# Moduli spaces, stacks and homotopy theory

For my final-year project (not this academic year but next) I'm hoping to write a relatively complete account of the basic theory of schemes used in modern algebraic geometry. My supervisor thinks that as long as I work hard I should have enough time to study some simple moduli spaces as well. It's quite an ambitious undergrad project but I'm hoping to use it as a stepping stone towards a PhD working with moduli, stacks and other "very general" geometric objects. However I'm also quite fond of topology and a lot of the homotopy theory that I've glanced over has really interested me, particularly that which uses higher categories and other very abstract machinery, and I'd like to work in an area which combines both geometry and topology with a lot of category theory thrown in. I've still got more than a year to think about which subfield to go into with the PhD but I'd like to get some ideas now so I can aim this project in the right direction.

Can anyone suggest some particular ideas or general areas in the intersection of algebraic geometry and topology which would be relevant to look into? I'm happy to accept answers which might be way too advanced for my knowledge at the moment - I'm just trying to get some ideas. Thanks in advance!

## 1 Answer

The Grassmannian is a moduli space and a projective variety. In topological K-theory one considers homotopy classes of maps from a space into an infinite-dimensional Grassmannian. The main connection here is that Grassmannians are involved; the topology of the vector bundles on arbitrary spaces isn't really related to the algebraic geometry of the Grassmannian, but one could say Grassmannians do lie at the "intersection."

• I've just read a little bit about it and K-theory in general seems like something I need to learn more about. The wide scope plus links across the board in geometry, topology, algebra, string theory etc. is exactly the sort of thing I'm interested in. Thanks very much! – Alex Saad Jan 2 '15 at 18:01
• @XeSaad you're welcome. I should warn you though that "topological K-theory" and "algebraic K-theory" are very different. Algebraic K-theory is the one most relevant to algebraic geometry. $K_0$ of a variety is the Grothendieck group of locally free modules over the sheaf of regular functions (which are essentially the same as vector bundles). $K_0$ of a ring is the Grothendieck group of finitely generated projective modules over the ring. – Matt Samuel Jan 2 '15 at 18:06