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.
Inspired by Jaeger’s composition formula for the HOMFLY polynomial, Turaev defined a coproduct on the HOMFLY skein algebra of a framed surface S, turning it into a bialgebra. Jaeger’s formula can be viewed as a universal version of the restriction of the defining representation from $\operatorname{GL}_{m+n}$ to $\operatorname{GL}_m × \operatorname{GL}_n$. The restriction functor, however, is not braided, and therefore there is a priori no reason for the induced linear map between the corresponding skein algebras to be multiplicative. In this talk, I will address this problem using defect skein theory and the formalism of parabolic restriction.
In the first part of the talk, I will introduce skein theory for 3-manifolds with both surface and line defects. Local relations near the defects are produced from the algebraic data of a central algebra (codimension 1) and a centred bimodule (codimension 2). Examples of such structures are provided by the formalism of parabolic restriction. In the second part of the talk, I will explain how to construct a universal version of this formalism. Finally, we will see how Turaev’s coproduct extends to the entire skein category using the previous constructions.
In this talk, I explore how internal deformations in spinless extended bodies, such as periodic pulsations or oscillations of the body’s shape, can trigger chaotic motion even when the background spacetime is fixed and the underlying geodesic dynamics is integrable. Using Dixon’s multipolar framework, I show how time-periodic finite-size effects can split separatrices and generate homoclinic chaos, diagnosed via the Melnikov method. I discuss Schwarzschild black holes for nearly spherical bodies with oscillating oblateness, as well as spherically symmetric pulsations in non-vacuum spacetimes, including electrovac/charged black holes and black holes embedded in dark-matter halos. Finally, I highlight a genuinely relativistic boundary of the point-particle idealization: in Ricci-flat vacuum, spinless spherical bodies move geodesically through quadrupole order, yet finite-size deviations can arise at hexadecapole order, where even vacuum Schwarzschild admits nontrivial corrections and chaos under pulsations.
I will first introduce the bigraded cohomology for real algebraic varieties developed by Johannes Huisman and Dewi Gleuher. This is a cohomology theory that refines the equivariant cohomology "à la Kahn-Krasnov" of the complex points of a real variety, the latter often being preferred (by the algebraic geometers) in the cohomological study of real algebraic varieties. Since the construction of this bigraded cohomology and its associated characteristic classes relies on the sheaf exponential morphism, I will explain how to produce an arithmetic (or algebraic) variant of these cohomology groups, whose main advantage is toeliminate topological or transcendental conditions. I will conclude by comparing these two versions of bigraded cohomology.
