Seminários

On a representation of the ${\mathfrak sl}_2$ Lie algebra by vector fields

Resumo: It is an easy exercise to verify that the vector space generated by the vector fields $$ f=-(t_1^2-\frac{1}{12}t_2)\frac{\partial}{\partial t_1}-(4t_1t_2-6t_3)\frac{\partial}{\partial t_2}- (6t_1t_3-\frac{1}{3}t_2^2)\frac{\partial}{\partial t_3}, $$ $$ h=-6t_3\frac{\partial }{\partial t_3}-4t_2\frac{\partial }{\partial t_2}-2t_1\frac{\partial}{\partial t_1},\ e=\frac{\partial}{\partial t_1} $$ together with the Lie bracket is isomorphic to the Lie algebra ${\mathfrak sl}_2$, that is, $[h,e] = 2e, \quad [h,f] = -2f, \quad [e,f] = h$. In this talk I will explain the origin of this exercise which is the Gauss-Manin connection of the three parameter family of elliptic curves $y^2=4(x-t_1)^3-t_2(x-t_1)-t_3$. The first vector field has a solution in terms of Eisenstein series.

As desigualdades de Chern-Kuiper

Resumo: Dada uma subvariedade Euclideana, o tensor de curvatura está determinado extrinsicamente pela famosa equação de Gauss. Estudar de que forma esta relação determinada as condições extrínseca a partir das intrínsecas é uma questão fundamental na teoria de subvariedades.

Nesta palestra vamos estudar a relação entre os kernels da curvatura e da segunda forma fundamental. Se bem estas distribuições estão relacionadas, elas não necessariamente coincidem, dificultando dar uma perspectiva intrínseca a problemas extrínsecos. Por este motivos, vamos estudar o caso em que elas não coincidem. A ferramenta principal para isto são as desigualdades de Chern-Kuiper.

Investigations of SL(4)-structures.

Resumo: The studies of convex-cocompact $SO(3,1),SO(2,2),$ and $SO(2,1)\times{\Bbb R}^{2,1}$ structures are the Lie-theoretic analogues of the studies of de Sitter, anti de Sitter, and Minkowski spacetimes respectively. The analytic treatments of these three theories are almost identical, suggesting a unified treatment involving Anosov SL(4)-structures. The main challenge in constructing such a unification is the absence of any invariant metric. It is thus necessary to enquire what structures may take their place. The purpose of this talk is to present some preliminary investigations into the theory of SL(4) structures aimed at resolving this problem.

Cascading upper bounds for triangle soup Pompeiu-Hausdorff distance

Resumo: We propose a new method to accurately approximate the Pompeiu-Hausdorff distance from a triangle soup A to another triangle soup B up to a given tolerance. Based on lower and upper bound computations, we discard triangles from A that do not contain the maximizer of the distance to B and subdivide the others for further processing. In contrast to previous methods, we use four upper bounds instead of only one, three of which newly proposed by us. Many triangles are discarded using the simpler bounds, while the most difficult cases are dealt with by the other bounds. Exhaustive testing determines the best ordering of the four upper bounds. A collection of experiments shows that our method is faster than all previous accurate methods in the literature.