Topics in algebraic number theory and Diophantine approximation

Date:   (12^th  – 17^th ) March, 2017

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

Sponsored by

  CIMPA  &     Salahaddin University/ Erbil

                                

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  Lecturers

Michel Waldschmidt

Emeritus Professor, Universit´e P. et M. Curie (Paris VI)

m[email protected] 

Pierre Cartier

Professor of Algebra

[email protected]

Francesco Pappalardi

 full Professor of Algebra, Università Roma Tre, Italy

[email protected]

Valerio Talamanca

Professore a contratto,Università Roma Tre, Italy

[email protected]

   Local Organizer Coordinator

Kawa M. A. MANMI

Head of the Mathematics Department, College of Science,

Salahaddin  University/Erbil-Kurdistan Region/IRAQ

 

   Local committee

      

     
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    

From 12^th-14^th of  March, The background of the main topic (Group, ring, polynomial ring and field) will be covered by the local lecturers as follows:

Time	Sunday, 12 March 2017	Lecturer
9:00-10:30	Group Theory	Neshtiman N. Sulaiman
10:30-11:00	Coffee Break
11:00-13:00	Ring Theoy	Abdullah Abdul-Jabbar
Monday, 13 March 2017
9:00-10:30	Polynomial Ring	Sanhan Khasraw
10:30-11:00	Coffee Break
11:00-13:00	Field	Parween A. Hummadi
Tuesday, 14 March 2017
9:00-10:30	Extension Field	Parween A. Hummadi
10:30-11:00	Coffee Break
11:00-13:00	Continuing Extension Field	Parween A. Hummadi

The main topics will be as follows 
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
1. 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 
– Kummer’s problem on cubic Gauss periods

2. 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 willuse 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

3. Introduction to Diophantine Approximation
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.