The following mini-courses will be offered during the week of July 17 to 21, 2017. The lectures will take place at the Department of Mathematics UFSC.

The stability of matter by Prof. Rafael Benguria, Departamento de Física, Pontificia Universidad Católica de Chile

Description: The course will cover the following topics: Historical facts: the origin of Quantum Mechanics, the Uncertainty Principle: The inequalities of Sobolev and Hardy, and their applications in Quantum Mechanics, the Birman--Schwinger principle, the Thomas--Fermi Model of atoms and molecules, and its extensions. Main properties. Teller's no binding theorem, many particle systems, and the definition of stability of first and second kind, Lieb--Thirring inequalities, electrostatic Inequalities, estimates on the indirect part of the Coulomb Energy. The Lieb--Oxford bound. I will also discuss some improved bounds found recently by G. Bley, M. Loss and Rafael Benguria, different proofs of the stability of nonrelativistic matter, stability of relativistic matter, magnetic Fields and the Pauli Operator, the Ionization Problem in Atomic and Molecular Physics.

X-ray tomography and boundary rigidity by Prof. Colin Guillarmou, École Normale Supérieure, Paris

Description: We will explain some problems in integral geometry related to Michel's conjecture, which asks if a Riemannian metric on a manifold with strictly convex boundary can be recovered from the Riemannian distance between boundary points. This involves analyzing the X-ray transform, which is a generalized Radon transform that consists integrating symmetric tensors along geodesics.

Fractal uncertainty principle and applications to open quantum chaos by Prof. Semyon Dyatlov, Department of Mathematics, Massachusetts Institute of Technology

Description: I describe a new approach to essential spectral gaps for open quantum systems with hyperbolic classical dynamics. An essential spectral gap gives exponential local energy decay of high frequency wavefunctions propagated by the quantum system. Such energy decay is possible since the system is open and thus energy can either leak out or escape to infinity. The approach I present is based on a "fractal uncertainty principle" (FUP) which quantifies the statement that no function can be localized too close to a fractal set in both position and frequency. The fact that FUP leads to spectral gaps relies on arguments from microlocal analysis, in particular propagation of singularities. On the other hand, to prove FUP one needs to use nonmicrolocal tools which go beyond the classical-quantum correspondence. I will first present FUP in the simplest setting of Cantor sets and discrete Fourier transform, where it can be established using only linear algebra and properties of polynomials. I will next explain how this version of FUP leads to spectral gaps for open quantum baker's maps, using integration by parts and basic properties of Fourier transform (no knowledge of microlocal analysis needed!). These results will be illustrated by numerical simulations. In the second half of the minicourse I will discuss FUP for the more general case of Ahlfors-David regular sets, whose proof makes use of advanced tools from harmonic analysis. As an application I show that each convex co-compact hyperbolic surface has an essential spectral gap, giving exponential local energy decay for the wave equation in the high frequency regime, as well as a strip where the Selberg zeta function has finitely many zeroes. Such gaps were previously known only under the pressure condition δ ≤ 1/2, where δ ∈ [0,1) is the dimension of the limit set of the group, as established in the works of Patterson, Sullivan, and Naud. This minicourse is based on joint works with Jean Bourgain, Long Jin, and Joshua Zahl.

Inverse scattering theory by Prof. Andreas Kirsch, Karlsruhe Institute of Technology (KIT)

Description: In this course we will introduce time harmonic scattering problems for acoustic waves; that is, scalar waves in frequency domain. We will concentrate on two types of problems. In the first case the scattering object consists of an impenetrable sound-soft scatterer (Dirichlet boundary considitions) while in the second case we will consider the scattering by an inhomogeneous medium (penetrable case). In both cases we will start with the mathematical analysis of the direct problems; that is, given the obstacle or the index of refraction, respectively, find the scattered field and the corresponding far field. In the second part of the course we will consider the corresponding inverse scattering problems: Given the far field pattern for one- or many incident waves, find the shape of the scattering obstacle or properties of the index of refraction, respectively.