Edward Frenkel is here this semester in the math department at Columbia, and he’s giving a series of lectures on a topic dear to my heart. Video of his lectures on The Langlands Program and Quantum Field Theory is starting to be available, courtesy of our graduate students Alex Waldron and Ioan Filip, as well as our staff member Nathan Schweer.
The first lecture last week was an overview, outlining the general picture of the Langlands program in the number field, function field and geometric cases, as well as two sorts of connections to QFT (to certain 2d conformal field theories, and to S-duality in 4d super Yang-Mills). This week he started to get more specific, giving some details about how the Langlands program works in the function field case, in preparation for moving next week to the geometric analog where a curve over a finite field gets replaced by a Riemann surface. As an indication of references covering much of the material to be discussed in lectures, Frenkel suggests this survey article and this Seminaire Bourbaki report.
Frenkel is also working on some other different but quite interesting projects. With Ngo and Langlands he has a program to “geometrize” the trace formula, for details see his very recent AMS Colloquium Lectures. With Losev and Nekrasov he has a fascinating program for studying certain field theories using instantons in a very different limit than the usual semi-classical one. See here, here, and here.