Conferencias

Conferencias plenarias

ENRIQUE FERNANDEZ CARA
Universidad De Sevilla
España

SOME FREE-BOUNDARY PROBLEMS AND THEIR CONTROL
This talk is concerned with the theoretical and numerical control of several free-boundary problems. Among them, we will consider the one-phase and two-phase Stefan problems with distributed or boundary controls. We will prove some local results. More precisely, it will be shown that there exist controls such that an associated dependent variable (resp. an associated interface) is steered to zero (resp. to a prescribed location) provided the initial data and interface position are sufficiently close to the targets. We will also present some numerical methods for the computation of null controls and will illustrate the techniques with several numerical experiments. Several parts of the work have been done in collaboration with J. Límaco, S. de Menezes, D.A. Souza and R.K.C. Araújo.

FERNANDO ANDRÉS QUINTANA
Pontificia Universidad Católica De Chile
Chile

THE SEMI-HIERARCHICAL DIRICHLET PROCESS AND ITS APPLICATIONS TO CLUSTERING HOMOGENEOUS DISTRIBUTIONS
Assessing homogeneity of distributions is an old problem that has received considerable attention, especially in the nonparametric Bayesian literature. To this effect, we propose the semihierarchical Dirichlet process, a novel hierarchical prior that extends the hierarchical Dirichlet process of Teh et al. (2006) and that avoids the degeneracy issues of nested processes recently described by Camerlenghi et al. (2019). We go beyond the simple yes/no answer to the homogeneity question and embed the proposed prior in a random partition model; this procedure allows us to give a more comprehensive response to the above question and in fact find groups of populations that are internally homogeneous when I ≥ 2 such populations are considered. We study theoretical properties of the semi-hierarchical Dirichlet process and of the Bayes factor for the homogeneity test when I = 2. Extensive simulation studies and applications to educational data are also discussed.

FABIÁN FLORES BAZÁN
Universidad De Concepción
Chile

QUASICONVEX OR QUASIMONOTONE FAMILIES WITH APPLICATIONS TO SUMS AND QUASICONVEX OPTIMIZATION
It is well-known that the sum of two quasiconvex functions is not quasiconvex in general, and the same occurs with the minimum. Although apparently these two state- ments (for the sum or minimum) have nothing in common, they are related, as we show in this paper. To develop our study, the notion of quasiconvex family is introduced, and we establish various characterizations of such a concept: one of them being the quasiconvexity of the pointwise infimum of arbitrary translations of quasi- convex functions in the family; another is the convexity of the union of any two of their sublevel sets; a third one is the quasiconvexity of the sum of the quasiconvex functions, composed with arbitrary nondecreasing functions. As a by-product, any of the aforementioned characterizations, besides providing quasiconvexity of the sum, also implies the semistrict quasiconvexity of the sum if every function in the family has the same property.

SALOMÓN ALARCÓN
Universidad Técnica Federico Santa María
Chile

ALGUNOS FENÓMENOS DE CONCENTRACIÓN EN ECUACIONES DIFERENCIALES PARCIALES ELÍPTICAS / SOME CONCENTRATION PHENOMENA IN ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
Esta charla trata sobre fenómenos de concentración que han sido ampliamente estudiados en el campo de las Ecuaciones Diferenciales Parciales (EDPs) elípticas durante las últimas décadas. Los problemas presentados involucran a las ecuaciones no lineales de Lane-Emden, Hénon y Schrödinger. Se muestran resultados de existencia y comportamiento asintótico de las soluciones que fueron obtenidos en algunas colaboraciones. Además, sin entrar en detalles técnicos, se destacan algunas de las principales herramientas provenientes del Análisis No lineal que han sido usadas para estudiar estos fenómenos en las EDPs no lineales previamente mencionadas.

Conferencias subplenarias

MARCELO BOURGUIGNON
Universidad Federal do Rio Grande do Norte
Brasil

REPARAMETERIZED REGRESSION MODELS
Regression models are typically constructed to model the mean of a distribution. However, the density of several distributions is not indexed by the mean. In this context, this work provides a collection of regression models considering new parameterizations in terms of the mean and precision parameters. The main advantage of our new parametrizations is the straightforward interpretation of the regression coefficients in terms of the expectation, as usual in the context of generalized linear models.

CRISTIAN GONZÁLEZ-AVILÉS
Universidad de La Serena
Chile

DUALITY THEOREMS FOR PSEUDO-PROPER ALGEBRAIC GROUPS OVER LOCAL FUNCTION FIELDS
If k is an imperfect field, the class of pseudo-proper commutative algebraic k-groups contains the class of all pseudo-abelian k-varieties introduced by B. Totaro in 2013, as well as some nonsmooth and/or non-connected algebraic k-groups. In this talk I will define these groups, as well as their dual generalized k-1-motives, and state a local duality theorem for these objects that extends the well-known Milne-Tate local duality theorems for abelian varieties. No previous familiarity with 1-motives will be required of the audience, but interested participants of this upcoming talk are invited to view the video https://www.youtube.com/watch?v=XDjWeRSPJ8k&t=86s for basic information about these objects.

