Abstracts
Ana Péon Nieto (University of Santiago de Compostela, Spain)
Title: Hodges bundles in rank 3
Abstract: In this talk, I will focus on some key loci inside the moduli space of Higgs bundles, consisting of fixed points under a C* action. These fixed points can be classified into very stable and wobbly depending on the dynamics of the C* action. After motivating the interest of these objects through their role towards understanding the geometry of the moduli space or towards mirror symmetry, I will turn to very stable fixed points of subregular type, whose downward flows yield a BAA brane, a conjectural mirror of which I will propose.
Catarina Faustino (University of Minho, Portugal)
Title: Topology of Higher-Dimensional Automata
Abstract: As amply demonstrated in the literature, concepts and methods from algebraic topology can be profitably employed in concurrency theory, the field of computer science that studies systems of simultaneously executing processes. A powerful combinatorial-topological model for concurrent systems is given by higher-dimensional automata, i.e., pointed labeled precubical sets. The first question we address is how complex the topology of an HDA model of a concurrent system can be. I will show that for every connected polyhedron there exists a concurrent system whose HDA model has the same homotopy type as the polyhedron. The second contribution I will present concerns the homology language of an HDA, which is a graded submodule of an exterior algebra and provides information about the independence structure of the modeled concurrent system. We prove that the homology language is actually a graded subcoalgebra of the exterior algebra. Finally, we introduce a new concept of directed homology for preordered spaces, called the homology digraph. The main result here is a Künneth formula, which enables one to compute the homology digraph of a product from the homology digraphs of the components.
Sandro Caeiro Oliveira (University of Vigo, Spain)
Title: Are products of critical metrics also critical?
Abstract: Einstein metrics are the critical metrics for the Hilbert-Einstein functional when restricted to constant volume variations. This functional is determined by the scalar curvature, which is the only scalar curvature invariant of order one. Taking the second-order invariants, the functionals obtained are the so-called quadratic curvature functionals. In the classification of critical homogeneous metrics for quadratic curvature functionals in dimension 4, products of lower-dimensional critical metrics naturally arise. In this talk, we will examine under what conditions this idea can be generalized and the limitations we encounter when working with these products.
Mehmood Ur Rehman (University of Beira Interior, Portugal)
Title: Primitive Immersion of Constant Curvature of Surfaces Into Flag Manifolds
Abstract: Following the seminal result by E. Calabi which establishes the local classification of complex submanifolds with constant holomorphic sectional curvature in complex space forms, several researchers have investigated minimal immersions with constant curvature of Riemann surfaces into symmetric spaces. For isometric immersions, minimality is equivalent to harmonicity, hence the rich theory of harmonic maps becomes highly relevant here. In recent years, the method of harmonic maps has been intensively used to classify such minimal immersions. There exists a well-established theory of twistor constructions of harmonic maps from Riemann surfaces into symmetric spaces. An important class of twistor lifts is that of primitive maps into k-symmetric spaces G/K. For k > 2, primitive maps are harmonic with respect to all G-invariant metrics. Thus, a natural problem arising from these observations is to classify primitive immersions of constant curvature of Riemann surfaces into k-symmetric spaces, when these are equipped with G-invariant metrics. In this talk, I will address this problem. This is a joint work with Rui Pacheco.
