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Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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What is a reasonable finitary analogue of the statement that harmonic functions are smooth?

There are a few tidbits in this blog post of Tao's from a few months back. He talks about Lipschitz harmonic functions on groups (it's in the context of Gromov's theorem about finitely generated group …
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