Tali Kaufman, Bar-Ilan University

Abstract

The spectral notion of high dimensional expansion that goes by the name "local spectral expansion" was instrumental in recent works on approximate counting. However, for state of the art counting results stronger notions of high dimensional  expansion, as spectral independence and entropic independence, had to be considered. The importance in these stronger expansion notions is that the expansion guaranty that they provide does not deteriorate with the sample size (i.e. with the dimension). Alas, these newly investigated stronger notions of high dimensional expansion are known to hold only for dense distributions, like strongly log concave distributions, where the simplicial complexes that describe them have unbounded degree (and there is a short path between every two faces in the simplicial complex). In light of these recent stronger notions of high dimensional expansion, we ask: what is a strong notion of high dimensional expansion that applies also for sparse distributions (as the Ramanujan complexes) that does not deteriorate with the dimension; We give some initial result in this direction.

This presentation is based on joint work with Roy Gotlib