Talks during the course 2019/2020

These are the lectures given during the academic course 2019/2020. On March 14th 2020, a lockdown was imposed and therefore the seminar was cancelled.

11 septiembre 2019, Seminar 2nd floor, 17:00-18:00

Speaker: Carlos Florentino (Universidad de Lisboa)

TItle: E-polynomials of SL(n,C)-character varieties of free groups

Abstract: Given a finitely generated group F and a complex reductive Lie group G, the G-character variety of F, X_F(G)=Hom(F,G)//G, is typically a singular algebraic variety whose geometric and topological properties can be studied via mixed Hodge structures (MHS); some of these properties are encoded in a polynomial generalization of the Euler-Poincaré characteristic called the E-polynomial, which also includes arithmetic information.

In this seminar, we address the problem of determining E-polynomials of character varieties in two main directions: (i) for the general linear group G=GL(n,ℂ), we present, for any F, a remarkable relation between the E-polynomials of X_F(G) and those of X^{irr}_F(G), the locus of irreducible representations of F into G. (ii) When F is a free group, we show that the E-polynomials of SL(n,ℂ) and of PGL(n,ℂ) character varieties coincide. This is joint work with A. Nozad, J. Silva and A. Zamora, and has been greatly inspired by previous work of V. Muñoz.

11 september 2019, Seminar 2nd floor, 18:00-19:00

Speaker: Jaime Silva (Universidad de Porto)

Title: Motives of Character varieties

Abstract: The theory of motives is an attempt to establish a universal cohomology theory for algebraic varieties. This has been completed in the case of smooth projective varieties, but the open case remains unsolved. In this seminar, I will talk about our attempt to adapt our previous work on Hodge structures of character varieties to the simplest type of motives, the so called naive motives. I will also try to convey some recent ideas on equivariant Chow motives to solve these problems. This is joint work with C. Florentino.

9 october 2019, Seminar 2nd floor, 18:00-19:00

Speaker: Antonio Viruel (Universidad de Málaga)

Title: Automorfismos de Dessins d'Enfants via combinatoria

Abstract: Recientemente, G. Jones y R. Hidalgo han demostrado de manera independiente que todo grupo finito puede ser representado como el grupo de automorfismos de un Dessins d'Enfants, y que de hecho existen infinitras maneras de hacerlo. En esta charla mostraremos como un sencillo argumento de combinatoria y geometría hiperbólica muestra que todo el retículo de subgrupos de cualquier grupo finito puede ser representado como los automorfismo de una colección de Dessins d'Enfants encajados en una superficie. Más aún, existen infinitas maneras de realizar esta representación. Es un trabajo conjunto con J. Turiel.

23 october 2019, Seminar 2nd floor, 18:00-19:00

Speaker: Ana J. Reguera (Universidad de Valladolid)

Title: Introducción al espacio de arcos en característica positiva

Abstract: En 1968, J. Nash comenzó el estudio del espacio de arcos X∞ de una variedad algebraica singular X definida sobre un cuerpo de característica cero, con objeto de entender las diversas resoluciones de singularidades de X. Realizó su trabajo poco después de la prueba de Hironaka de Resolución de Singularidades en característica cero. Demostró, utilizando la Resolución de Singularidades, que el espacio de arcos centrados en Sing X (que denotamos por XSing∞) tiene un número finito de componenetes irreducibles.

El programa de Nash se extiende a cuerpos base perfectos de característica p > 0. Pero la Resolución de Singularidades sigue siendo un problema abierto cuando car k = p > 0 y dim X ≥ 4. En la charla, haremos una introducción al espacio de arcos desde este punto de vista. Propondremos diversas preguntas que tendrían respuesta afirmativa si existiera una resolución de singularidades:

Q1 : ¿Tiene XSing∞ un número finito de componentes irreducibles?

Q2 : Dada una variedad X, ¿existe un morfismo birracional y propio Y → X tal que Y∞ es irreducible?

Daremos algunas respuestas parciales y explicaremos el estado de estos problemas.

6 november 2019, Seminar 2nd floor, 18:00-19:00

Speaker: Peter Symonds (University of Manchester)

Title: Rank, Coclass and Cohomology.

Abstract: We prove that for any prime p the finite p-groups of fixed coclass have only finitely many different mod-p cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of fixed rank.

13 november 2019, Seminar 2nd floor, 18:00-19:00

Speaker: Ángel González-Prieto (Instituto de Ciencias Matemáticas ICMAT)

Title: Bordism theory for homotopy groups of knots

Abstract: It is a well-known fact that, for n ≥ 4, all the knots in Rⁿ are equivalent (or, in other words, the space of embeddings of S¹ into Rⁿ is connected). However, far from being trivial, the topology of the space of embeddings of S¹ into R⁴ is extremely rich. Indeed, almost nothing is known about its higher homotopy groups.

