Title: Hypergeometric functions and Langlands correspondences
Abstract: The Modularity Theorem, established by Wiles ('95) and Breuil, Conrad, Diamond, and Taylor ('01), is one of the most significant results in modern number theory, giving deep connections between the seemingly disparate fields of algebraic geometry, complex analysis, and algebraic number theory, as well as finally resolving Fermat's Last Theorem. It is also one of the simplest cases of a far broader spectrum of conjectures comprising the Langlands Program. We give a brief overview of the ideas captured by modularity and the Langlands program. We will also discuss a recent approach to establishing Langlands correspondences between abelian varieties and automorphic forms by studying p-adic congruences of specialized hypergeometric functions.