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SUMMARY:Geometric Langlands duality with generalized coefficients (VIRTUAL
 ) - Sanath Devalapurkar (Harvard University)
DTSTART:20250619T143000Z
DTEND:20250619T153000Z
UID:TALK230704@talks.cam.ac.uk
DESCRIPTION:Recent years have seen a flurry of developments relating homot
 opy theoretic constructions with results in arithmetic geometry. In this t
 alk\, I will discuss another place where homotopy theory plays an importan
 t organizational role\, namely geometric representation theory.&nbsp\; I w
 ill explain some conjectures surrounding the derived geometric Satake equi
 valence\, which relates a category of constructible sheaves of k-modules o
 n the affine Grassmannian of a complex Lie group G to the algebraic geomet
 ry of its Langlands dual group G^. When k is the sphere spectrum and G is 
 the trivial group\, this relationship is just the Adams-Novikov spectral s
 equence\; and when k is an ordinary commutative ring (and G can be nontriv
 ial)\, this reduces to Bezrukavnikov-Finkelberg's derived Satake theorem.&
 nbsp\; Time permitting\, I will explain how these ideas allow one to const
 ruct an associative algebra U_F(G) associated to any 1-dimensional formal 
 group law F such that when F is the additive formal group law\, U_F(G) is 
 the usual enveloping algebra U(g)\, and when F is the multiplicative forma
 l group law\, U_F(G) is the quantum group U_q(G) (closely related to q-de 
 Rham cohomology\, etc.). There are many questions in this area that I do n
 ot know how to address\, and I hope to share some of them.
LOCATION:Seminar Room 1\, Newton Institute
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