Departamento de Matematica

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Coloquios del Departamento 2009

Coloquios del Departamento de Matemática - 2009

Jueguitos de tira y afloja y ecuaciones diferenciales

Resumen:

Se trata de una charla informal, sin demostraciones, sobre algunas relaciones que existen entre juegos de suma cero entre dos jugadores y soluciones de ciertas ecuaciones diferenciales.

No se espera que el publico tenga nociones de ecuaciones diuferenciales ni de probabilidad.

El lenguaje de las formas(La cinta de Möbius, las álgebras de Kac-Moody afines y los dibujos de niños)

Resumen:
Uno de los temas más recurrentes en la Matemática y la Física moderna es el estudio y la construcción de objetos que "localmente son isomorfos". En esta charla, especialmente por medio de ejemplos, veremos cómo el significado del concepto "localmente" evolucionó (en gran parte por ideas hermosas de J.-P. Serre y A. Grothendieck) para adaptarse a la Geometría Algebraica. Este punto de vista permite aplicaciones inesperadas a la teoría de Lie en dimensión infinita que también vamos a ilustrar durante la charla.

Combinatorial Algebraic Topology

Resumen:
Combinatorial algebraic topology is a fascinating and dynamic field at the crossroads of algebraic topology and discrete mathematics. The subject of Combinatorial Algebraic Topology is in a certain sense a classical one, as modern Algebraic Topology derives its roots from dealing with various combinatorially defined complexes and with combinatorial operations on them. Yet, the aspects of the theory which we consider are far from classical and have been brought to the attention of the general mathematical public fairly recently. In this talk we will give examples of structures, methods, as well as applications of combinatorial algebraic topology.

Cálculo del grupo K1 y los funtores determinantes

Resumen:
En esta charla explicaremos cómo un cierto enfoque para el cálculo de los dos primeros grupos de homotopía de un espacio nos lleva a un cálculo nuevo de los grupos K0 y K1 de la teoría K. La exposición será autocontenida y apta para todo público. Daremos también una aplicación de nuestra teoría a los llamados 'funtores determinantes universales'. Fue Deligne quién axiomatizó en 1987 la noción de funtor determinante, y observó una correspondencia con la teoría K. Podremos dar un contexto general para la correspondencia entre los funtores determinantes y la teoría K de categorías exactas, trianguladas, y de Waldhausen, recuperando las diferentes generalizaciones ya conocidas. (Trabajo conjunto con F Muro y M Witte)

Macdonald polynomials and explicit commuting operators diagonalized by them

Resumen:
In the 1980's Ian Macdonald formulated a series of conjectures concerning the value of the constant term of certain power series indexed by parameters related to a semisimple Lie algebra (more concretely to its root system $R$). The conjectures when they first appeared seemed to be isolated curiosities and it was not clear what lay behind them. That became clear a few years later with the introduction of (nowadays called) Macdonald polynomials. These are polynomials $P^{q,t}_\lambda$ in several variables, depending on parameters $q$ and $t$, indexed by the dominant weights $\lambda$ for the above root system $R$. Their various $q,t$-specialization yield for example the monomial symmetric functions, Jack's symmetric functions, zonal spherical functions on certain symmetric spaces for $p$-adic groups and other classical families of functions. Macdonald constant term conjectures in their more general form also predict the specialization of $P^{q,t}_\lambda$ and a certain duality between them. These conjectures and related problems concerning Macdonald polynomials generated huge activity in the last twenty years in representation theory, combinatorics and theory of quantum integrable systems, amongst others. In this talk I will give an overview of the Macdonald polynomials and talk about the above mentioned conjectures of Macdonald. These conjecture are actually all theorems now, although I will not say much about the proofs. If time permits I will talk about recent joint work with Jan Felipe van Diejen concerning explicit commuting difference operators diagonalized by the Macdonald polynomials.

An Adaptive and Information Theoretic Method For Compressed Sampling

Resumen:
By considering an s-sparse x signal to be an instance of vector random variable X=(X_1,...,X_n)^t we determine a sequence of binary sampling vectors for characterizing the signal x and completely determining it from the samples. Unlike the standard approaches, tis one is adaptive and is inspired by ideas from the theory of Huffman codes. The method seeks to minimize the number of steps needed for the sampling and reconstruction of any sparse vector x which is an instance of X. We prove that the expected total cost (number of measurements and reconstruction combined) that we need for an s-sparse vector in R^n is no more than slog n + 2s.

Perfect Graphs - structure and recognition

Resumen:
A graph is called perfect if for every induced subgraph, the size of its largest clique equals the minimum number of colors needed to color its vertices. As it turns out, the notion of perfect graphs generalizes a large number of phenomena, both in graph theory and in combinatorial optimization. Therefore, the problems of charactering perfect (or minimal imperfect) graphs and finding an efficient recognition algorithm have become well known in both communities. In 1960's Claude Berge made a onjecture that any graph with no induced odd cycles of length greater than three or their complements is perfect (thus, odd cycles of length greater than three and their complements are the only minimal imperfect graphs). This conjecture is know as the Strong Perfect Graph Conjecture. We call graphs containing no induced odd cycles of length greater than three or their complements Berge graphs. A stronger conjecture was made by Conforti, Cornuejols and Vuskovic, that any Berge graph either belongs to one of a few well understood basic classes or has a decomposition that can not occur in a minimal counterexample to Berge's Conjecture.
In joint work with Neil Robertson, Paul Seymour and Robin Thomas we were able to prove this conjecture and consequently the Strong Perfect Graph Theorem.
Later, in joint work with G. Cornuejols, X, Lui, P.Seymour and K. Vuskovic, we found an algorithm that tests in polynomial time whether a graph is Berge, and therefore perfect.
In my talk I will give an overview of both these results.

Geometría algebraica de espacios topológicos

Resumen:
La charla versará sobre un trabajo conjunto con A. Thom. En ese trabajo probamos un teorema de invarianza homotópica para funtores de $\mathbb{C}$-álgebras conmutativas en grupos abelianos. El teorema dice que si $F$ satisface ciertas condiciones algebraicas, entonces el funtor que manda un espacio compacto de Hausdorff $X$ a $F(C(X))$, es invariante homotópico. Aquí $C(X)$ es el álgebra de funciones continuas $X\to \mathbb{C}$. La demostración del teorema utiliza técnicas de geometría algebraica. En la charla mostraremos algunas aplicaciones de este teorema, como por ejemplo la confirmación de una conjetura formulada por Rosenberg en 1990: para todo $n<0$, el funtor $X\mapsto K_n(C(X))$ que envía a $X$ en la $K$-teoría algebraica negativa de $C(X)$, es invariante homotópico.
La charla es apta para todo público.

Coeficientes de Fourier de formas automorfas

Resumen:
Se hará una introducción a la teoría de formas automorfas en conexión con la teoría de funciones elípticas. Posteriormente se darán resultados sobre coeficientes de Fourier de formas automorfas holomorfas y analíticas reales, en particular se enunciará la fórmula de Kuznetsov, y algunas generalizaciones y aplicaciones de la misma.

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