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Seminar & Colloquium

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제목 [AORC] A satellite conference of the Joint Meeting of the Korean Mathematical Society and the German Mathematical Societ
작성자 관리자 등록일 2018-10-08 조회수 132


Workshop on

barycenters, convexity on metric spaces


positive operators

SungKyunKwan University, October 8-9, 2018




In the last decade many results of functional analysis in Hilbert and Riemannian spaces have been extended to more general metric spaces that lack the smooth structure of Riemannian manifolds. In particular spaces with abstract curvature bounds became important objects to study. Alexandrov spaces, for instance CAT(k) spaces where the first non-smooth metric spaces which have been studied from the point of view of geometric analysis, then metric spaces with synthetic Ricci curvature bounds emerged from the works of Lott-Villani and Sturm. One of the key tools in this setting is to study convex functions and their gradient flows, in particular the relative entropy on the Wasserstein space. Another example is the squared distance function on a CAT(0) space which is uniformly convex, so one obtains a generalization of the arithmetic average of points in a Euclidean space by studying a convex optimization problem of squared distance functions that admits a unique solution, thus defines a barycenter for probability measures.

This barycenter has been successfully applied to the cone of positive definite matrices which is a CAT(0) space and lead to a useful non-commutative generalization of the geometric mean that satisfies all the desired properties from the point of view of matrix and operator theory. Techniques from gradient flow theory were applied to prove monotonicity properties of this mean. Currently possible extensions of this theory are being investigated in the infinite dimensional setting of positive operators, however this turns out to be challenging, since the metric spaces in that case, no longer admit sensible Riemannian structures. Instead they have Finslerian and Busemann NPC structure. This lack of inner product-like structures has lead to similar challenges in establishing a nice gradient flow theory in general on metric spaces as well.

This workshop aims to bring experts closer together working on non-smooth geometric analysis, probability and matrix/operator theory to exchange thoughts on the above topics.

Seminar & Colloquium 게시판의 이전글 다음글
이전글 [2018/10/08-12] Seminars and Talks of this week
다음글 [2018/09/20] AORC Monthly seminar