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 strengthen 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.

He changes 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 exits bounded degree simplicial complexes with the above geometric/topological properties, i.e. "bounded degree geometric/topological expanders".

The geometric problem was answered in 2013 by Fox-Gromov-Lafforgue-Naor-Pach who showed that the Ramanujan complexes constructed by Lubotzky-Samuels-Vishne give rise to such geometric expanders, but left open the more difficult topological question.

This question was answer affirmatively for dimension d=2, by Kaufman, Kazhdan and Lubotzky. By extending the method of proof of Kaufman, Kazhdan and Lubotzky: we show that the (d-1)-skeletons of the d-dimensional Ramanujan complexes give bounded degree topological expanders.

This is a joint work with Tali Kaufman.

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 strengthen 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.

He changes 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 exits bounded degree simplicial complexes with the above geometric/topological properties, i.e. "bounded degree geometric/topological expanders".

The geometric problem was answered in 2013 by Fox-Gromov-Lafforgue-Naor-Pach who showed that the Ramanujan complexes constructed by Lubotzky-Samuels-Vishne give rise to such geometric expanders, but left open the more difficult topological question.

This question was answer affirmatively for dimension d=2, by Kaufman, Kazhdan and Lubotzky. By extending the method of proof of Kaufman, Kazhdan and Lubotzky: we show that the (d-1)-skeletons of the d-dimensional Ramanujan complexes give bounded degree topological expanders.

This is a joint work with Tali Kaufman.

## Date:

Thu, 26/11/2015 - 14:30 to 15:30

## Location:

Manchester Building (Hall 2), Hebrew University Jerusalem