Exponential sums, differential equations and geometric Langlands correspondence

In 1970s, Dwork established a relationship between the Bessel differential equation and the Kloosterman sums. Such a relationship can be regarded as an instance of the geometric Langlands correspondence for $\mathrm{GL}_2$. In this talk, we will first review some classical results on exponential sums and differential equations, and then discuss some recent progress on generalizations of Dwork's result from the perspective of geometric Langlands correspondence. It is based on joint works with Xinwen Zhu and Kamgarpour-Yi.