In this talk, we will review some of the classical techniques that have being applied to the computation of these homotopy groups in the low degree case, like functor calculus. However, as we will show, these methods fail when we consider high dimensional families. To tackle this problem, we will explain a new approach based on the interpretation of the underlying obstruction theory as a decorated bordism class. This will allow us to easily replicate the known results and suffers almost no dimensional course. (Joint work with E. Fernández and F. Presas)

4 december 2019, Seminar 2nd floor, 18:00-19:00

Speaker: Robert Low (Coventry University, UK)

Title: The topology and geometry of the space of null geodesics

Abstract: This talk will provide a general introduction to the space of null geodesics. I will begin with a discussion of the case of Minkowski space, a special case with a great deal of structure. After this, I will move on to the more general case of the space of null geodesics of space-time, modelled as a Lorentz manifold, beginning with the fundamental topological and differentiable structure. With this in place, we can introduce additional structure, and I will describe (some of) the interplay between the geometrical and causal structure of the original space-time and its space of null geodesics.

18 december 2019, Seminar 2nd floor, 18:00-19:00

Speaker: Luis J. Alías (Universidad de Murcia)

Title: Subvariedades atrapadas en el cono de luz

Abstract: (Extracto) El concepto de superficie atrapada fue formulado orginalmente por Penrose en 1965 para el caso de superficies espaciales bidimensionales en espaciotiempos tetradimensionales, en términos de los signos o de la anulación de las llamadas expansiones nulas. Esto está íntimamente relacionado con la orientación causal del vector curvatura media de la superficie, lo que proporciona una caracterización mejor y más potente de las superficies atrapadas, permitiendo la generalización de dicho concepto al caso de subvariedades espaciales de codimensión dos y dimensión arbitraria en espaciotiempos generales.

En esta conferencia consideraremos subvariedades espaciales de codimensión dos contenidas en el cono de luz del espaciotiempo de de Sitter así como en el cono de luz del espaciotiempo de Lorentz-Minkowski.

22 january 2020, Seminar 2nd floor, 18:00-19:00

Speaker: Ferdinand Ngakeu. (University of Douala, Cameroon)

Title: KV-Cohomology and some applications

Abstract: Two versions of the KV-cohomology are presented and some algebraic and geometrical applications are given.

12 february 2020, Seminar 2nd floor, 17:00-18:00

Speaker: Ivan P. Costa e Silva (Federal University of Santa Catarina, Brasil)

Title: Topological restrictions in Lorentzian geometry: a survey

Abstract: It is well know that globally hyperbolic solutions (M,g) of the Einstein field equations in general relativity may have initial data Cauchy hypersurfaces with any topology. However, some restrictions on the fundamental group of M can arise from the causal structure if either all inextendible causal geodesics in (M,g) are complete or if one assumes that M has a boundary with suitable properties. I shall review a number of such "topological censorship" results and discuss some open issues.

12 february 2020, Seminar 2nd floor, 18:00-19:00

Speaker: Ettore Minguzzi (University of Florence, Italy )

Title: A gravitational collapse singularity theorem that improves Penrose's

Abstract: The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work I show that the causality conditions in Penrose's theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in absence of predictability and hence compatibly with quatum field theory and black hole evaporation.

19 february 2020, Seminar 2nd floor, 18:00-19:00

Speaker: Ilya Kossovskiy (Masaryk University, Brno, Czech Republic )

Title: Real-analytic coordinates for smooth strictly pseudoconvex CR-hypersurfaces

Abstract: CR (Cauchy-Riemann) geometry is one of the classical examples of a Parabolic Geometry. It initially arises in the study of holomorphic functions of several variables, and has connections with many areas of mathematics, including Partial Differential Equations, Dynamical Systems, Lorentzian manifolds.

In this talk, I will present a necessary and sufficient condition for an integrable strictly pseudoconvex CR-structure of hypersurface type to be equivalent to a real-analytic CR-structure.

11 march 2020, Seminar 2nd floor, 18:00-19:00

Speaker: Marek Golasinski (University of Warmia and Mazury in Olsztyn, Poland)

Title: Homotopy nilpotency and co-nilpotency of spaces

Abstract: We review known and state some new results on homotopy nilpotency and co-nilpotency of spaces. Next, we take up the systematic study of homotopy nilpotency of homogenous spaces G/K for a Lie group G and its closed subgroup K < G. Then, the homotopy nilpotency of the loop spaces Ω(Gn,m(K)) and Ω(Vn,m(K)) of Grassmann Gn,m(K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is investigated.

22 april 2020, Seminar 2nd floor, 18:00-19:00 (Cancelled due to lockdown)

Speaker: Miguel Sánchez (Universidad de Granada)

Title: Sobre los fundamentos relativistas de la Desigualdad de Penrose

Abstract: En 1973, Penrose propuso una célebre desigualdad geométrica para testar el punto de vista establecido sobre la teoría de los agujeros negros, la cual iría más allá de la desigualdad de masa positiva (que sería demostrada unos años más tarde). La desigualdad de Penrose sugería una relación inesperada entre elementos geométricos aparentemente no relacionados tanto en una clase de variedades lorentzianas como en una de riemannianas. Las demostraciones independientes de las desigualdades puramente riemannianas por Huisken e Ilmanen y por Bray hace 20 años supusieron un hito en Relatividad Matemática y Geometría Diferencial. A lo largo de la charla, haremos un repaso informal de los argumentos que dieron lugar a la conjetura y su estado actual.