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Tverberg-style theorems over lattices and other discrete sets.

Banff International Research Station for Mathematical Innovation and Discovery

This year we celebrate 50 year of the lovely theorem of Helge Tverberg! Let $a_{1},\ldots,a_{n}$ be points in $\mathbb{R}^{d}$. If the number of points $n$ satisfies $n >(d+1)(m-1)$, then they can be partitioned into $m$ disjoint parts $A_{1},\ldots,A_{m}$ in such a way that the $m$ convex hulls $Conv A_1, \ldots, Conv A_m$ have a point in common. Over the years many generalizations and extensions, including colorful, fractional, and topological versions, have been developed and are a bounty for discrete geometers. My talk will discusses yet another fascinating way to interpret Tverberg's theorem, now with a view toward number theory, lattices, integer programming, all things discrete not continuous nor topological. Given a discrete set $S$ of $\mathbb{R}^d$ (e.g., a lattice, or the Cartesian product of the prime numbers), we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points $A_{1},\ldots,A_{m}$ such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantitative integer versions of Helly's and Carath\'eodory's theorems. Joint work with subsets of the following wonderful people Reuben La Haye, David Rolnick, and Pablo Soberon, Frederic Meunier and Nabil Mustafa.

Mathematics; Convex and discrete geometry; Algebraic topology; Combinatorics

video/mp4

Author affiliation: University of California

2017-06-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61861

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/