Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, May 08, 2014, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: May Szedlák

Discrete Higher Dimensional Cheeger Inequalities

For graphs there exists a strong connection between spectral expansion and edge expansion. This is expressed, e.g., by the Cheeger inequality, which states that \lambda(G) \leq h(G), where \lambda(G) is the second smallest eigenvalue of the Laplacian of G and h(G) the Cheeger constant measuring the edge expansion of G. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension. Whereas for simplicial complexes, higher dimensional Laplacians were introduced already in 1945 by Eckmann, the generalization of edge expansion is not straightforward. Recently, a topologically motivated notion analogous to edge expansion was introduced by Gromov and independently by Linial, Meshulam and Wallach and by Newman and Rabinovich. It is known that for this generalization there is no higher dimensional analogue of the lower bound of the Cheeger inequality. A different, combinatorially motivated generalization of the Cheeger constant, denoted by h(X), was studied by Parzanchevski, Rosenthal and Tessler. They showed that indeed \lambda(X) \leq h(X), where \lambda(X) is the smallest non-trivial eigenvalue of the Laplacian, for the case of k-dimensional simplicial complexes X with complete (k-1)-skeleton. Whether this inequality also holds for k-dimensional complexes with non-complete (k-1)-skeleton has been an open question. We give two different strengthenings of the inequality for arbitrary complexes.

Upcoming talks     |     All previous talks     |     Talks by speaker     |     Upcoming talks in iCal format (beta version!)

Previous talks by year:   2024  2023  2022  2021  2020  2019  2018  2017  2016  2015  2014  2013  2012  2011  2010  2009  2008  2007  2006  2005  2004  2003  2002  2001  2000  1999  1998  1997  1996  

Information for students and suggested topics for student talks

Automatic MiSe System Software Version 1.4803M   |   admin login