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DTSTART;TZID=Asia/Seoul:20220519T161500
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SUMMARY:Gil Kalai\, The Cascade Conjecture and other Helly-type Problems
DESCRIPTION:[Colloquium\, Department of Mathematical Sciences\, KAIST] \nFor a set $X$ of points $x(1)$\, $x(2)$\, $\ldots$\, $x(n)$ in some real vector space $V$ we denote by $T(X\,r)$ the set of points in $X$ that belong to the convex hulls of r pairwise disjoint subsets of $X$.\nWe let $t(X\,r)=1+\dim(T(X\,r))$. \nRadon’s theorem asserts that\nIf $t(X\,1)< |X|$\, then $t(X\, 2) >0$. \nThe first open case of the cascade conjecture asserts that\nIf $t(X\,1)+t(X\,2) < |X|$\, then $t(X\,3) >0$. \nIn the lecture\, I will discuss connections with topology and with various problems in graph theory. I will also mention questions regarding dimensions of intersection of convex sets. \nSome related material:\n1) A lecture (from 1999): An invitation to Tverberg Theorem: https://youtu.be/Wjg1_QwjUos\n2) A paper on Helly type problems by Barany and me https://arxiv.org/abs/2108.08804\n3) A link to Barany’s book: Combinatorial convexity https://www.amazon.com/Combinatorial-Convexity-University-Lecture-77/dp/1470467097
URL:https://dimag.ibs.re.kr/event/2022-05-19/
LOCATION:Zoom ID: 868 7549 9085
CATEGORIES:Colloquium
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