CCOR mini-course and workshop

Mini-course 

Dates:

7 September, 10:00-11:30, 13:00-14:30 

8 September, 9:30-11:00 

Place: Corvinus University, Building C 428 and Zoom 

Abstract: 

The mini-course focuses on certain optimization problems on curved spaces. In the first part, I recall those geometric objects which are crucial in our investigations (basic elements from Riemannian and Finsler geometry, including geodesics, Jacobi fields, curvature, Busemann convexity, etc). In the second part I present the proximal point algorithm on Hadamard manifolds (i.e., on complete, simply connected Riemannian manifolds with nonpositive sectional curvature), emphasizing the role of the curvature constraint. I also mention from the literature some incorrect extensions of the proximal point algorithm on Hadamard manifolds arising from not well-defined convexity notions. In addition, I present a variational approach to find Nash equilibria on Hadamard manifolds by using fine properties of the projection map; as a surprising byproduct, it turns out that Hadamard manifolds represent the optimal geometric framework for the theory of Nash equilibria.  In the third part, I intend to present some elements from the optimal mass transport theory, – which have their roots in linear programming, – by providing simple proofs of the famous Brunn-Minkowski, Prékopa-Leindler and Borell-Brascamp-Lieb inequalities. 

Workshop 

Date: 8 September, 13:00–15:30 

Place: Corvinus University, Building C 510 and Zoom 

Miklós Pálfia: TBA 

Sándor Kristály: Sharp geometric inequalities on non-Euclidean settings: an optimal mass transport approach 

Based on optimal mass transportation, we present a quick introduction to the theory of CD(K,N) spaces, defined by J. Lott, K.-T. Sturm and C. Villani. [In the setting of Riemannian manifolds the CD(K,N) notion characterizes the objects having Ricci curvature at least K and dimension at most N.] We then prove a sharp isoperimetric inequality in CD(0,N) metric measure spaces assuming an asymptotic volume growth at infinity. As applications of this isoperimetric inequality, we establish Sobolev and Rayleigh-Faber-Krahn inequalities with explicit sharp constants in Riemannian manifolds with nonnegative Ricci curvature; here we also use appropriate symmetrization techniques and optimal volume non-collapsing properties. We also intend to discuss a few aspects of the above problems on Heisenberg groups, which are the simplest sub-Riemannian objects. 

You can inquire about the possibility of connecting online at the marianna.eisenberg-nagy@uni-corvinus.hu email address.  

We encourage all interested colleagues to participate in these interesting events.