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High-dimensional expanders are used in a few areas of TCS (coding theory, sampling, probably some others). While I'm not too familiar with their usage, I know that in sampling they can be useful to analyze Markov Chains, and there are some useful local-to-global theorems concerning them. I was wondering if, since these HDXs are simplicial complexes, if tools like homology and cohomology can be used to say anything interesting about HDXs or something in that vein.

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High-dimensional expanders come in several flavors. The most common ones in TCS are spectral expanders, coboundary expanders, and cosystolic expanders. The latter two are defined using the algebraic topology setup. See Yotam Dikstein and Irit Dinur, Coboundary and cosystolic expansion without dependence on dimension or degree, for some pointers.

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  • $\begingroup$ Thank you, I hadn't heard of coboundary and cosystolic expanders before. $\endgroup$
    – p-adict
    Commented Nov 15, 2023 at 8:47

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