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# User:Sanath Devalapurkar

I am a high-school student at West High School, Torrance, California. I like to do mathematics, especially homotopy theory and related fields. More information about me is at the about page (though, admittedly, it's not a lot of additional information). My email address is devalapurkarsanath [at] gmail [dot] com. My homepage is at https://sites.google.com/site/homotopytheory/.

Currently, I'm interested in higher category theory, abstract homotopy theory, chromatic homotopy theory, structured ring spectra (stable homotopy theory in general), and derived algebraic geometry. I must say, though, that I'm also interested in mathematical physics, like quantum gravity and topological quantum field theories, but do not know nearly enough to contribute anything to those subjects.

I'll describe some of my current projects:

- Providing an answer to Gunnar Carlsson's question, which asks when the K-theory of spectrum of a permutative category can be written as a module spectrum over another K-theory spectrum of a bipermutative category. I want to use higher categorical methods to provide an answer to this. So the question I'm asking is, when is the K-theory of a unital ∞-operad equivalent to the ∞-category of O-modules over an O-algebra A over the K-theory of another ∞-operad (here O⊗is an ∞-operad)? In the form of an equation, when is K (C⊗∞) equivalent to ModOA(K(D⊗∞))? I've written up something about this at my website, https://sites.google.com/site/homotopytheory/.

- Creating a theory of (∞,n)-stacks to study some analogue of tmf in (∞,n)-category theory. What I mean is this: tmf arises as (connective cover of) the global sections of a sheaf of commutative ring spectra over the moduli stack of generalized elliptic curves. I want to study generalized elliptic curves in the derived context, to define the moduli (∞,n)-stack of these derived elliptic curves, and then define a sheaf of E∞-monoidal stable ∞-categories on this moduli (∞,n)-stack. But I probably can only do this in n=1, and for n>1 I probably need something that I don't know about.