# Analytic and Geometric Aspects of Spectral Theory

Casa Matemática in Oaxaca, 14. - 19.08.2021

Spectral theory deals with the study of spectra of linear operators acting in Banach or Hilbert spaces. It is a wide area of research in mathematics with many applications to mathematical physics, chemistry, engineering, and medicine among others. Very often, linear operators are used to model physical systems, and their spectra contain important information on the behaviour of the system, for example its stability. In applications to physics and geometry, the type of the operators that one considers may be differential, pseudodifferential or Fourier integral operators. The techniques used to study their spectra depend on the type of the operator under consideration.

Spectral invariants are quantities that depend only on the spectrum of an operator in the sense that if two operators have the same spectrum, the associated quantity is the same for both operators. Examples of spectral invariants on a compact manifold are the first eigenvalue, the trace of the heat operator, the determinant of the Laplace operator, and analytic torsion when it is defined. Direct spectral theory studies how these quantities change when the Riemannian manifold changes.

The aim of the proposed workshop is to bring together researchers with different backgrounds and from different countries who work in the exciting and versatile field of spectral theory. We want to encourage interaction between the participants who might otherwise not meet in a relaxed and stimulating atmosphere as provided by the Casa Matemática in Oaxaca.

Small groups will be formed in advance who will interact and work on specific problems so that the time of personal interaction during the workshop can be used efficiently for discussions.

These groups will be organized well in advance of the workshop, so that at the time of arrival participants already have some common ground and can immediately start collaborative work. The groups will mix researchers of different backgrounds and at different stages of their careers with one or two members of the group being identified as group leaders who propose an initial research topic. The ultimate goal is that the intensive weeklong workshop will jump-start long term collaborations, leading to future publications.