When: Sunday, March 6, 10am
Where: Schreiber 309
Speaker: Shai Evra, Hebrew University
Title: Topological expanders
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.