Michel Waldschmidt

Enseignement

Université Pierre et Marie Curie  Paris 6 , UFR 929


Enseignement 2009/2010

Premier semestre / First term

Inde

Du 2 au 24 Décembre 2009 et du 11 au 29 janvier 2010:
    Chennai Mathematical Institute CMI, Chennai (=Madras), Inde.

Modular Algebraic Independence

Professor M. Waldschmidt will be visiting CMI for the period December 3–23 2009 and January 11–29, 2010. He shall be giving a course of lectures on transcendence. The central goal of his lectures is to introduce the basic tools for the remarkable works of Nesterenko. His works can be regarded as one of the first "truly modular" theorems on algebraic independence, viz.: Let q be a non-zero complex (or p-adic) number of absolute value <1; then, at least three of the numbers q, P(q), Q(q), R(q) are algebraically independent. Here P, Q, R are Ramanujan's notation for the Eisenstein series E2, E4 and E6 respectively. A striking consequence is an answer to a folklore conjecture that π and eπ are algebraically independent.
    Lecture schedule for December
        Dates: December 3, 7, 10, 14, 16 and 21.
        Time: 3.30 pm – 5:00 pm
        Venue: Lecture Hall 6, CMI
    Reference: Yu. V. Nesterenko, Algebraic Independence, TIFR Mumbai - Narosa, 2009.
    Courses of 03/12/2009, 07/12/2009, 10/12/2009 and 14/12/2009 (29 pages).
    Lecture schedule for January
    Dates: January 13 (Wednesday), 11:30 am - 1 pm,
        January 19 (Tuesday), 2:00 - 3:15 pm,
        January 21 (Thursday) 11:30 am - 1:00 pm,
        January 25 (Monday) 3:30 - 5:00 pm,
        January 27 (Wednesday) 11:30 am - 1 pm,
        January 29 (Friday) 3:30 - 5:00 pm
    Venue: Room 123, The Institute of Mathematical Sciences (IMSc)

Deuxième semestre / Second term

Brésil

IMPA - Instituto Nacional de Matematica Pura e Aplicada , Rio de Janeiro   April 12 - June 29, 2010 (rapport en français). Introduction to Diophantine approximation and transcendental number theory

Content:
  The purpose of this course is to give an elementary introduction to some of the main results in Diophantine approximation and transcendence. The emphasis will be on the ideas, while technical details will be reduced to the minimum.
  The course will start with rational approximation to real numbers. The main tool is the theory of continued fractions. As an application of this theory, we will study the so-called Pell-Fermat equation. We explain the relation with the units of quadratic number fields.
  Rational approximation theory yields irrationality results. We give an overview of the main results of irrationality and their proofs.
  Liouville introduced what is called now, in dynamical systems theory, a Diophantine condition. We study the proof of Liouville and the properties of the set of Liouville numbers.
  After irrationality, the next step is transcendence. This course will include a survey of the main ideas which are used in the proofs of transcendence and algebraic independence, including the proof by Nesterenko in 1996 of the algebraic independence of values of modular functions.

Remark:
  Few prerequisites are necessary to follow this course. Depending on the audience, the necessary background will be recalled whenever it is necessary.
  This course will provide the interested students necessary bases if they wish to pursue on this topic and continue their studies for a thesis, for instance in Paris VI.

References:
    N.I. Fel'dman and Y.V. Nesterenko -Transcendental numbers, in Number Theory, IV, Encyclopaedia Math. Sci., vol. 44, Springer, Berlin, 1998, p.1-345.
    Irrationality of some constants coming from analysis.     Historical introduction to transcendence.
    Auxiliary functions in transcendence proofs.

Lecture schedule
    Mondays and Wednesdays, 09:30-11:30, Sala 347. First course April 14, last course June 28.

Notes
    Course 1, April 14, 2010 (15 pages).
        Chap.1: Introduction: Irrationality of $\sqrt {2}$; Continued fractions; Irrational numbers; History of irrationality; Variation on a proof by Fourier (1815).
    Course 2, April 19, 2010 (14 pages).
        Chap.2: Irrationality Criteria: Statement of the criterion; Geometry of numbers; Irrationality of at least one number.
    Course 3, April 26, 2010 (20 pages).
        Chap.3: Criteria for linear independence; Hermite's method; Rational approximations; Linear forms.
        Chap.4: Criteria for transcendence: Liouville's inequality; Transcendence criterion of A.Durand.
        Chap.5: Criteria for algebraic independence: Small transcendence degree, Gel'fond's criterion; Large transcendence degree
    Course 4, April 28, 2010 (10 pages).
        Liouville's inequalities. A short historical survey on Diophantine approximation.
    Course 5, May 3, 2010 (12 pages).
        Chap.6: Continued fractions.
    Course 6, May 5, 2010 (11 pages).
        Pell's equation.
    Course 7, May 10, 2010 (11 pages).
        Simple continued fraction of the square root of a positive integer.
    Course 8, May 12, 2010 (12 pages).
        Continued fractions (continued).
    Course 9, May 17, 2010 (3 pages).
        The ring of integers of an algebraic number field.
    Course 10, May 19, 2010.
        On the Markoff equation x^2+y^2+z^2=3xyz.   screen pdf 4.2 Mo, 162 p.   print pdf 3.3 Mo, 20 p.
    Course 11, May 24, 2010 (10 pages).
        Farey series
    Course 12, May 26, 2010 (13 pages).
        Chap.7: Approximation of functions
    Course 13, May 31, 2010 (10 pages).
        Chap.8: Hermite's method (I)
    Course 14, June 2, 2010 (10 pages).
        Hermite's method (II)
    Course 15, June 7, 2010 (8 pages).
        End of Hermite's proof.
        Chap.9: Interpolation.
    Course 16, June 9, 2010 (13 pages).
        Integer valued entire functions.
    Course 17, June 14, 2010 (8 pages).
        Transcendance of e^{pi}, following A.O. Gel'fond (1929).
    Course 18, June 16, 2010 (2 pages).
        The Schneider-Lang Theorem.
    Course 19, June 21, 2010 (17 pages).
        Elliptic functions: introduction.
        Consequences of the Schneider-Lang Theorem.
    Course 20, June 23, 2010 (3 pages).
        Endomorphisms of elliptic curves, complex multiplication.
        Chudnovski's algebraic independence results.
    Course 21, June 28, 2010 (5 pages).
        Algebraic independence of values of modular functions.