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AMSS-YMSC-BIMSA Joint Seminar on Progress of Topology and Its Applications
Depth in arrangements: Dehn--Sommerville--Euler relations with applications
Depth in arrangements: Dehn--Sommerville--Euler relations with applications
Organizers
Haibao Duan
, Fei Han
, Yong Lin
, Jianzhong Pan
, Guo-Wei Wei
,
Jie Wu
, Kelin Xia
,
Shing Toung Yau
, Chao Zhou
Speaker
Herbert Edelsbrunner
Time
Thursday, May 26, 2022 4:00 PM - 5:00 PM
Venue
1110
Online
Zoom 361 038 6975
(BIMSA)
Abstract
The depth of a cell in an arrangement of $n$ (non-vertical) great-spheres in $S^d$ is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn--Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.
This is work with Ranita Biswas, Sebastiano Cultrera, and Morteza Saghafian.
Speaker Intro
Herbert Edelsbrunner is one of pioneers for topological and geometric approaches to data science, by his excellent research on the topic during the decades, from his PhD thesis on computational geometry in 1982 and assistant professorship in Information Processing at the Graz University of Technology during 1981-1985, to his professorship positions at UIUC, Duke and ISTA. Professor Edelsbrunner received Alan T. Waterman Award, he is members of the Academia Europaea, the American Academy of Arts and Sciences and the German Academy of Sciences Leopoldina. Professor Edelsbrunner is widely recognized as one of founding fathers of the new-born area of topological data analysis (TDA).