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I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow.

I've just come across a Twitter thread by Laurent Fargues explaining a work by Dustin Clausen & Peter Scholze under the name condensed mathematics and how condensed mathematics can algebrize topology and analysis. I find it somewhat surprising (or hard to believe) that you can do such a thing. So my question is can condensed mathematics completely algebrize topology and analysis, or can it only algebrize some analytic object close to algebraic geometry? Can we reprove theorem and inequalities in analysis such as mean value theorem, Hölder's inequality, dominated convergence theorem, and so on using condensed mathematics?

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    $\begingroup$ You're absolutely right. There's no way condensed math can "completely algebrize" topology and analysis, nor even close. I prefer to say that what it's good for is mixing topology and analysis with algebra. It does this by recasting topology and certain aspects of analysis in a way which is more friendly to algebra. $\endgroup$
    – Dustin
    Apr 14, 2022 at 12:41
  • $\begingroup$ Thanks for the reply, so I guess it is probably very useful for algebraists, but not much of an interest for an analyst then? $\endgroup$
    – Ythyb
    Apr 14, 2022 at 12:44
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    $\begingroup$ Reading that Twitter thread made me wonder if future math textbooks will have emojis scattered throughout them. After all, en.wikipedia.org/wiki/Bourbaki_dangerous_bend_symbol is a kind of proto-emoji... $\endgroup$ Apr 14, 2022 at 13:21
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    $\begingroup$ The title of Dustin Clausen's talk New foundations for functional analysis (a video is, e.g., here youtube.com/watch?v=qKC0fciQkfU) is not as modest as the comment of the user Dustin above. $\endgroup$ Apr 15, 2022 at 13:48

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