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