Homogeneous Adjacency Spectra of Random and Complete Hypergraphs

Combinatorics Seminar
Friday, February 7, 2014 - 15:05
1 hour (actually 50 minutes)
Skiles 005
University of South Carolina
Abstract: There has been a recent flurry of interest in the spectral theory of tensors and hypergraphs as new ideas have faithfully analogized spectral graph theory to uniform hypergraphs.  However, even in their simplest incarnation -- the homogeneous adjacency spectrum -- a large number of seemingly basic questions about hypergraph spectra remain out of reach.  One of the problems that has yet to be resolved is the (asymptotically almost sure) spectrum of a random hypergraph in the Erd\H{o}s-R\'{e}nyi sense, and we still don't know the spectrum of complete hypergraphs (other than a kind of implicit description for 3-uniform).  We introduce the requisite theoretical framework and discuss some progress in this area that involves tools from commutative algebra, eigenvalue stability, and large deviations.