Palestras de Geometria
Palestrantes: Daniel Fadel (IMPA), Matias del Hoyo (UFF) e Federico Quallbrunn (Universidad CAECE e CONICET, Arg.)
Data: 10/03/2023, às 10:30h
Local: Sala C116 do CT
10:00h - Café
10:30h - Daniel Fadel (IMPA) - On the harmonic flow of geometric structures
11:30h - Café
13:30h - Matias del Hoyo (UFF) - Completeness of metrics and linearization of groupoids
14:30h - Café
15:00h - Federico Quallbrunn (Universidad CAECE e CONICET, Arg.) - Singularidades Persistentes de Folheações.
16:00h - Café
Palestrante: Daniel Fadel (IMPA)
Título: On the harmonic flow of geometric structures
Resumo: In this talk, I will report on recent results of an ongoing collaboration with Éric Loubeau, Andrés Moreno and Henrique Sá Earp on the study of the harmonic flow of H-structures. This is the negative gradient flow of a natural Dirichlet-type energy functional on a given isometric class of H-structures on a closed Riemannian n-manifold, where H is the stabilizer in SO of a finite collection of tensors in Rn. Using general Bianchi-type identities of H-structures, we are able to prove monotonicity formulas for scale-invariant local versions of the energy, similar to the classic formulas proved by Struwe and Chen (1988-89) in the theory of harmonic map heat flow. We then deduce a general epsilon-regularity result along the harmonic flow and, more importantly, we get long-time existence and finite-time singularity results in parallel to the classical results proved by Chen-Ding (1990) in harmonic map theory. In particular, we show that if the energy of the initial H-structure is small enough, depending on the C^0-norm of its torsion, then the harmonic flow exists for all time and converges to a torsion-free H-structure. Moreover, we prove that the harmonic flow of H-structures develops a finite time singularity if the initial energy is sufficiently small but there is no torsion-free H-structure in the homotopy class of the initial H-structure. Finally, based on the analogous work of He-Li (2021) for almost complex structures, we give a general construction of examples where the later finite-time singularity result applies on the flat n-torus, provided the n-th homotopy group of the quotient SO/H is non-trivial; e.g. when n=7 and H=G2, or when n=8 and H=Spin(7). Reference: arXiv:2211.05197
Palestrante: Matias del Hoyo (UFF)
Título: Completeness of metrics and linearization of groupoids
Resumo: Every smooth fiber bundle admits a complete Ehresmann connection. I will talk about the story of this theorem and its relation with Riemannian submersions. Then, after discussing some foundations of Riemannian geometry of Lie groupoids and stacks, I will present a generalization of the theorem into this framework, which somehow answers an open problem on linearization. Talk based on collaborations with my former student M. de Melo.
Palestrante: Federico Quallbrunn (Universidade CAECE e CONICET, Arg.)
Título: Singularidades Persistentes de Folheações.
Resumo: Falaremos das propriedades notáveis de certos tipos especiais de singularidades de Folheações nas variedades algébricas que chamamos persistentes."