# Combinatorics Seminar

When: Sunday, March 6, 10am

Where: Schreiber 309

Speaker:
Shai Evra, Hebrew University

Title:
Topological expanders

## Abstract:

A classical result of Boros-Furedi (for d=2) and Barany (for d>=2) from
the 80's, asserts that given any n points in R^d, there exists a point in
R^d which is covered by a constant fraction (independent of n) of all the
geometric (=affine) d-simplices defined by the n points. In 2010, Gromov
strengthened this result, by allowing to take topological d-simplices
as well, i.e. drawing continuous lines between the n points, rather then
straight lines and similarly continuous simplices rather than affine.

Gromov changed the perspective of these questions, by considering the
above results as a result about geometric/topological expansion properties
of the complete d-dimensional simplicial complex on n vertices. He asked
whether there exist bounded degree simplicial complexes with the above
topological properties, i.e. "bounded degree topological expanders".

This question was answered affirmatively for dimension d=2 by Kaufman,
Kazhdan and Lubotzky. By extending the method of proof of Kaufman, Kazhdan
and Lubotzky, we gave a solution to the general problem, showing that the
(d-1)-skeletons of the d-dimensional Ramanujan complexes give bounded
degree topological expanders.

This is a joint work with Tali Kaufman.

redesigned by barak soreq