Colóquio de Geometria e Aritmética
Rio de Janeiro


Programa: 2015.2

25/9, UFRJ. Sala C 116

10:30-11:30 Matias del Hoyo (IMPA), Fibred categories in differential geometry
Resumo. Fibred categories were introduced by Grothendieck as a device to generalize glueing techniques of geometric objects. Their theory is rather abstract and since its appearance in algebraic geometry they have been applied in several other areas. There is a correspondence between fibred categories and pseudo-functors, which allow us to think of them as families of categories, the fibers, parametrized by another one, the base. In this talk I plan to review the general theory, to discuss fibred Lie groupoids as an incarnation in differential geometry, and to present a rigidity theorem for fibred Lie groupoids developed in a joint work with Fernandes. This structural stability generalizes several classic results on families of actions and foliations, obtained by Palais and Rosenberg among others.
12:00-13:00 Cícero Carvalho (UFU), Calculando a distância mínima em certos códigos de avaliação
Resumo. Um dos parâmetros importantes de um código corretor de erros linear é a chamada distância mínima. Em boa parte da literatura sobre códigos encontramos métodos para se estabelecer cotas para esse parâmetro. Nesta palestra vamos apresentar um método de cálculo da distância mínima para determinados códigos de avaliação, ilustrando-o com um exemplo onde ele produziu os valores exatos da distância mínima.

30/10, IMPA. Sala 228

10:30-11:30 Charles Favre (École Polytechnique, Centre de Mathématiques Laurent Schwartz), Non-Archimedean links of normal surface singularities.
Resumo. (j.w. with Lorenzo Fantini and Matteo Ruggiero) One can associate to any complex normal surface singularity a non-Archimedean analog of its classical link. This non-Archimedean link carries a natural analytic structure that is locally modelled on Berkovich analytic spaces over C((t)). We shall explain how to obtain a characterization of sandwich singularities in term of self-similar properties of this link.
12:00-13:00 Ben Smith (École Polytechnique, INRIA), Applications of arithmetic geometry in contemporary cryptology.
Resumo. Elliptic curves and low-dimensional Jacobian varieties over finite fields are an important tool in contemporary public-key cryptography. Formally---that is, from the point of view of cryptographic protocols---they can often be used as a drop-in replacement for the multiplicative group of a finite field, where they offer higher levels of security with much more compact keys. But this purely formal point of view ignores the rich arithmetic structure of elliptic curves, Jacobians, and their Kummer varieties, which we can exploit to create practical improvements (and even some cryptographic attacks) that have no analogues in conventional finite field-based cryptosystems.

In this talk, we will survey some explicit applications of the arithmetic of low-dimensional abelian varieties in cryptography. These techniques include
* the construction of models with more efficient group laws and scalar multiplication operations,
* using isogenies and endomorphisms to accelerate encryption and decryption algorithms,
* using endomorphism ring structures to accelerate point counting (ie, zeta function computation) for elliptic and genus 2 curves, and
* using isogenies to attack discrete logarithm problems.

27/11, UFF. Campus do Gragoatá, Bloco H, sala 407 (como de chegar)

10:30-11:30 Emanuel Carneiro (IMPA), Extremal entire functions and bounds in the theory of the Riemann zeta-function
Resumo. In this talk I will describe how to bound some objects related to the Riemann zeta-function (and general L-functions), under the Riemann hypothesis, using certain special entire functions of order 1. Among the particular objects that we are interested in we highlight: the modulus on the critical line, the argument (and its antiderivative) on the critical line and the pair correlation of the nontrivial zeros.
12:00-13:00 Joe Rabinoff (Georgia Tech), Attacking uniform Mordell and uniform Manin--Mumford via p-adic integration
Resumo. Coleman's effective version of Chabauty's method of attacking the Mordell conjecture involves counting zeros of certain p-adic integrals on p-adic open discs, using a Newton polygon argument. Recently, Stoll extended the Chabauty-Coleman method using integration on open discs as well as open annuli, to prove a uniform Mordell conjecture for hyperelliptic curves of fixed genus and small Mordell--Weil rank. We use potential theory on Berkovich curves and the Baker--Norine theory of linear systems on metric graphs in order to extend Stoll's methods to all curves of small Mordell--Weil rank. We also give an application to the uniform Manin--Mumford conjecture for curves of a fixed genus with sufficiently degenerate reduction type. This work is joint with Eric Katz and David Zureick-Brown.

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