Workshop: Topics in algebraic number theory and Diophantine approximation
Date: March 12th – March 22nd 2017
Place: Mathematics Department, College of Science, Salahaddin University/Erbil-Kurdistan Region/IRAQ.
Lecturers
Michel Waldschmidt
Emeritus Professor, Universit´e P. et M. Curie (Paris VI)
Francesco Pappalardi, full Professor of Algebra
Università Roma Tre, Italy
Valerio Talamanca, Professore a contratto,
Università Roma Tre, Italy
local organizer coordinator
Kawa M. A. MANMI
Head of the Mathematics Department, College of Science,
Salahaddin University/Erbil-Kurdistan Region/IRAQ
Scientific committee
1. Herish O. Abdullah, The Dean of College of Sciences, department of Mathematics,
Salahaddin University/Erbil-Kurdistan Region/ – IRAQ
2. Kawa M. A. Manmi, Head of the Mathematics Department,
College of Science, Salahaddin University, /Erbil-Kurdistan Region/IRAQ.
3. Valerio Talamanca, Università Roma Tre
4. Michel Waldschmidt, Université Paris 6
The school will consist of the following four courses:
1. Topics in algebra, Pierre Cartier
2. Cyclotomy, Francesco Pappalardi
3. Mahler measure of polynomials, Valerio Talamanca
4. Introduction to Diophantine Approximation, Michel Waldschmidt
Description of each course
Cyclotomy
A Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods.
* Reminders of the Galois Theory of cyclotomic polynomial
* Gauss sums and Gauss periods
* the determination of the quadratic Gauss Sum
* Gauss Theorem for expression of roots of unity via nested radicals
* The computation of cos 2=17
* Kummer’s problem on cubic Gauss periods
Mahler measure of polynomials The Mahler measure of a polynomial (with integer coefficients) is a measure of its arithmetic complexity as well as the arithmetic complexity of its roots. Lehmer’s conjecture asserts that the Mahler measure of non cyclotomic polynomials is bounded below by an absolute constant. We will
use this very explicit problem to introduce a few concept of algebraic number theory.
* Mahler measure of a polynomial
* Algebraic integers and their minimal polynomial
* Absolute values and the height of algebraic numbers
* Lower bounds for the Mahler measure of totally real polynomials
Introduction to Diophantine Approximation Rational approximation to a real number: continued fractions and applications. Rational approximation to an algebraic number: transcendence theorem of Thue,
Siegel, Roth. Simultaneous approximation. Schmidt Subspace Theorem. Diophantine approximation in fields of power series.
