Colóquio de Geometria e Aritmética
Rio de Janeiro


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Programa: 2014.2

29/08, UFRJ. Sala: C 116

10:30-11:30 John Alexander Cruz Morales (IMPA), The geometric semantics for algebraic quantum mechanics.
Resumo. In this talk I will report a work in progress with Boris Zilber (Oxford University) about the construction of a sheaf (of Zariski geometries) associated with (rational) Weyl algebras and its application in constructing a geometric semantics for the algebraic quantum mechanics.
12:00-13:00 Marc Hindry (Paris 7), Análogos do teorema de Brauer-Siegel para curvas elípticas e variedades abelianas
Resumo. O teorema clássico de Brauer-Siegel afirma que, para uma familia de corpos de números de grau conhecido, o produto do regulador das unidades pelo número de classes d'ideais cresce asintoticamente como a raiz quadrada do discriminante. O problema análogo para uma familia de variedades abelianas de dimensão dada, definidas sobre um corpo global é de comparar o produto do regulador de Néron-Tate pelo cardinal do grupo de Shafarevich-Tate com a altura da variedade abeliana.
Apresentaremos os objetos, a motivação principal do problema e resultados (condicionais sobre corpos de números, incondicionais sobre corpos de funções) que mostram que o comportamento é parcialmente diferente do caso clássico : existem famílias com comportamento assintotico similar ao teorema de Brauer-Siegel clássico, mais existem outros casos, nos quais o comportamento é diferente.
Trabalho em colaboração com Amílcar Pacheco (UFRJ)


26/09, IMPA. Sala 232

10:30-11:30 Aftab Pande (UFRJ), p-adic families of Hilbert modular forms.
Resumo. We will give a brief introduction about modular forms with examples. We will then define the notion of a p-adic family of modular forms and explain a generalization to Hilbert Modular Forms.
12:00-13:00 Alice Garbagnati (Milano), Families of elliptic K3 surfaces with finite Mordell--Weil group and sandwich theorems.
Resumo. The aim of this talk is to describe the families of K3 surfaces with torsion Mordell-Weil group in terms of $L$-polarized K3 surfaces. Moreover we will prove that the generic elliptic K3 surface with a torsion group as Mordell--Weil group is an example of K3 surface X which is both a cover and a quotient of the same K3 surface Y, indeed. there exists 2 finite rational map $Y\rightarrow S\rightarrow Y$. This pheonomen is in general called "sandwich of K3 surfaces".

31/10, UFF. Sala de seminários da pós-graduação (7o. andar)

10:30-11:30 Mikhail Belolipetsky, Arithmetic hyperbolic reflection groups
Resumo. A group of isometries of the hyperbolic n-space is called a reflection group if it has a finite generating set which consists of reflections in hyperplanes. The study of hyperbolic reflection groups has a long and remarkable history going back to the papers of Makarov and Vinberg. In recent years there has been a wave of activity in this area which has led to a solution to the open question of the finiteness of these groups and to some quantitative results towards their classification. In the talk I will review the recent results and discuss some open problems concerning arithmetic hyperbolic reflection groups.
12:00-13:00 Aron Simis, Redes homalóides planas e ideais de pontos grossos
Resumo. O estudo de ideais de pontos planos com multiplicidades virtuais arbitrárias (fat points) é de grande interesse em geometria algébrica e álgebra comutativa, desde a famosa conjectura de Nagata. Nesta palestra, analisaremos o entrelaçamento do ideal de base de um mapa plano de Cremona com o ideal de pontos ``grossos'' associado aos pontos e às multiplicidades virtuais do mapa. O foco é em aspectos de natureza homológica, tais como sizigias, resoluções livres, Cohen-Macaulay, etc.

28/11, IMPA.

10:30-11:30 Renato Vidal Martins,
Resumo.
12:00-13:00 Ana-Maria Castravet (Ohio State University), Mori Dream Spaces and moduli spaces of stable rational curves
Resumo. Mori Dream Spaces form an ideal class of algebraic varieties in which, as the name suggests, the Minimal Model Program works very well. Introduced by Hu and Keel, Mori Dream Spaces can be algebraically characterized as varieties whose total coordinate ring (or Cox ring) is finitely generated.
Cox rings appear naturally in the context of Hilbert's 14th Problem. One can use the geometry of algebraic varieties to answer questions about finite generation of invariant rings (work of Nagata and Mukai, among others). In their original paper, motivated by questions about the birational geometry of moduli spaces of stable curves, Hu and Keel asked whether the Grothendieck-Knudsen moduli space of stable, $n$-pointed, rational curves is a Mori Dream Space. Along with the Fulton-Faber conjecture, this question was one of the main open problems in the area, generating a flurry of activity in the last couple of years.
In this talk I will explain the connections between Cox rings and invariant rings, introduce the Grothendieck-Knudsen moduli space and the main questions about its geometry, and finally, present joint work with Jenia Tevelev, answering negatively Hu and Keel's question for $n$ large.

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