Lecture Notes
Asymptotic analysis
- Lecture 00
(January 8, 2024)
Notion of resummation. Cesaro, Abel and Euler summations.
- Lecture 01
(January 10, 2024)
Euler and Exponential-integral series. Borel resummation.
- Lecture 02
(January 12, 2024)
Laplace transforms. Stokes' phenomenon.
- Lecture 03
(January 17, 2024)
Asymptotic power series. Watson's lemma.
- Lecture 04
(January 19, 2024)
Stokes' automorphism - imprecise version.
- Lecture 05
(January 22, 2024)
Linear differential equations, summary and motivation.
- Lecture 06
(January 24, 2024)
Endlessly continuable germs. Convolution product.
- Lecture 07
(January 26, 2024)
Another look at (median) resummation and Stokes' automorphism.
- Lecture 08
(January 29, 2024)
Non-linear equations, Riccati equation.
- Lecture 09
(January 31, 2024)
Mellin transform and the zeta function.
- Lecture 10
(February 2, 2024)
Euler-Maclaurin summation formula. Asymptotics of infinite sums.
- Lecture 11
(February 5, 2024)
Bernoulli polynomials. Asymptotic formula for infinite
sums. Partition and divisor functions.
- Lecture 12
(February 7, 2024)
Duplication formula for gamma function. Derivatives
of zeta function.
- Lecture 13
(February 9, 2024)
Hardy-Ramanujan-Rademacher asymptotic formula.
- Lecture 14
(February 12, 2024)
Hypergeometric equation. Barnes' integral. Inverse
Mellin transform.
Elliptic integrals and functions
- Lecture 15
(February 14, 2024)
Elliptic integrals in various contexts.
- Lecture 16
(February 16, 2024)
Weierstrass' P-function.
- Lecture 17
(February 19, 2024)
P-function continued. Addition theorem.
- Lecture 18
(February 21, 2024)
SL(2,Z) action on the upper half plane.
- Lecture 19
(February 23, 2024)
Fourier series of modular functions.
- Lecture 20
(February 26, 2024)
Weight counting formula for modular forms.
- Lecture 21
(February 28, 2024)
Hecke operators and their algebraic properties.
- Lecture 22
(March 1, 2024)
Eigenvalues of Hecke operators. From modular
forms to Dirichlet series.
Factorization problems and singular
integral equations
- Lecture 23
(March 4, 2024)
Riemann-Hilbert factorization problem. Plemelj formulae.
- Lecture 24
(March 6, 2024)
Application to (scalar) Riemann--Hilbert problems.
Cauchy's inversion formula.
- Lecture 25
(March 8, 2024)
Integral equations - Fredholm theory.
- Lecture 26
(March 18 and 20, 2024)
Fredholm equations continued.
- Lecture 27
(March 22, 2024)
Birkhoff factorization theorem.
Riemann surfaces
-
- Lecture 28
(March 25, 2024)
Riemann surfaces, maps between them. Local behaviour
of holomorphic maps.
- Lecture 29
(March 27, 2024)
Genus and Riemann-Hurwitz formula.
- Lecture 30
(March 29, 2024)
Sheaf of germs of holomorphic functions. Riemann
surface of analytic continuation.
- Lecture 31
(April 1, 2024)
Compact coverings vs algebraic germs.
- Lecture 32
(April 3, 2024)
Differential forms: holomorphic and meromorphic.
Line integrals.