2025-12-26 yt Peter Scholze is Brightest Mathematician in a Generation
claims that topological spaces are the wrong definition, and that one should replace them with the slightly different notion of condensed sets.
Let me now explain what condensed sets are. Clausen and I came to it from different trajectories (and Barwick and Haine yet from a different one); there are some precursors in the literature, notably Spanier’s quasi-topological spaces (in algebraic topology) and Waelbroeck’s compactological spaces (in functional analysis). For me, it started (unsurprisingly…) in relation to my work on perfectoid geometry.
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perfectoid spaces
yt Terry Tao on the future of mathematics
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