Modul:   MAT677  Applied Algebraic Topology

Topology of random simplicial complexes

Vortrag von Dr. Matthew Kahle

Sprecher eingeladen von: Prof. Dr. Thomas Kappeler

Datum: 26.03.10  Zeit: 15.00 - 17.00  Raum: Y27H28

Abstract: We will discuss three different kinds of random simplicial complex and what is known about the expected topological properties of each. (1) The Linial-Meshulam 2-complex. Nati Linial and Roy Meshulam found the threshold for vanishing of first homology, a result analogous to the Erdos-Renyi theorem. Eric Babson, Chris Hoffman, and I found the threshold for vanishing of the fundamental group to be very different, and the result is analogous to the threshold for vanishing of density (or triangular) random groups. Our main tool are ideas about negative curvature in groups and complexes due to Gromov. (2) The random clique complex. In my thesis I investigated a d-dimensional analogue of the random graph. Homology is no longer monotone in the underlying parameter, but we are nevertheless able to identify thresholds for vanishing and nonvanishing, compute the expected dimension of homology, and in more recent work,prove central limit theorems for Betti numbers. (3) Random geometric complexes. Trying to extend the results on random clique complexes to a geometric setting presents has several potential applications and presents new challenges. I will mention a few recent results in this area, including central limit theorems for Betti numbers within a certain range of parameters. (Part of this is ongoing joint work with Elizabeth Meckes.)