Non Linear Analysis, Geometry and Applications

Invited Speakers

Invited Speakers.

1. Yuri Tschinkel

Affiliation: Professor at the Courant Institute of Mathematical Sciences New York university
Webpage: https://math.nyu.edu/~tschinke/
Conference Title: Equivariant Birational Types
Abstract: I will discuss new invariants of actions of finite groups on function fields of algebraic varieties, developed in joint work with A. Kresch.

2. Ahmed Zeriahi

Institution: Emeritus Professor at the University of Toulouse 3-Paul Sabatier in France, Attached to the Institute of Mathematics of Toulouse (IMT).
Webpage: http://www.math.univ-toulouse.fr/~zeriahi/
Conference Title: Geometric Pluripotential Theory : introduction and applications.
Abstract: We will give an elementary introduction to recent developments in Kähler Geometry based on geometric aspects of Pluripotential Theory developed in the last two decades by many researchers, including the author and his collaborators. In particular we will show how Geometric Pluripotential Theory can be applied to solve degenerate complex Monge-Ampère equations on Kähler manifolds, generalizing deep results obtained by S.T. Yau and T. Aubin in the late seventies on the solution to the Calabi conjecture and the existence of Kähler-Einstein metrics. As an application we will discuss the existence of singular Kähler-Einstein metrics on Calabi-Yau spaces or projective varieties of general type with mild singularities in the sense of the Minimal Model Program (MMP) in Algebraic Geometry. These projective varieties arise naturally as canonical models in the birational classification of projective algebraic manifolds.

3. Jimmy Lamboley

Affiliation: Full Professor at Sorbonne university, in the laboratory IMJ-PRG, in Paris
Web page: https://webusers.imj-prg.fr/~jimmy.lamboley/
Conference Title: Blaschke-Santalo diagram and eigenvalues of the Laplace operator
Abstract: the notion of Blaschke-Santalo diagram has mainly been used in the context of geometry of planar convex sets, in order to describe all possible inequalities involving geometrical quantities like the area, the perimeter, the diameter, etc…In this talk, we will start by defining this tool, and then will show it can be applied to the study of eigenvalues of the Laplace operator with Dirichlet boundary conditions: more precisely, we will review the possible inequalities involving the first Dirichlet eigenvalue of the Laplace operator, the perimeter and the area of the domain, when this one is either an open set, or a planar convex body.

4. Alexandre Girouard

Affiliation : Professor at Laval University, Quebec Canada
Webpage: https://archimede.mat.ulaval.ca/agirouard/
Conference Title: A tale of isoperimetry and eigenvalues
Abstract: It has been known since classical antiquity that disks have the largest area among planar figures of prescribed perimeter. Nevertheless, a complete proof was only given around the end of the 19th century! During the 20th century, area and perimeter were replaced by several new analytic and geometric quantities, such as the heat content, torsional rigidity and natural frequencies of vibrations. In this talk, I will survey recent results on isoperimetric bounds for eigenvalues of the Dirichlet-to-Neumann map. This pseudodifferential operator arises naturally in the study of inverse problems that are linked to geophysical and medical imaging. The main question that we will investigate is to find, in various geometric contexts, how large specific eigenvalues can be under isoperimetric type constraints. We will see that a vast number of methods can be used, ranging from abstract discretization and expander graphs to homogenization theory, which comes from applied mathematics.

5. Gilles Courtois

Affiliation: Research Director, CNRS, Institute of Mathematics of Jussieu, Sorbonne University, Paris
Webpage: https://webusers.imj-prg.fr/~gilles.courtois/
Conference Title: Cheeger-type inequality for differential forms
Abstract: On a compact Riemannian manifold, the Cheeger's inequality relates the first non zero eigenvalue of the Laplacian of functions with an isoperimetric constant of the manifold. J. Cheeger asked if an analogous inequality would hold for the first non zero eigenvalue of differential forms. We will discuss the case of 1-differential forms. (Joint work with Adrien Boulanger).

6. Volker Schulz

Affiliation: Professor, Department of Mathematics Building E Trier University, Germany
Web page: https://www.math.uni-trier.de/~schulz/
Conference Title: From shape optimization to preshape calculus
Abstract: Shape optimization is a very active field of research. This talk discusses, on the one hand, fast methods for the solution of PDE constrained shape optimization problems in certain fields of application, based on the shape calculus. On the other hand, algorithmic shape optimization is usually geared to the generation of successive normal deformations of shapes towards optimality criteria. Often, this stretches surfaces in tangential direction leading to misbalanced surface meshes. Indeed, the shape derivative does not give any information in tangential direction at all, which means that tangential deformations can be used for other practical purposes without compromising optimality. A broader perspective thus leads to a natural generalization of the classical shape calculus to pre-shape calculus, as we would like to call it. Motivation of this concept together with first theoretical and numerical results are presented.

7. Gérard Besson

Affiliation: Research Director, CNRS,Institut Fourier Université de grenoble
Web page: https://www-fourier.ujf-grenoble.fr/~besson/
Conference Title: A finiteness theorem for hyperbolic groups
Abstract: This is a joint work with G. Courtois, S. Gallot and A. Sambusetti. We will show that given two positive numbers $\delta$ and $H$ there is only a finite number of marked groups $(\Gamma , \Sigma)$, $\delta$-hyperbolic, torsion-free and non-cyclic, which satisfy ${\rm Entropy}(\Gamma , \Sigma) \le H$, to the nearest isometry (of marked groups). Here a marked group is a finite generation $\Gamma$ group with a finite and symmetric $\Sigma$ generating system. All these notions will be precisely defined and we will try to describe the principles of the proof.