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Wadim Zudilin from Moscow Lomonsov State University is going to visit School of Mathematics of IPM during January 17-February 6, 2005. During his visit, Prof. Zudilin will present two lectures and will conduct a short course in "Special Functions in Number Theory" at IPM.
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Short Course
|
| Special Functions in Number Theory |
| 1. |
Hypergeometric Series |
| Jan. 22, 10:00-12:00 |
|
Gauss hypergeometric series. Integral representations. Gauss'
summation formula. Irrationality of
log(2). Generalized
hypergeometric series. Summation and transformation formulae.
|
| 2. |
Hypergeometric Differential Equation |
| Jan. 23, 10:00-12:00 |
| Hypergeometric equation. Monodromy representation. Solutions around singular points. Schwarz map and Schwarz triangles. Schwarz's reflection principle.
|
| 3. |
Automorphic and Modular Functions |
| Jan. 30, 10:00-12:00 |
| Non-linear differential equations. Modular group. Modular forms.
Eisenstein series and Ramanujan's system of differential
equations. Modular invariant. Modular polynomial. Algebraic
independence results for modular functions. |
| 4. |
Ellptic Functions |
| Feb. 1, 10:00-12:00 |
| Lattices in $\mathbb C$ and elliptic curves. Isomorphism classes
of elliptic curves. Elliptic functions, basic properties.
Weierstrass $\wp$-function. Laurent expansion of the
$\wp$-function. Algebraic differential equation for the
$\wp$-function. Cubic plane curves. Realization of elliptic curves
in terms of the $\wp$-function. Applications in number theory. |
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Lectures
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| Computing Mathematical Constants |
| Jan. 24, 10:00-12:00 |
| Abstract. Mathematical constants, like $\pi$, Euler's constant $\gamma$,
Catalan's constant $G$, and values of the Riemann zeta function
at positive integers, are a curious part of mathematics.
We present two major (pure mathematical!)
methods in the high-precision evaluation of such constants, namely,
(1) accelerated convergence series, and
(2) polynomial recurrences. We also explain interaction
of the methods with the study of arithmetic properties
of the constants and indicate some open problems
in this respect. |
| Hypergeometric Integrals and Irrationality Proofs |
| Jan. 25, 10:00-12:00 |
Abstract. One of the hard problems in number theory is
proving that values of the Riemann zeta function
$$
\zeta(s)=\sum_{n=1}^\infty\frac1{n^s}
$$
at odd integers $s\ge3$ are irrational.
Several attempts were made to generalize
Ap\'ery's remarkable proof (1978) of the
irrationality of $\zeta(3)$, and to the moment
we dispose of some new partial arithmetic
results for $\zeta(5),\zeta(7),\zeta(9), ....
The aim of the talk is to show that hypergeometric
integrals appear quite naturally in the
corresponding irrationality proofs. |
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