Coloquio de Matemáticas

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Uno de los problemas de interés al trabajar con grandes volúmenes de datos es el de encontrar estrategias eficientes para agrupar o clasificar muchos datos en espacios posiblemente de gran dimensión o de naturaleza compleja. Una de las alternativas más usadas es la de encontrar esquemas de muestreo o más recientemente de proyecciones aleatorias que permitan encontrar la información relevante sin tener que considerar todo el conjunto de datos. Por otro lado, los métodos basados en kernels se basan en la construcción de un espacio de características sobre el cual resulta más natural realizar esta tarea mediante un producto interno definido sobre el nuevo espacio. Una pregunta natural es estudiar estrategias de proyecciones aleatorias para este nuevo espacio, teniendo en cuenta la existencia de direcciones principales definidas por el kernel. En esta charla presentaremos algunos resultados sobre proyecciones aleatorias para el problema de clasificación y agrupamiento usando kernels.

A lo largo del siglo 20 el movimiento Browniano se convertió en uno de los objetos sin duda más estudiados y interesantes en probabilidad. En esta charla vamos a conocer los orígenes de éste proceso estocástico, su construcción matemática, varias de sus propiedades sorpendentes y aparentemente paradoxales y preguntas actuales de investigación inspiradas en este objeto.

Consider the unit sphere in R^3. On the one hand, a real projective curve is the zero set of a homogeneous polynomial restricted to the sphere (up to the quotient by the antipodal map). On the other hand, the sphere is a compact surface, on which the Laplacian has a discrete spectrum with finite dimensional eigenspaces. It is well known that its eigenfunctions are smooth. Their zero set, also called nodal set, has been an object of great interest in analysis for many years. In both cases, many questions remain open about the possible, or typical shapes of these curves. One way to attack these questions is to consider random instances of the functions defining them. This approach has been quite fruitful over the past ten years. More recently, following a conjecture made by the physicists Bogomolny and Schmit, a link has been established with Planar Bernoulli Percolation, which is the study of large scale connectivity properties of certain random subgraphs of Z^2. We will first give a precise definition of the functions we consider as well as the choice of randomization procedure. Next we will give an overview of (planar) Percolation theory and explain how it adapts to this context.

When studying representation theory of a finite group, we are naturally lead to study its irreducible representations. When we have a nice family of groups (think here all the symmetric groups $S_n$ for $n \ge 0$), we can bundle all the representations together and define interesting operations on them: tensors, inductions, restrictions. Sometime, this defines nice algebraic structures and in the case of the family of symmetric groups, Frobenius characteristics shows that it is equivalent to symmetric functions. But for some family of groups it impossible to classify all their irreducibles. For example the family $U_n(q)$ of upper triangular matrices over a finite field, $n\ge 0$, it is impossible to classify the irreducibles. This is a known wild problem. Supercharacter theory is a framework that approximate the notion of irreducibles. In the context of the family $U_n(q)$, we can define a supercharacter theory that is classifiable, tamed, and we can bundle them together with interesting algebraic operation. The result is an algebraic structure that is equivalent to symmetric functions in non-commutative variables.

En los últimos 20 años me he interesado por encontrar buenas parametrizaciones para conjuntos que estan bien aproximados por planos, conos y mas recientemente grafos de funciones de Lipschitz. En esta charla describire una serie de resultados obtenidos con Guy David en esta area. Indicare algunas de las aplicaciones de estos resultados a varias areas de analisis geométrico.

En esta charla, discutiremos la definición de la función Zeta de Riemann, su conexión con la distribución de los números primos, y demostraremos la Hipótesis de Riemann (no, es broma!)...y hablaremos sobre la Hipótesis de Riemann y algunos resultados relacionados con la misma.

I will first explain the idea of quantum spaces. It is well known that under some general assumptions on the topology, the (commutative) algebra of functions recovers the points of the space. The algebraic statements of the topology can be extrapolated to generalize to noncommutative algebras, which can be viewed as a generalization of spaces. Quantum groups are examples of quantum spaces, which are endowed with a product, an inverse and a unit just as much as groups are endowed with these basic operations. I will explain exactly how to generalize ordinary spaces to quantum spaces then explain how quantum groups can act by a group action on these spaces including my recent results.

We say that an algebraic variety is unirational if it can be parametrized by rational functions, rational if moreover the parametrization can be chosen to be one-to-one. The Lüroth problem asks whether a unirational variety is necessarily rational. This holds for curves (Lüroth, 1875) and for surfaces (Castelnuovo, 1894); after various unsuccessful attempts, it was shown in 1971 that the answer is quite negative in dimension 3: there are many examples of unirational varieties which are not rational. Up to last year the known examples in dimension >3 were quite particular, but a new idea of Claire Voisin has significantly improved the situation. I will survey the colorful history of the problem, then explain Voisin's idea, and how it leads to a number of new results.

En esta charla daremos una introducción a los ultraproductos de espacios métricos. También daremos una descripción de la lógica continua, una manera de adaptar la teoría de modelos a espacios métricos y lo que dice acerca de los ultraproductos.

En esta charla estaremos interesados en resolver tres problemas, relacionados, en el contexto de cadenas de Markov controladas: Hallar una política de control que maximice la probabilidad de llegar a un conjunto A, evitando llegar a un conjunto prohibido B. Una versión restringida del anterior problema: Hallar una política de control que maximice la probabilidad de llegar a un conjunto A, de forma que la probabilidad de llegar a un conjunto B sea menor que un error epsilon. Caracterizar el dominio de atracción de las cadenas de Markov controladas. Mostramos cómo dichos problemas pueden resolverse usando los denominados Markov Decision Process, en particular esto implica que pueden usarse programas lineales para hallar políticas óptimas.

TBA

Let G be a reductive affine algebraic group over an algebraically closed field and V and W finite dimensional G-modules. These data induce a G-action on IP(V)xIP(W) and, for any pair (m,n) of positive integers, a linearization of this action in the very ample line bundle O(m,n). When the ratio n/m becomes sufficiently large, the GIT notion of (semi)stability with respect to the linearization in O(m,n) becomes independent of (m,n) and has a nice description. This elementary fact plays an important role in understanding semistability for decorated sheaves, e.g., quiver sheaves. We will firstly review these facts and give, as an application, a proof of the recent Hilbert-Mumford criterion of Gulbrandsen, Halle and Hulek in relative GIT when the base is of finite type. By means of an example, we will show that these results do not extend to arbitrary quasi-projective varieties. Secondly, we will explain a related result on the instability flag and review how this is applied to moduli problems.

El documento "Investigación en Uniandes" Las trasparencias de la charla

We provide explicitly an upper bound of the interpolating constant for a weighted Bergman space of infinite order on the open unit complex ball.