CCOR mini-course and workshopSándor Kristály, professor at Babes-Bolyai University and at Óbuda University will visit Corvinus Institute for Advanced Studies (CIAS) at the beginning of September. CIAS organizes a workshop entitled “Optimization on curved spaces” with two speakers, Sándor Kristály and Miklós Pálfia. Sándor Kristály will also hold an introductory mini-course including three presentations in this topic.
7 September, 10:00-11:30, 13:00-14:30
8 September, 9:30-11:00
Place: Corvinus University, Building C 428 and Zoom
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.
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 firstname.lastname@example.org email address.
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