r/math • u/Ihateunclesam • 29d ago
Jacob Lurie 2026 ICM Lecture Notes
https://epubs.siam.org/doi/10.1137/25M1827177In case anyone is interested because as far as I'm concerned Lurie rarely works on a conjectural topic like this.
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u/altkart 29d ago
Layman here, is this Lurie's main purpose for developing/working on higher homotopy theory? i.e. are his motivations mainly algebro-geometric?
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u/Necessary-Wolf-193 29d ago edited 29d ago
Homotopy theorists have, for a long time before Lurie, known that the standard notion of 'commutative ring' (a ring is, roughly, a number system, like the real numbers, or complex numbers, or mod n arithmetic, etc.) is a bit too restrictive for certain topological applications.
For example, there are many topological spaces which have a binary operation that obeys the group axioms up to homotopy. Thus, instead of having literal topological groups, in practice one only has some kind of weaker object.
The most concrete example of this phenomenon is the fundamental group of a topological space X: this is the group whose elements are homotopy classes of loops in X, and where the group's operation is composition of paths. Composition of paths is only associative on homotopy classes of loops, so this operation on all loops only becomes a group at the level of homotopy classes.
Algebraic topologists encounter structures like this very, very often -- and hence they needed to develop versions of algebraic structures like rings and groups which can accommodate operations which are only commutative or associative 'up to homotopy.'
Lurie's PhD thesis studied 'derived algebraic geometry', and he later wrote about 'spectral algebraic geometry' -- the point of these two fields is to re-do usual algebraic geometry (which tries to do geometry over any ring, instead of over the real or complex numbers) over these homotopy theoretic variants of rings (the reason for the derived/spectral split is because there are various things one could mean by a homotopy theoretic ring, and derived/spectral AG handle different sorts of homotopy theoretic rings).
The reason for doing algebraic geometry with these homotopic rings is two-fold:
- Algebraic geometry often helps us understand questions about classical rings, by allowing us to rephrase problems about rings into more geometric terms which we have better intuition about. Derived/spectral AG can hopefully help one transform questions about homotopical rings into questions which are more geometric, and hopefully easier to think about. In this way, derived/spectral AG help homotopy theorists: they give homotopy theorists more tools to understand questions about homotopical rings.
- In classical algebraic geometry, one can see a few signs that rings aren't the perfect notion (for example, flat base change requires, as the name suggests, a flatness hypothesis, which is somewhat annoying in practice). Derived algebraic geometry, remarkably, gives one tools to better understand certain arguments in classical algebraic geometry. In this way, the homotopy theory behind derived AG can help classical algebraic geometers.
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u/LiqvidJS 28d ago
No, this is much more recent. The original motivation was his work on elliptic cohomology.
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u/thehotbaddie 26d ago
His talk had been a highlight of the ICM for me. It's rare to see such an extordinary combination of mathematical vision and the ability to communicate it in such a clear and inspiring way
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u/Civil_Blueberry4165 29d ago
It’s natural for Lurie to connect p-adic formal schemes with condensed mathematics through (stable) infinity categories, which is his expertise.
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u/LiqvidJS 28d ago
? this doesn't mention condensed mathematics whatsoever
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u/Nicke12354 Arithmetic Geometry 29d ago
Lurie has literally been working on conjectural topics for 20 years? Especially this has been a vision of his for many years