High Dimensional Expanders Luis Kumanduri MIT 1 / 3 What is an - PowerPoint PPT Presentation

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High Dimensional Expanders Luis Kumanduri MIT 1 / 3 What is an expander? Definition Let X be a d -dimensional simplicial complex. X is an -topological expander if for every continuous F : X R d , there is a point p R d so that F

High Dimensional Expanders Luis Kumanduri MIT 1 / 3
What is an expander? Definition Let X be a d -dimensional simplicial complex. X is an ǫ -topological expander if for every continuous F : X → R d , there is a point p ∈ R d so that F − 1 ( p ) meets an ǫ fraction of the d -dimensional faces of X . Theorem (Gromov) If X has large cosystoles, satisfies a co-isoperimetric inequality and is sparse, then X is a topological expander. 2 / 3
Questions Question Can we develop better tests for expansion? In particular, what topological/geometric properties does expansion imply? Question Can we algorithmically estimate the expansion constant for a given complex? Somewhat relatedly, can we improve the bounds on the constant in Gromov’s theorem? 3 / 3

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