Braga - October 2024

October 18, 2024, Mathematics Department / CMAT of Universidade do Minho

Speaker: Viviana del Barco (Universidade Estadual de Campinas, Brasil)
Title: G_2-instantons on nilpotent Lie groups
Abstract: In this talk we will discuss recent advancements on G_2-instantons on 7-dimensional 2-step nilpotent Lie groups endowed with a left-invariant coclosed G_2-structures. I will present necessary and sufficient conditions for the characteristic connection of the G_2-structure to be an instanton, in terms of the torsion of the G_2-structure, the torsion of the connection and the Lie group structure. These conditions allow to show that the metrics corresponding to the G_2-instantons define a naturally reductive structure on the simply connected 2-step nilpotent Lie group with left-invariant Riemannian metric. This is a joint work with Andrew Clarke and Andrés Moreno.

Speaker: Pedro M. Silva (Universidade de Lisboa)
Title: Projective structures and the geometry of the conformal limit
Abstract: The classical non-abelian Hodge correspondence is a gauge-theoretic construction that has allowed for the use of complex geometric methods in the study of representations of the fundamental group of a closed surface. The conformal limit was introduced by Gaiotto as a parameterized variation of this classical correspondence. In this talk, we will explore how, in the case of representations into SL(2,C), this limit is related to complex projective structures. We will also use this relation to further our geometric understanding of the limiting process. This is joint work with Peter B. Gothen.

Speaker: Dario di Pinto (Universidade de Coimbra)
Title: Geometry and topology of anti-quasi-Sasakian manifolds
Abstract: In the present talk I will introduce a new class of almost contact metric manifolds, called anti-quasi-Sasakian (aqS for short). They are non-normal almost contact metric manifolds , locally fibering along the 1-dimensional foliation generated by onto Kähler manifolds endowed with a closed 2-form of type . Various examples of anti-quasi-Sasakian manifolds will be provided, including compact nilmanifolds, -bundles and manifolds admitting a -reduction of the structural group of the frame bundle. Then, I will discuss some geometric obstructions to the existence of aqS structures, mainly related to curvature and topological properties. In particular, I will focus on compact manifolds endowed with aqS structures of maximal rank, showing that they cannot be homogeneous and they must satisfy some restrictions on the Betti numbers. This is based on joint works with Giulia Dileo (Bari) and Ivan Yudin (Coimbra).

Speaker: Henrik Winther (Artic University of Norway)
Title: The Gap Phenomenon for Automorphisms of Parabolic Geometries
Abstract: We consider and resolve the gap problem for global automorphisms of complex or real parabolic geometries. Concretely, the automorphism group of a parabolic geometry of type (G,P) is largest for the flat model G/P. The symmetry dimension is maximal in this case and is equal to dim(G). We prove that the next realizable, so-called submaximal dimension of the automorphism group of a (G,P) type geometry is the dimension of a (specific) maximal parabolic subgroup in G. We also discuss maximal and submaximal dimensions of the automorphism group of compact models and provide several examples. Joint work with B. Kruglikov.