Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Friday, May 04, 2018, 12:15 pm

Duration: 45 minutes

Location: CAB G11

Speaker: Nati Linial (The Hebrew University of Jerusalem)

Random simplicial complexes

About 50 years ago Erdos and Renyi have started the systematic study of random graphs. Their results concern mostly the G(n,p) model of random graphs. Such a random graph G has n vertices and every pair of vertices is independently made adjacent with probability p. They have discovered many important phenomena in this realm and have, in particular determined the critical p at which G becomes connected, where it has for the first time a "ginat" connected component of linear size and more.
Over a decade ago Roy Meshulam and I initiated the study of high-dimensional counterparts X=X(d,n,p) of G(n,p) graph. Here X is a set system (technically it is a simplicial complex) on a set V of n vertices. It contains every set of cardinality d or less and it contains every subset of V of cardinality d+1 independently and with probability p. Note that X(1,n,p) is identical to G(n,p). There are natural analogs of connectivity, cycles cuts etc. that come from basic topology (not to worry - NO knowledge of topology is assumed....). We are now able to answer questions analogous to the Erdos-Renyi theory, although the theory is much richer and many unexpected complications arise.
The only required background to follow this talk includes basic combinatorics linear algebra and probability. This line of research is by now an active area of research.
My list of collaborators in this endeavor includes: R. Meshulam, Y. Peled, L. Aronshtam, T. Luczak, M. Rosenthal, I. Newman and Y. Rabinovich.

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