Spectra of Hypergraphs

报告题目:Spectra of Hypergraphs

报告人:Joshua Cooper Associate Professor, Department of Mathematics, University of South Carolina



摘要: We present a spectral theory of hypergraphs that closely parallels graph spectral theory. Classic work by Gel'fand-Kapranov-Zelevinsky and Canny, as well as more recent developments by Chang, Lim, Pearson, Qi, Zhang, and others has led to a rich understanding of  "hyperdeterminants'' of hypermatrices, a.k.a. multidimensional arrays. Hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantially more complicated than the linear algebra required to understand spectral graph theory (i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of a tensor via its characteristic polynomial and variationally. We apply this notion to the "adjacency hypermatrix'' of a uniform hypergraph, and prove a number of natural analogues of graph theoretic results. Computations are particularly cumbersome with hyperdeterminants, so we discuss software developed in Sage which can perform basic calculations on small hypergraphs. Open problems abound, and we present a few directions for further research.

       Joint work with Aaron Dutle of the University of South Carolina.

报告人简介: Joshua Cooper obtained his BS in Mathematics from the Massachusetts Institute of Technology (MIT, Cambridge, MA, 1999), and his PhD in Combinatorics from the University of California, San Diego (UCSD, San Diego, CA, 2003) under advisers Fan Chung and Ron Graham. He spent time as a postdoctoral researcher at Microsoft Research (Redmond, WA), New York University (NYU, New York, NY), and the Swiss Federal Institute of Technology (ETH-Z, Zürich, Switzerland) before assuming his current position at the University of South Carolina (Columbia, SC), where he is an Associate Professor in the Mathematics Department. His research interests include quasirandomness, spectral hypergraph theory, discrete geometry, universal cycles, combinatorial number theory, coding theory, extremal graph theory, and permutation patterns.



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