CRISTIAN ORTIZ
Universidade De São Paulo
Brasil

GEOMETRIC STRUCTURES ON DIFFERENTIABLE STACKS
Differentiable stacks are fibered categories satisfying a sheaf condition. One can think of a differentiable stack as a singular quotient of a differentiable manifold, e.g. orbifolds. In this talk I will give an overview on recent developments on the differential geometry of stacks, including connections with representation theory and symplectic geometry.

CARLOS MARIJUÁN LÓPEZ
Universidad de Valladolid
España

UNIVERSAL REALIZABILITY IN LOW DIMENSION
We say that a list Λ = { λ_1 , . . . , λ_n } of complex numbers is realizable, if it is the spectrum of a nonnegative matrix A (a realizing matrix). We say that Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. This work studies the universal realizability of spectra in low dimension, that is, realizable spectra of size n ≤ 5. It is clear that for n ≤ 3 the concepts of universally realizable and realizable are equivalent. We characterize the universal realizability of real spectra of size 4 and of size 5 with trace zero, and we describe a region for the universal realizability of nonreal 5-spectra with trace zero. As an importan by-product of our study, we also show that realizable lists on the left half-plane, that is, lists Λ = { λ_1 , . . . , λ_n } , where λ 1 is the Perron eigenvalue and Re λ i ≤ 0, for i = 2, . . . , n, are not necessarily universally realizable.

ELIZABETH GASPARIM
Universidad Católica Del Norte
Chile

SIMETRÍA ESPECULAR Y DEFORMACIONES
La teoría de deformaciones es un tema clásico de la geometría algebraica. La conjetura de la simetría especular es un tema nuevo que vive naturalmente en  2 mundos: geometría algebraica y  geometría simpléctica, que se puede expresar de muchas formas diferentes. A cada modelo algebraico se hace corresponder un espejo simpléctico. Explicaré algunas de las formulaciones principales de esta conjetura, juntamente con ejemplos que ilustran efectos de las deformaciones sobre un objeto y su espejo.

RAFAEL STERN
Universidade Federal de São Carlos
Brasil

INDEMNITY FOR A LOST CHANCE
This presentation will discuss the legal quantification of lost chances. Civil liability for a lost chance occurs when a tortious action decreases the probability of the victim obtaining desirable outcomes. Such a situation might occur, for instance, when medical malpractice reduces a patient’s chance of survival. Quantification of damages in these situations poses challenging questions, which lie in the borderline between Law and Statistics. This presentation will be centered around three legal questions regarding the meaning of indemnity in the context of lost chances. I will discuss possible legal answers to these questions and translations of these answers into mathematical statements. Based on these translations, it is possible to obtain general rules for quantifying lost chances. These rules generalize formulae that have been proposed previously in jurisprudence and show that each formula is consistent with different qualitative interpretations of the law.

DANIEL A. JAUME
Universidad Nacional de San Luis
Argentina

2-SWITCH: TRANSITION AND STABILITY ON FORESTS, AND PSEUDOFORESTS
Given any two forests (pseudoforests) with the same degree sequence, we show in an algorithmic way that one can be transformed into the other by a finite sequence of 2-switches in such a way that all the intermediate graphs of the transformation are forests (pseudoforests). We also prove that the 2-switch operation perturbs minimally some well-known integer parameters in families of graphs with the same degree sequence. Then, we apply these results to conclude that the studied parameters have the “interval property” (a discrete analogous to intermediate value property) on forests (pseudoforests).

GIANCARLO LUCCHINI ARTECHE
Universidad De Chile
Chile

SOBRE EXTENSIONES DE GRUPOS
En teoría de grupos existe un resultado clásico que relaciona el conjunto Ext(G,H) de (clases de isomorfismo de) extensiones de un grupo G por un grupo H con el conjunto Ext(G,Z) de (clases de isomorfismo de) extensiones de G por el centro Z de H. Cuando uno pasa a cualquier teoría de grupos “con apellido” (grupos algebraicos, grupos de Lie, grupos topológicos, etc.), la noción de extensión cambia naturalmente de forma ad hoc y por lo tanto la generalización de este tipo de resultados se plantea de forma natural. Pero lamentablemente la demostración clásica no se extiende a todos estos contextos. En esta charla revisitaremos este resultado sobre grupos abstractos de una manera más “conceptual”, lo que nos permite generalizarla a muchos otros contextos, en particular al marco de grupos algebraicos (¡sin ninguna modificación!).

ENRIQUE OTAROLA
Universidad Técnica Federico Santa María
Chile

MÉTODOS NUMÉRICOS PARA DIFUSIÓN FRACCIONARIA
El objetivo de la charla es presentar dos definiciones no equivalentes de difusión fraccionaria en dominios acotados, discutir nociones básicas de regularidad de soluciones para problemas elípticos lineales que involucran estas definiciones, presentar métodos numéricos elementales y ejemplificar como estimaciones de regularidad y propiedades intrínsecas de los métodos propuestos permiten obtener estimaciones del error.