Coloquio de Matemáticas

: La teor�a de matroides es una teor�a combinatoria de la independencia, que tiene sus or�genes en el �lgebra lineal y la teor�a de grafos, y que resulta tener conexiones importantes con muchas otras �reas. Con el tiempo, las ra�ces geom�tricas de esta teor�a se han hecho bastante m�s profundas, dando muchos frutos nuevos.<br>Recientemente, el acercamiento geom�trico a la teor�a de matroides ha llevado al desarrollo de matem�tica fascinante en la intersecci�n de la combinatoria, el �lgebra, y la geometr�a, y a la solucion de varias conjeturas cl�sicas. Esta charla resumir� algunos logros recientes.

Moduli spaces are fascinating spaces that appear in various guises across mathematics. Vaguely, a moduli space is a space (e.g. topological space, manifold or algebraic variety) whose points are in bijective correspondence with isomorphism classes of certain topological, geometric or algebraic objects (e.g. curves or vector bundles). We will provide an introduction to moduli spaces that appear in algebraic geometry with an emphasis on examples. We will see that the properties of a moduli space depend crucially on properties of the automorphism groups of the objects. We will first explain how one can construct a moduli space as a projective variety in the case when the automorphism groups are finite. Our final goal is to show how recent breakthroughs in the foundations of moduli theory allow for similar constructions even when the automorphism groups are not finite.

Although macrophages are part of the human immune system, it has been remarkably observed in laboratory experiments that decreasing its number can slow down the tumor progression. We analyze through a recently mathematical model proposed inthe literature, necessary conditions for aggregation of tumor cells and macrophages. In order to do so, we prove the possibility of having blow-up in finite time. Next, we study if the aggregation of macrophages can occur when having a low density of tumor cells, and vice versa. With this purpose, we consider the problem of analyzing the existence or not of a simultaneous blow-up. We achieve this goal thanks to a novel process that allows us to compare the entropy functional associated with the density of each population, which turns out to be also a method to find enough conditions for having a simultaneous blow-up

A classical problem in the study of computer code experiments is the evaluation of the relative influence of the input variables on some numerical result obtained by a computer code. In this context, the output is seen as a function f of random inputs (generally assumed independent) and a sensitivity analysis is performed using the so-called Hoeffding decomposition. In this functional decomposition, f is expanded as an L�-sum of uncorrelated functions involving only a part of the random inputs. This leads to the Sobol index that measures the amount of randomness (the part of the variance) of the output due to one or more input variables. It remains then to estimate these Sobol indices to rank the variables with respect to their influence on the output. Nevertheless, the Sobol indices and their Monte-Carlo estimation are order two methods: thus they are well adapted to measure the contribution of an input on the deviation around the output mean and it seems very intuitive that the sensitivity of an extreme quantile of the output could depend on sets of variables that cannot be captured using only the variances. One may generalize them with higher order methods. Indices based on contrast functions depending on the quantity of interest is a nice alternative when one considers quantiles or medians. Another promising possibility consists in defining indices depending on the whole distribution of the output conditioned by the input whose influence must be quantified

Uno de los problemas fundamentales en teor�a de representaciones es entender como una representaci�n se puede escribir en t�rminos de sus componentes irreducibles. Esta no es una tarea f�cil, y aunque en principio es un problema puramente algebraico, su soluci�n involucra una mezcla de t�cnicas tanto algebraicas como geom�tricas. En esta charla se mostrar� c�mo atacar este problema en diferentes categor�as de representaciones, pasando desde grupos finitos hasta grupos cu�nticos