Seminários, para a disseminação informal de resultados de investigação, trabalho exploratório de equipas de investigação, actividades de difusão, etc., constituem a forma mais simples de encontros num centro de investigação de matemática.
O CAMGSD regista e publica o calendário dos seus seminários há bastante tempo, servindo páginas como esta não só como um método de anúncio dessas actividades mas também como um registo histórico.
The subject of counting curves on 3-folds is extremely rich. Modern tools allow us to define several sophisticated invariants that we think of as "curve counts", such as Gromov-Witten, Donaldson-Thomas, Pandharipande-Thomas and Gopakumar-Vafa invariants. They are proven or expected to be equivalent to each other, in a precise sense. In this talk, I will give an idea of how these invariants are defined and what is known about them, focusing on these correspondences and rationality properties. Along the way, I'll explain a result of myself with Ivan Karpov concerning the rationality of PT generating series on Fano 3-folds; other than that, the talk will be mostly expository.
I will discuss a localized Big Bang formation result, for the Einstein-nonlinear scalar field equations, without symmetry assumptions and that does not require the initial data to be close to any fixed background solution. The key new ingredient for the proof is the introduction of a new gauge for Einstein's equations. This gauge has two advantages over the one used in previous localized Big Bang stability results. First, it is independent of the matter model, so it is possible that it can be used to prove Big Bang formation in the absence of a scalar field. And second, it is suitable to prove that the solutions induce geometric initial data on the singularity, thus providing a complete description of the asymptotics near the Big Bang.
