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added pointer to today’s
The introduction makes the interesting claim that the duality between geometry and commutative algebra is due to Descartes’ La Géométrie from 1637.
I didn’t know this. What does Descartes actually say there?
I’m pretty sure he doesn’t speak of duality. It’s a method of translating between geometry and algebra. This allows him to go beyond the Ancients and solve Pappus’s problem.
I believed that I could borrow all that was best both in geometrical analysis and in algebra, and correct all the defects of the one by help of the other. (The Discourse on Method)
There’s a lot of discussion at that time of what counts as a proper geometric curve. Algebra and the degree of equations helps here, but there’s little of the modern sense of a totality which could feature in a duality.
Connes suggests that a full understanding of the RH will require both the notion of non-commutative space and the concept of topos (due to Grothendieck) as the ever more general definitions of what a geometric space must mean. This would entail a third paradigm shift. (Lupercio)
Do we know what Connes takes to be what NCG brings to topos theory? Is there anything that isn’t taken up in higher topos theory?
I don’t know what that quote is after (heard some things, but don’t want to be guessing), maybe somebody else here does.
(I had lost interest in this approach to RH after reading Connes-Bost and Connes-Marcolli. The promising geometric attack on RH seems, to me, to be Yau’s.)
But I do know what the right unification of higher topos theory and non-commutative geometry is:
Tabuada-style non-commutative motives, namely simply well-behaved stable $\infty$-categories. By the Schwede-Shipley-style theorems (here) these are $\infty$-categories of modules over $A_\infty$-algebras(algebroids) and as such are the higher “toposes” of abelian $\infty$-stacks on non-commutative $\infty$-spaces.
There is a zoom conference that may be of interest although I do not know if NCG will be mentioned: here
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