Beijing Institute of Mathematical Sciences and Applications

The classical hypergeometric function was introduced by Euler in the 18th century, and was extensively studied in the 19th century by Gauss, Riemann, Schwarz and Klein, among others. For its frequent occurrences in many branches of science, it was then generalized naturally in two directions: with more parameters and in more variables. It turns out that these functions pack a lot of information: geometric, algebraic, arithmetic and so on. To name a few, they, as a guiding example, lead to the formulation of the Riemann–Hilbert Problem (#21 in Hilbert’s famous list of problems), and they even have a connection with the Prime Number Theorem.
In this course, I will give a basic introduction to the properties of various hypergeometric functions, with an emphasis on the monodromy of their accompanying hypergeometric equations.