[Binghamton Geometry Topology Seminar] Special joint meeting

[jwilliams at math.binghamton.edu]

All, I forgot about a joint GT-Combinatorics seminar meeting tomorrow.

Special date and time: May 1, 1:15pm in WH-100E

Speaker: Boris Bukh (Carnegie Mellon)

Title: Topological Version of Pach's Overlap Theorem

Abstract: Consider the collection of all the simplices spanned by some
n-point set in $\mathbb{R}^d$. There are several results showing that
simplices defined in this way must overlap very much. In this talk I focus
on the generalization of these results to 'curvy' simplices.

Specifically, Pach showed that every $d+1$ sets of points $Q_1, ...,
Q_{d+1}$ in $\mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$
such that all the transversal simplices that they span intersect. In joint
work with Alfredo Hubard, we show, by means of an example, that a
topological extension of Pach's theorem does not hold with subsets of size
$C(\log n)^{1/(d-1)}$. We show that this is tight in dimension 2, for all
surfaces other than $S^2$. Surprisingly, the optimal bound for $S^2$ is
$(\log n)^{1/2}$. This improves upon results of Bárány, Meshulam, Nevo,
and Tancer.

Combinatorics announcement:
http://seminars.math.binghamton.edu/ComboSem/abstract.201805buk.html