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

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Sep 05, 2022 168 likes •213 views

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

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High Dimensional Expanders

Luis Kumanduri

MIT

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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 → Rd, there is a point p ∈ Rd 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.

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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?

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