Talks during the course 2023/2024

These lectures were delivered during the 2023/2024 academic year, in reverse chronological order. 

May 13th, 2024, Room "Seminario de Álgebra", 12:00-13:00

Speaker:  Vicente Muñoz (Universidad Complutense de Madrid) 

Title: Inflexible manifolds and mapping degree sets     

Abstract: Let M, N be two oriented closed connected manifolds of dimension n. We define the mapping degree set as deg(M,N)={deg(f)| f:M → N}. It is very relevant to construct inflexible manifolds M, when deg(M,M) is bounded and strongly inflexible manifolds M, those for which for all N, deg(N,M) is bounded. They serve to produce functorial seminorms on n-manifolds.

On the other hand, one may ask which sets of integers can appear as deg(M,N) for some M,N. By cardinality reasons, not all sets can. Here we shall prove that any finite set of integers A, containing 0, is a mapping degree set for some choice. We extend this question to the rational homotopy theory setting, where an affirmative answer is also given, by using Sullivan models. (Joint work with C. Costoya and A. Viruel)

April 30th, 2024, Room "Seminario de Álgebra", 16:00-17:00

Speaker:   Rossella Bartolo (Politecnico di Bari) 

Title: Geodesic connectedness of a spacetime with a causal Killing vector field   

Abstract: We study the geodesic connectedness of a globally hyperbolic spacetime (M, g) admitting a complete smooth Cauchy hypersurface S and endowed with a complete causal Killing vector field K. The main assumptions are that the kernel distribution D of the one-form induced by K on S is non-integrable and that the gradient of g(K, K) is orthogonal to D. We approximate the metric g by metrics gε smoothly depending on a real parameter ε and admitting K as a timelike Killing vector field. A known existence result for geodesics of such type of metrics provides a sequence of approximating solutions, joining two givenpoints, of the geodesic equations of (M, g) and whose Lorentzian energy turns out to be bounded thanks to an argument involving trajectories of some affine control systems related with D.

February 27th, 2024, Room "Seminario de Álgebra", 12:30-13:30

Speaker:  Martin Markl (Institute for Mathematics of the Czech Academy of Sciences) 

Title: Transfers of strongly homotopy structures as Grothendieck bifibrations      

Abstract: It is well-known that strongly homotopy structures can be transferred over chain homotopy equivalences. Using the uniqueness results of Markl & Rogers we show that the transfers could be organized into a discrete Grothendieck bifibration. An immediate aplication is e.g. functoriality up to isotopy. 

February 12th, 2024, Room "Seminario de Álgebra", 12:30-13:30

Speaker:  Shun Wakatsuki (Nagoya University) 

Title: BV exactness and computation in rational homotopy theory      

Abstract: It is difficult to compute the S^1-equivariant cohomology of free loop spaces, even with rational coefficients. In this talk, we introduce a new notion "BV exactness", which enables us to compute it only from cohomological information. BV exactness is closely related to other notions such as formality, positive weights on Sullivan models and the periodicity operator on cyclic homology. Moreover, we will explain an application of computer in rational homotopy theory and show an example concerning BV exactness. This is a joint work with Katsuhiko Kuribayashi, Takahito Naito, and Toshihiro Yamaguchi. 

November 2nd, 2023, Room "Q1", 11:30-12:30

Speaker:  Oisín Flynn-Connolly (Université Sorbonne Paris Nord ) 

Title: p-adic homotopy theory     

Abstract: In this talk, we discuss the theory of strictly commutative algebras over the p-adic numbers. We discuss homotopy theory in the p-adic context and show it is not possible to construct strictly commutative models for non-trivial spaces. We define E_n formality and discuss some conjectures of Mandell in this context. We construct a “best approximation functor” for the singular cochains in commutative algebras and discuss some conjectures relating to it. 

October 27th, 2023, Room "B8", 11:30-12:30

Speaker:  Calvin M. Brice (University of Douala, Cameroon) 

Title: Statistical structures arising in null submanifolds    

Abstract: We show a link between affine differential geometry and null sub-manifolds in a semi-Riemannian manifold via statistical structures. Once a rigging for a null submanifold is fixed, we can construct a semi-Riemannian metric on it. This metric and the induced connection constitute a statistical structure on the null submanifold in some cases. We study the statistical structures arising in this way. 

September 29th, 2023, Room "Q2", 10:30-12:30

Speaker: Eduardo Sáenz de Cabezón  (Universidad de La Rioja) 

Title: Introducción a la fiabilidad algebraica   

Abstract: La teoría de fiabilidad se encarga de analizar la disponibilidad y fiabilidad de procesos y sistemas industriales, redes, etc. Existen muchos métodos para el cálculo de fiabilidad de sistemas, es un área en continuo desarrollo. En esta charla hablaremos de un acercamiento algebraico al problema. La idea central de este acercamiento es codificar el sistema como un objeto algebraico (un ideal monomial) y analizar sus propiedades algebraicas para así obtener información sobre el sistema y su fiabilidad. Este acercamiento nos permite además implementar algoritmos algebraicos para el cálculo de fiabilidad. 

September 13th, 2023, Room "M4", 10:00-12:00

Speaker: Kevin Iván Piterman (Philipps-Universität Marburg) 

Title: Avances en la conjetura de Quillen   

Abstract: El estudio de los complejos de p-subgrupos comenzó en los años 70 motivado por trabajos de K. Brown y D. Quillen, y guarda conexión con cohomología equivariante "módulo el primo p", teoría de grupos, incluyendo la clasificación de grupos simples finitos, categorías de fusión, geometrías finitas, representaciones, etc.

En el célebre artículo de D. Quillen del año 1978, el autor prueba que si un grupo finito G posee un p-subgrupo normal no trivial, entonces su poset de p-subgrupos no triviales, con la topología inducida por su order-complex, es contráctil. Quillen conjeturó la recíproca dando lugar a la conjetura de Quillen. Él mismo demostró la conjetura para los grupos resolubles y también para ciertos grupos de matrices como PGL(n,q). El mayor progreso en dirección a la resolución de la conjetura fue logrado por M. Aschbacher y S.D. Smith en los años 90. Básicamente ellos probaron que si p>5 y el grupo G no contiene ciertos subgrupos unitarios PSU(n,q), con p dividiendo a q+1, entonces la conjetura vale para G.

En esta charla, os contaré sobre el progreso de la conjetura obtenido en colaboración con S.D. Smith. En nuestro trabajo hemos extendido métodos del artículo original de Aschbacher-Smith a todo primo p ("evadiendo" mayormente el uso de la clasificación de grupos simples), y en particular mostramos que su resultado principal vale también para p=3,5. Más aún, un trabajo reciente de Antonio Díaz Ramos propone una resolución a la restricción de los grupos unitarios, lo cual demostraría completamente la conjetura para primos impares. También daremos detalles sobre lo que ocurre para el primo p=2.