Gauge Theory and the Geometric Langlands Program
Langlands program consists of a collection of surprising conjectures which relate algebraic geometry, algebraic number theory, harmonic analysis, and representation theory. On the other hand, Quantum Field Theory (and Gauge Theory) is the best existing framework of physics which describes our universe on the micro scale. In recent years there have been surprising applications of QFT to Geometric Langlands Program, the geometric version of Langlands program.
In this course, we will learn the mathematical ingredients composing the Geometric Langlands Program. At the same time, we will also learn the physical ingredients that make sense of supersymmetric gauge theories mathematically and find their relations to the original program, partially following the influential work of Kapustin and Witten. Towards the end, we will discuss some applications of the physics perspective that lead to new mathematical results.