Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, June 03, 2010, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Yuri Rabinovich (University of Haifa)

Finite volume spaces and related issues

We introduce and study finite volumes, the high dimensional generalization of finite metrics. Notions, concepts and methods of the theory of finite metric spaces often extend to finite volume spaces, leading to new intriguing problems, as well as to new perspectives in the theory of simplicial complices. For example, introducing the class of L_1 volumes (an analogue of L_1 metrics), and studying how well can they approximate an arbitrary volume function, we naturally arrive at the notion of expansion of a simplicial complex. High dimensional analogues of network flows, graph spanners and spectral gaps also arrise rather naturally in this contex. We shall discuss these notions, and present some related results.

A separate issue is the dimension reduction for L_1 metrics and volumes. We introduce new tools (related to the sparsification techniques of Karger et al., and Spielman et al.), and show that there is indeed a dimension reduction phenomenon for L_1 volumes, although not to the same striking degree as in the Euclidean case. For L_1 metrics we show a linear upper bound on number of dimensions, improving the previously best known O(n log n) bound due to Schechtman.

Based on a joint work with Ilan Newman.

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