El problema del isomorfismo de grafos: trabajos de Babai, Luks, Weisfeiler-Leman
por Harald Helfgott, el 22, 23, 24 y 25 de agosto, de 10:00 a 12:00 (minicurso en castellano)
Sean Γ1, Γ2 dos grafos con n vértices. ¿Son isomorfos? Es decir: ¿son en verdad el mismo grafo, con una simple permutacion de los vértices? El desafío de encontrar un algoritmo que siempre dé una respuesta rápida y correcta a esta pregunta ha permanecido abierto durante largo tiempo. Babai dio recientemente un algoritmo que corre en tiempo cuasipolinomial, i.e., acotado por exp(C log n), donde C es una constante. Su estrategia, muy innovadora, utiliza ideas precedentes de Luks (1980/1982) y un método de Weisfeiler y Leman. Daremos un recorrido por las nociones principales de la prueba.
Dirichlet series and zeta functions of several variables and applications
por Driss Essouabri, el 28, 29, 31 de agosto, y el 1 de setiembre, de 10:00 a 12:00 (minicurso en inglés)
In this course, we will give a brief overview of some of the methods used to prove several important properties of Dirichlet series and zeta functions of several variables (meromorphic continuation, moderate growth, localization of singularities, special values, etc.) We will also give some applications of this theory to the arith- metic of number fields, Manin conjecture for Toric varieties, fractal geometry, etc.
Schedule of the course:
- Some tools from analytic number theory and complex geometry: Integral representation formulas in one and in several variables; Resolution of singularities.
- Dirichlet series associeted to polynomial of several variables: Analytical properties (meromorphic continuation, moderate growth, lo- calization of singularities,etc.); Special values, periods, etc.; Application to the Arithmetic of number fields: Shintani’s Method, Special values of Dedekind zeta function of a number field, etc.
- Multiple Euler products and Mixed zeta functions: Some results on Multiple Euler products; Mixed zeta functions; Height zeta functions of Toric varieties and application to Manin conjecture on rational points (for Toric varieties).
- Fractal zeta function and applications: Fractal zeta functions; Application to the geometry of discrete fractals.
Iwasawa Theory
por Jean-Robert Belliard, el 4, 5, 6, 7 y 8 de setiembre, de 10:00 a 12:00 (minicurso en inglés)
Content: Originally algebraic number theory grew from Kummer's programm which aimed to prove Fermat's last theorem using the arithmetic of cyclotomic num- bers. From one point of view this program failed because this problem is now solved by A. Wiles (amongs many other) using modular method and today nobody knows whether or not a proof in the spirit of Kummer may exist. On the other hand these attempts of proof generated an impressive amount of fascinating mathematics with contributions of essentially all great mathematicians of this part of history. Kum- mer program leaves us with ideal class group: a still to be understood invariant, as the main algebraic invariant of number theory. On the analytic side complex Dede- kind zeta functions are known to encapsulate most of the arithmetic of their related number elds. By using p-adic method the idea behind all the "main conjectures" of Iwasawa theory is to state properly then prove that these two invariants are in fact equal in some sense that will be explained during this course. Prerequisite: All this course is contained in Larry Washington's book "Intro- duction to Cyclotomic Fields" which is himself fairly self contained. The few needed facts from basic algebraic number theory will be (brie y) recalled during the rst course and any student able to read alone Pierre Samuel's book "Algebraic Number Theory" will have no di culty to follow these lectures.