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

Date and Time: Thursday, May 24, 2012, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Anna Gundert

Expansion for 2-complexes

For graphs, combinatorial expansion properties are closely related to the eigenvalues of the adjacency matrix and the Laplacian. This relationship is expressed by the Cheeger Inequality, which states in particular that spectral expansion (a large spectral gap) implies combinatorial expansion. We consider generalizations of these graph matrices to higher dimensional simplicial complexes as well as a higher dimensional analogue of edge expansion, recently introduced independently by Gromov, Linial and Meshulam and Newman and Rabinovich.
We will see that in higher dimensions the most straightforward attempt at an analogue of Cheeger's Inequality fails: A large spectral gap for the generalized Laplacian doesn't imply combinatorial expansion. We furthermore present concentration results for the spectra of random complexes which match the corresponding results on spectra of random graphs.
Joint work with Uli Wagner

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