Hypergeometric | Dali Shen's Homepage

Hypergeometric Functions (Fall 2023)

What is this course about?

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.

Prerequisite

Complex analysis

Literature

[1] F. Beukers, Gauss’ hypergeometric function, 2009, available at https://webspace.science.uu.nl/~beuke106/GaussHF.pdf
[2] F. Beukers, Hypergeometric functions in one variable, 2008, available at https://webspace.science.uu.nl/~beuke106/springschool99.pdf
[3] G. Heckman, Tsinghua lectures on hypergeometric functions, 2015, available at https://www.math.ru.nl/~heckman/tsinghua.pdf
[4] E. Looijenga, Uniformization by Lauricella functions – an overview of the theory of Deligne-Mostow, in: Arithmetic and geometry around hypergeometric functions, Progress in Mathematics 260, Birkhäuser Verlag Basel, 2007, 207–244.

Place and time

Mondays and Wednesdays 09:50-11:25
Venue: A3-4-101

ZOOM: 559 700 6085, PW: BIMSA

Starting from Sep. 18

WeChat Group

We have created a WeChat Group for communication. You could write to my TA Dr. Yi Liu (刘熠) if you would like to join.

Material covered

Sep. 18: An introduction to hypergeometric functions.

Sep. 20: The global theory of a linear differential equation.

Sep. 25: The local theory of a LDE.

Sep. 27: The monodromy representation.

Oct. 9: The monodromy representation, regular singular points.

Oct. 11: Regular singular points (continued).

Oct. 16: Regular singular points (continued), Fuchsian equations.

Oct. 18: Fuchsian equations (continued).

Oct. 23: The Riemann-Hilbert Problem.

Oct. 25: The Riemann-Hilbert Problem (continued).

Oct. 30: The Euler-Gauss hypergeometric functions.

Nov. 1: The Euler-Gauss hypergeometric functions (continued).

Nov. 6: The monodromy of the Euler-Gauss HF.

Nov. 8: The monodromy of the Euler-Gauss HF (continued).

Nov. 13: The Euler integral revisited.

Nov. 15: The Euler integral revisited (continued).

Nov. 20: The Clausen-Thomae hypergeometric functions.

Nov. 22: The Clausen-Thomae hypergeometric functions (continued).

Nov. 27: The monodromy of nFn-1.

Nov. 29: The criterion of Beukers-Heckman.

Dec. 4: A glimpse of Coxeter groups.

Dec. 6: Hyperbolic hypergeometric groups.

Dec. 11: Hyperbolic hypergeometric groups (continued), Monodromy of Lauricella functions.

Dec. 13: Monodromy of Lauricella functions (continued).

There is also a webpage at BIMSA dedicated to this course where you could find all the videos and notes for this course.