Workshop: Topics in algebraic number theory and Diophantine approximation

 

 

wams1
 
                                             Sponsered by  cimpa 

 


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)

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Francesco Pappalardi, full Professor of Algebra
Università Roma Tre, Italy
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Valerio Talamanca, Professore a contratto,
Università Roma Tre, Italy
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local organizer coordinator

Kawa M. A. MANMI
Head of the Mathematics Department, College of Science,
Salahaddin  University/Erbil-Kurdistan Region/IRAQ 

   Local committee (5 members at most), specifying the role of each member

Director: Kawa M. A. MANMI

Head of the Mathematics Department, College of Science,Salahaddin  University/Erbil-Kurdistan Region/IRAQ.

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Co-chair: Prof. Dr. Rostam Karim Saeed

Mathematics Department, College of Science,Salahaddin  University/Erbil-Kurdistan Region/IRAQ.

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Member: Andam Ali Mustafa

Mathematics Department, College of Science,Salahaddin  University/Erbil-Kurdistan Region/IRAQ.

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Member: Karzan Ahmed Perdawud

Mathematics Department, College of Science,Salahaddin  University/Erbil-Kurdistan Region/IRAQ.

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Member: Ms Evar Lutfalla Sadraddin

Mathematics Department, College of Science,Salahaddin  University/Erbil-Kurdistan Region/IRAQ.

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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.