Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 Spring 2024

Speaker:  Michael Dobbins, Binghamton University
Title: Colorful intersections and Tverberg partitions
Time: 2:30 PM, Monday, March 4, 2024
Place:  Malott 206

Abstract: Consider 6 convex bodies in $3$-space, 3 red and 3 blue, such that each red-blue pair intersects. Then, either there must be a line through all 3 red bodies or through all 3 blue bodies. With this observation as a starting example, we show that if m families of $k+r$ convex bodies each in $d$-space have the colorful intersection property, and if $d<(r+1)m/(k-1)$ and $k$ is a prime power, then one of the families is intersected by an affine $r$-flat. Moreover, we prove an interpolation between the colorful Helly theorem and Tverberg's theorem. As part of the proof we use discrete Morse theory to analyse the connectivity of a certain simplicial complex of partitions. This is joint work with Andreas Holmsen and Dohyeon Lee.

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