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
It 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
We 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.