Zhiwei Yun 恽之玮 (MIT)

Hitchin moduli spaces and ramified geometric Langlands (week 1)

I will start with a general introduction of Hitchin moduli space and its role in geometric representation theory. Then I will focus on the specific Hitchin moduli spaces for Higgs bundles on ${\mathbb{P}}^1$ with level structures that were introduced and studied in joint work with Bezrukavnikov, Boixeda-Alvarez and McBreen. I will explain their role in formulating special cases of the ramified geometric Langlands conjecture, and give evidence for the conjecture. (recommended reading)

Copilot: Zeyu Wang 王泽宇

Tsao-Hsien Chen 陈朝铣 (U. of Minnesota)

Langlands Duality and Symmetric Varieties (week 1)

In this course I will discuss recent progress on Langlands duality for symmetric varieties. I will review the structure theory for symmetric varieties $X=G/K$ for a complex connected reductive group $G$ and explain combinatorial constructions of dual groups $G_X$ of symmetric varieties. Then I will explain the work of Gaitsgory-Nadler on relative geometric Satake equivalence which provides a geometric construction $G_X$ using perverse sheaves on the loop space of $X$. Finally, I will discuss the derived version of the story and its connections to Coulomb branches and geometric Langlands for real groups. (recommended reading)

Sam Raskin (Yale)

Geometric Langlands for projective curves (week 2)

The geometric Langlands equivalence asserts that (suitable) sheaves on the space of $G$-bundles on $X$ and on the space of $\check{G}$-local systems are equivalent categories. I will provide some background on the conjecture(s) and highlight some aspects of its proof. (recommended reading)

Roman Bezrukavnikov (MIT)

Some applications of affine Springer theory (week 2)

Springer fibers are subvarieties in a flag variety of a reductive group, playing an important role in some questions of representation theory. Their loop group counterparts known as affine Springer fibers are more complicated geometrically, they turn out to be connected to problems about characters of $p$-adic groups, to quantum groups etc. I will survey some of these connections. (recommended reading)