Hypergeometric Functions (Fall 2015)
Teachers: F.Beukers (UU), G.Heckman (RU)
The classical hypergeometric function was introduced by Euler in the eighteenth century,
and was well developed in the nineteenth century, notably by Riemann, Schwarz and Klein.
Hypergeometric functions appear in many areas of mathematics and its applications.
Very often in mathematical physics in the 19th century, but also algebraic geometry in the 20th century.
In this course we start with a brief introduction of linear differential equations in the complex plane
and then focus on the hypergeometric differential equation and its solutions.
We start with the following classical subjects:
- Holomorphic ODE (ordinary differential equation), local theory
- Holomorphic ODE, global theory
- Monodromy, Riemann-Hilbert problem
- Euler-Gaus 2F1, first properties
- Monodromy and Schwarz-Klein theory (including Schwarz's list and criterium for arithmeticity of the monodromy group)
- Connection with modular forms and functions
Then we continue with more recent work:
- Clausen-Thomae hypergeometric function
- Levelt's theorem, Beukers-Heckman theorem
- Intermezzo on reflection groups
We end with a discussion why this admittedly old subject of hypergeometric functions
still attracts mathematicians of recent times
- Harmonic analysis for hyperbolic spaces
- Mirror symmetry
- Analogies in algebraic geometry and number theory
We will use course notes. In principle we follow the
notes by Gert Heckman.
As a backup you may use the notes
by Frits Beukers. However, they should still be checked for inaccuracies, etc,
hence the disclaimer on the first page of the notes.
Classes take place on Mondays 13:30 - 16:30 in room BBG165 (Buys Ballot Gebouw),
the first two hours will be a regular
lecture, the last one an exercise class.
Homework:
This will be assigned every two weeks and graded.
Exams: The final grade will be the average of a final exam(60%) and the
home work assignments (40%).
Material covered:
- September 7
Lecture, Chapter 1 (Linear differential equations, section 1.1 (The local
existence problem), 1.2 (The fundamental group), 1.3 (The monodromy representation).
Exercises: 1.9, 1.11
Backup notes: Section 1.1 and 1.2 (first half).
- September 14
Lecture, Chapter 1 (Linear differential equations, section 1.4 (Regular singular
points) 1.5 (Fuchs theorem).
Exercises: 1.13, 1.14, 1.16, 1.17
Home work assignment 1: 1.18, 1.19, 1.20 (deadline for hand in: September 28, 1:30 PM).
Backup notes: Section 1.2 (second half) and 1.3.
- September 21
Lecture, Chapter 2.1 Introduction Gauss hypergeometric function.
Exercises: see pdf.
Backup notes: Sections 1.5 and 2.1.
- September 28
Lecture, Chapter 2.2 (Riemann mapping, Schwarz derivative, Schwarz mapping theorem)
Exercises: first three problems of pdf.
Home work assignment 2 (deadline for hand in: October 12, 1:30 PM).
Backup notes: p 27,28,29 and Notes for today
- October 5
Lecture, Chapter 2.2, contiguity, triangle maps, algebraic hypergeometric functions
Exercises: problem 2, then problem 1 of pdf.
Backup notes: p29, 30
- October 12
Lecture on the Riemann-Hilbert problem, Chapter 1.6
There is (probably) no exercise class, as the lecture may take up some
more time.
Homework assignment 3: Exercises 2.7, 2.8, 2.9, 2.10 of
lecture notes by Gert Heckman
(deadline for hand in: October 26, 1:30 PM)
- October 19
Lecture, modular forms and the Schwarz mapping. For modular forms see sections 1 and 2 of
Modular form notes,
(just for reference, they are not part of the course),
for the Schwarz mapping: backup notes pp 29-30
Exercises: To be given in class
- October 26
Lecture, hypergeometric functions of higher order (Clausen-Thomae), chapter 3.1, 3.2
Exercises: 3.1 and 3.2
Home work assignment 4 (deadline for hand in: November 9, 1:30 PM).
- November 2
Lecture, hypergeometric functions of higher order (Clausen-Thomae), chapter 3.2, 3.3.
Exercises: 3.6, 3.7.
- November 9
Lecture, hypergeometric functions of higher order, algebraicity, Coxeter groups, section 3.3, 3.4
Home work assignment 5: Exercises 3.8, 3.10, 3.11 (deadline for hand in: November 23, 1:30 PM).
- November 16
Lecture, Euler integrals and De Rham cohomology, see notes
(unfortunately the pictures are still not in place).
Exercises 1.1 and 1.2 of the notes.
- November 23
Lecture, Integer monodromy, mirror maps, mirror symmetry. Unfortunately no notes yet,
but here is the pdf of an advanced paper by David Morrison
with a mathematician's interpretation of the work of Candelas et al.
Exercises: see pdf.
Home work assignment 6 (deadline for hand in: December 7, 1:30 PM).