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Từ ngày 12/9 đến ngày 5/10, GS Waldschmidt giảng bài cho SV, học viên cao học và các đối tượng khác tại phòng F207 (chiều 2, sáng 3, chiều 4, sáng 6) với nội dung Irrationality and Transdentality, trước hết kể về các kết quả kinh điển từ thời Liuoville (tính vô tỷ của e, e^r, pi ...) đến các kết quả liên quan đến bài toán Hilbert và kết thúc bằng 1 số vấn đề mở. GS giảng rất dễ hiểu, rất sư phạm. Dưới đây xin giới thiệu 4 bài giảng đầu tiên của GS.

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[namdung]

Các bạn có thể download trực tiếp các bài giảng tiếp theo, cũng như hai bài nói chuyện của GS Waldschmidt tại trường PTNK (20/9) và Seminar các PP toán sơ cấp (30/9) tại địa chỉ sau:

http://www.institut....HCMUNS2007.html

Introduction to Diophantine methods: irrationality and transcendence
Ho Chi Minh University of Natural Sciences, September 12 - October 4, 2007.
Content
Diophantine approximation is a chapter in number theory which has witnessed outstanding progress together with a number of deep applications during the recent years. The proofs have long been considered as technically difficult. However, we understand better now the underlying ideas, hence it becomes possible to introduce the basic methods and the fundamental tools in a more clear way.

We start with irrationality proofs. Historically, the first ones concerned irrational algebraic numbers, like the square roots of non square positive integers. Next, the theory of continued fraction expansion provided a very useful tool. Among the first proofs of irrationality for numbers which are now known to be transcendental are the ones by H. Lambert and L. Euler, in the XVIIIth century, for the numbers e and pi. Later, in 1815, J. Fourier gave a simple proof for the irrationality of e.

We first give this proof by Fourier and explain how J. Liouville extended it in 1840 (four years before his outstanding achievement, where he produced the first examples of transcendental numbers). Such arguments are very nice but quite limited, as we shall see. Next we explain how C. Hermite was able in 1873 to go much further by proving the transcendence of the number e. We introduce these new ideas of Hermite in several steps: first we prove the irrationality of the exponential of r for non-zero rational r as well as the irrationality of pi. Next we relate these simple proofs with Hermite's integral formula, following C.L. Siegel (1929 and 1949). Hermite's arguments led to the theory of Padé Approximants. They also enable Lindemann to settle the problem of the quadrature of the circle in 1882, by proving the transcendence of pi.

One of the next important steps in transcendental number theory came with the solution by A.O. Gel'fond and Th. Schneider of the seventh of the 23 problems raised by D. Hilbert at the International Congress of Mathematicians in Paris in 1900: for algebraic alpha and beta with alpha not 0 nor 1 and and beta irrational, the number alpha^beta is transcendental. An example is 2^{\sqrt{2}}, another less obvious example is e^pi. The proofs of Gel'fond and Schneider came after the study, by G. Polya, in 1914, of integer valued entire functions, using interpolation formulae going back to Hermite. We introduce these formulae as well as some variants for meromorphic functions due to R. Lagrange (1935) and recently rehabilitated by T. Rivoal (2006). The end of the course will be devoted to a survey of the most recent irrationality and transcendence results, including results of algebraic independence. We shall also introduce the main conjectures on this topic.

Notes of the course
Lecture 1: September 12, 2007 (11 pages, pdf file 196 Ko).
Lecture 2: September 14, 2007 (9 pages, pdf file 196 Ko).
Lecture 3: September 17, 2007 (10 pages, pdf file 208 Ko).
Lecture 4: September 18, 2007 (11 pages, pdf file 192 Ko).
Lecture 5: September 19, 2007 (8 pages, pdf file 188 Ko).
Lecture 6: September 21, 2007 (7 pages, pdf file 212 Ko).
Lecture 7: September 24, 2007 (6 pages, pdf file 172 Ko).
Lecture 8: September 25, 2007 (2 pages, pdf file 156 Ko).
Lecture 9: September 26, 2007 (8 pages, pdf file 216 Ko).
Lecture 10: September 28, 2007 (9 pages, pdf file 196 Ko).
Lectures 11 (October 1), 12 (October 2) and 13 (October 3): correction of the exercises.

Exercises - deadline for solutions: October 1, 2007 (5 pages, pdf file 152 Ko).

Thursday, September 20, 2007.
Lecture at High School For Gifted Students, Ho Chi Minh City, Arithmetic and secret codes (ppt file 3,4 Mo)

Sunday, September 30, 2007.
Lecture at High School For Gifted Students, Ho Chi Minh City, Coding Theory, Card Tricks and Hat Problems (ppt file 2,2 Mo)
mail : Michel Waldschmidt
http://www.institut....HCMUNS2007.html
Update: September 29, 2007

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