Here are the places you can find me and my work around the ‘net:

Recently I have taken to writing notes for talks that I give and/or topics I am learning about. I do my best to include references and make them as clear as possible for my future self, should I need to go back to them. Hopefully they can help other people as well!

Brown Representability (pdf, github)

Notes for a talk I gave to my homological algebra course that built up to the results in Amnon Neeman’s The Grothendieck Duality Theorem via Bousfield’s Techniques and Brown Representability. Along the way we develop the notion of triangulated categories and of Brown Representability in its original context of (stable) homotopy theory. Eventually we show how Neeman recreated Brown’s original argument in the language of triangulated categories to establish important results regarding representability and adjointness of certain classes of functors.

Notes on the Grassmannian (pdf, github)

I was asked to participate in a reading group on the Grassmannian. My role was to describe the construction via smooth manifold theory of the Grassmannian as a quotient of GLn\mathrm{GL}_n. After motivating how we endow this set with a topology and smooth structure, I finish the discussion with a quick argument that this manifold is compact.

Along the way, I discovered that there had been a significant amount of work into recovering this classical geometric objects in the world of noncommutative algebraic geometry. Thus I dedicated the second half of my research to a paper of Taft and Towber in which they use the natural action of GL(n)\mathrm{GL}(n) on Gr(k,n)\operatorname{Gr}(k,n) and the quantized GLq(n)\mathrm{GL}_q(n) to construct a comodule which plays the role of a qq-analog of Gr(k,n).\operatorname{Gr}(k,n).

Schur Duality and Strict Polynomial Functors (pdf, github)

This is the paper I prepared for my general exam (and subsequent masters degree). I began from the foundational work of Schur in his 1901 thesis on the representations of the group GLn(k)\mathrm{GL}_n(k) and his discovery of the connection to representations of the symmetric group SrS_r via the so-called Schur-Weyl functor.

From there, we generalize this theory to a statement about (polynomial) representations of the (affine) group scheme GLn\mathrm{GL}_n over an arbitrary infinite field and show how this representation theory is properly encoded in terms of the representations of Schur algebras S(n,r)S(n,r). We then discuss strict polynomial functors developed by Friedlander & Suslin as well as Krause and Aquilino & Reischuk. We catalog their efforts towards establishing that the Schur-Weyl functor is monoidal under an appropriate monoidal structure on strict polynomial functors developed by Krause.

Finally we turn towards techniques of analyzing the representation theory of algebras and sketch a possible program for analyzing the structure of the bounded derived category Db(S(p,p))D^b(S(p,p)) for a prime pp over a field of characteristic pp.