Coloquios 16-17

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Recent advances in mathematical general relativity

Niky Kamran (McGill University (Canada))
Viernes, 2 de junio, ICMAT, Aula Naranja, 11:30h.
Abstract Einstein's general theory of relativity provides a geometrical description of gravity in terms of space-time curvature. The Einstein field equations pose fascinating challenges that have stimulated a great deal of research in geometry and partial differential equations. Important questions include the well-posedness of the initial value problem, the linear and non-linear stability of space-times, the formation of black holes, and the boundary value problems arising from the classical aspects of the AdS/CFT correspondence. I will give a survey of some significant advances and open problems pertaining to these questions. No background knowledge of General Relativity will be assumed.
A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago. Via such an approach one can give a precise definition of what means for a non-smooth space to have Ricci curvature bounded below. Such an approach has been refined in the last years giving new insights to the theory and yielding applications which seems to be new even for smooth Riemannian manifolds.

Smooth and non-smooth aspects of Ricci curvature lower bounds

Andrea Mondino (Warwixk Mathematics Institute, University of Warwick)
Viernes, 28 de abril, módulo 17, Aula 520, 11:30h.
Cartel

Abstract After recalling the basic notions coming from differential geometry, the colloquium will be focused on spaces satisfying Ricci curvature lower bounds. The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the ‘80s and was pushed by Cheeger and Colding in the ‘90s who investigated the fine structure of possibly non-smooth limit spaces.
A completely new approach via optimal transportation was proposed by Lott-Villani and Sturm almost ten years ago. Via such an approach one can give a precise definition of what means for a non-smooth space to have Ricci curvature bounded below. Such an approach has been refined in the last years giving new insights to the theory and yielding applications which seems to be new even for smooth Riemannian manifolds

 

Topology of zero sets of smooth random functions

Mikhail Sodin (Tel Aviv University)
Viernes, 3 de marzo, ICMAT, Aula Naranja, 11:30h.
Cartel

AbstractWe describe recent progress in understanding the topology of zero sets of smooth random functions of several real variables and discuss a number of basic questions which remain widely open.

The h-principle in fluid mechanics: non-uniqueness and energy dissipation

László Székelyhidi (Universidad  de Leipzig)
Viernes, 10 de febrero, Módulo 17, Aula 520, 12:30h.
Cartel

AbstractIt is known since the pioneering work of Scheffer and Shnirelman in the 1990s that weak solutions of the incompressible Euler equations behave very differently from classical solutions, in a way that is very difficult to interpret from a physical point of view. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to Kolmogorov’s K41 theory of turbulence. In joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we can interpret the wild behaviour of weak solutions of the Euler equations as an instance of Gromov's celebrated h-principle. In this lecture I will explain this connection and outline the most recent progress concerning Onsager's.

 

 

Analysis on quantum tori

Quanhua Xu  (Université de Franche-Comté and Harbin Institute of Technology)
Viernes, 27 de enero, ICMAT, Aula Naranja, 11:30h.  

Grupos de Artin

Carlos Paris (Institut de Mathématiques de Bourgogne, Université de Bourgogne)
Jueves, 26 de enero, Módulo 17, Aula 520, 11:30h.  

A Rough Path Perspective on Renormalizatio

Peter Friz (Institut für Mathematik, Technische Universität Berlin)
Martes, 8 de noviembre, 11:30h. Aula Naranja, ICMAT
Cartel
AbstractWe shall introduce geometric and branched rough paths, following T. Lyons (1998) and M. Gubinelli (2010) respectively. In particular, we revisit / extend the notation of translation operator on these objects. As in Hairer's work (2015) on the renormalization of SDPEs subjected to stationary, if ill-defined noise, we propose a purely algebraic view on the matter. Recent advances in the theory of regularity structures, especially the Hopf algebraic interplay of positive and negative renormalization, are seen to have precise counter-parts in the rough path context. At last, we consider some finite-dimensional stochastic examples which do need renormalization.