MINI- COURSES JULY 17 to 21 |
LAST UPDATED JUNE 20th |
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Time |
Monday |
Tuesday |
Wednesday |
Thursday |
Friday |

9:30 to 10:30 |
Guillarmou |
Guillarmou |
Guillarmou |
Guillarmou |
Guillarmou |

10:30 to 11:00 |
Coffee |
Coffee |
Coffee |
Coffee |
Coffee |

11:00 to 12:00 |
Dyatlov |
Dyatlov |
Dyatlov |
Dyatlov |
Dyatlov |

12:00 to 2:00 |
Lunch |
Lunch |
Lunch |
Lunch |
Lunch |

2:00 to 3:00 |
Benguria |
Benguria |
Benguria |
Benguria |
Benguria |

3:00 to 3:30 |
Coffee |
Coffee |
Coffee |
Coffee |
Coffee |

3:30 to 4:30 |
Kirsch |
Kirsch |
Kirsch |
Kirsch |
Kirsch |

4:45 to 5:30** |
Sa Barreto |
Sa Barreto |
Sa Barreto |
Sa Barreto |
Sa Barreto |

** Mini-course in Portuguese for undergraduate students |

**Prof. Rafael Benguria,
Departamento de F’sica, Pontificia Universidad Cat—lica de Chile: The stability of Matter.**

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.

**Prof. Colin Guillarmou,
Universitˇ de Paris Sud: X-ray
Tomography and Boundary Rigidity.**

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.

**Prof. Semyon Dyatlov, Department of Mathematics, Massachusetts Institute
of Technology: Fractal Uncertainty Principle and Applications to Open Quantum
Chaos.**

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 ² 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.

**Prof. Andreas Kirsch, Karlsruhe
Institute of Technology (KIT): Inverse Scattering Theory **

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.