Geometric Langlands theory originates from trying to understand certain aspects of Langlands program for number fields in a more geometric setting. This turns out to be a surprisingly rich topic, having to do with various subjects ranging from number theory to quantum field theory in many different guises.
In this course, we focus on a global unramified categorical conjecture over the complex number field, following Beilinson and Drinfeld's ideas in 90's. There have been many important developments since then, but in its most recent formulation, the conjecture involves heavy use of derived algebraic geometry.
In the first third of the course, we review some of the old ideas (which tend to be concrete) in the subject. In the second third, we develop necessary background for derived algebraic geometry and explain the statement of the main conjecture as formulated by Arinkin and Gaitsgory. Finally, in the last part, we discuss some of the important recent ideas toward proving the conjecture.
The focus of the lectures is not necessarily just on formulating and proving the conjecture; We will aim to discuss some of the central topics in geometric representation theory along the way.
Course notes by Alex Atanasov and by Yau Wing Li