is based on a joint paper with Tom ter Elst (Auckland, New Zealand) and Fritz Gesztesy (Baylor, US).

• Friday, September 23, 2:30pm - 3:30pm, Keck 370
  Palle Jorgensen, University of Iowa
  Harmonic analysis and frames for fractal IFS L2 spaces via Infinite products of projections

The talk begins with an idea/algorithm, originating initially with a fundamental recursive iteration scheme (usually referred as “the” Kaczmarz algorithm). It is shown to admit new and important applications in such infinite-dimensional, and non-commutative, settings as are central to spectral theory of operators in Hilbert space, to optimization, to large sparse systems, and to fractal harmonic analysis of iterated function systems (IFS). A new recursive iteration scheme involving as input infinite products of selfadjoint projections, is presented.

• Friday, November 4, 2:00pm - 3:00pm, Keck 370
  João Pedro Gonçalves Ramos, ETH Zürich
  Time-frequency localisation operators, their eigenvalues and relationship to elliptic PDE

In the classical realm of time-frequency analysis, a classical object of interest is the short-time Fourier transform of a function. This object is a modified Fourier transform of a signal $f(x),$ modified by a certain 'window function', in order to make joint time-frequency analysis of functions more feasible.

Since the pioneering work of Daubechies, time-frequency localisation operators have been of extreme importance in that analysis. These are defined through $V^* 1_{\Omega} V f = P_{\Omega} f,$ where $V$ denotes the short-time Fourier transform with some fixed window. These operators seek to measure how much a function concentrates in the time-frequency plane, and thus the study of their eigenvalues and eigenfunctions is intimately connected to the previous questions.

In this talk, we will explore the case of a Gaussian window function $\varphi(x) = e^{-\pi x^2}$, and the operators thus obtained. We will discuss some classical and recent results on domains of maximal time-frequency concentration, their eigenvalues, and inverse problems associated with such properties. During this investigation, we shall see that many of these problems possess some rather unexpected connections with overdetermined elliptic boundary value problems and free boundary problems in general. This is based on recent joint work with Paolo Tilli.

• Friday, December 2, 2:00pm - 3:00pm, Keck 370
  Vanja Wagner, University of Zagreb
  Nonlocal quadratic forms with visibility constraint

Given a subset $D$ of the Euclidean space, we study nonlocal quadratic forms that take into account tuples $(x, y) \in D \times D$ if and only if the line segment between $x$ and $y$ is contained in $D$. We discuss regularity of the corresponding Dirichlet form leading to the existence of a pure-jump process with visibility constraint. Our main aim is to investigate corresponding Poincar\' e inequalities and their scaling properties. For dumbbell shaped domains we show that the forms satisfy a Poincar\' e inequality with diffusive scaling.