Ali Caglayan / is on Nostr

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Last Notes

2025-01-12 12:30:37

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@nprofile…fa4k There are some algebraic geometers that study varieties in a similar way to the way you mention. Their approach works by noting that the derived category of quasicoherent sheaves on a scheme holds quite a lot of important geometric information. I think there are conjectures that it is a birational invariant even, but I am not certain since I don't recall the details.
This derived category has some nice structure that you can exploit in the case for projective spaces. For instance, you can pick a generating object (known as a tilting object) for the entire category. You can then use representation theory of other objects like quivers to gain a handle on those kinds of objects. Quivers are particularly nice because they have well understood moduli spaces of representations which make them suitable as a "presentation" of a derived category.
I asked a question about this on MO a while ago: https://mathoverflow.net/questions/285955/why-are-coherent-sheaves-on-bbb-p1-derived-equivalent-to-representations-of
I wonder if the POV with 2-rigs is in some way related to this. It has been a while since I've done AG however, so perhaps the relation is superficial at best.

2025-01-11 10:38:51

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@nprofile…fa4k could you elaborate on how projective geometry appears here?

2025-01-05 21:10:07

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@nprofile…fa4k I think another thing that ought to be mentioned is the brain drain that string theory has created in physics from the 80s onwards. This is of course all opinion, but a common saying I've heard is that big developments follow from funerals. There are some who believe that effort in physics is being misdirected at dead ends due to structural flaws in how academia works. I like mathematics and I do think string theory introduces interesting ideas, but it is not the kind of physics that was being practiced 70 years ago.

2025-01-05 21:10:07

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2025-01-04 19:38:06

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@nprofile…fa4k Isn't dim SO(10) = (20 + 25) rather?