Colóquio de Geometria e Aritmética
Rio de Janeiro


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Programa: 2013.1

22/03, IMPA. Sala: 228

10:30-11:30 René Schoof, Abelian varieties over Q with only one prime of bad reduction
Resumo. In 1985 J.-M. Fontaine and, independently, V.A. Abrashkin proved the following conjecture by Safarevic: there do not exist any non-zero abelian varieties over Q with good reduction everywhere. In this talk we discuss generalizations of this result. See http://alturl.com/hwb4i.
12:00-13:00 Gunther Cornelissen, Graphs and diophantine equations
Resumo. Li and Yau have given a lower bound on the minimal degree of a morphism from a compact Riemann surface to the Riemann sphere, in terms of the first eigenvalue of the Laplacian and the volume of the Riemann surface. In this talk, we present a graph theoretical analogue of this inequality, its relation to the algebraic geometry of curves over non-archimedean fields, and applications to certain diophantine problems. (Joint work with Fumiharu Kato and Janne Kool.)

26/04, UFRJ. Sala:

10:30-11:30 Alex Abreu, Mapas de Abel para curvas nodais com duas componentes
Resumo. O mapa de Abel de grau d de uma curva lisa é um morfismo que associa a uma d-upla de pontos da curva, o fibrado induzido por esses d pontos. Um problema muito estudado nos ultimos anos é a construção de mapas de Abel para curvas singulares. Nesta palestra mostraremos como construir mapas de Abel para curvas nodais com duas componentes.
12:00-13:00 Fernando Cukierman, On deformations of varieties and foliations
Resumo. In this talk we shall review some aspects of the deformation theory of smooth and of singular varieties. Then we plan to discuss the extension of these ideas to deformations of foliations and some first steps towards deformations of exterior differential systems.

31/05, IMPA. Sala: 232

10:30-11:30 Dave Anderson, Okounkov bodies and toric degenerations
Resumo. Given a projective variety X of dimension d, a "flag" of subvarieties Y_i, and a big divisor D, Okounkov showed how to construct a convex body in R^d that captures interesting aspects of the geometry of D. In the last five years, this construction has been developed further in work of Kaveh-Khovanskii and Lazarsfeld-Mustata. In general, this "Okounkov body" is quite hard to understand, but when X is a toric variety, it is just the polytope associated to D via the standard yoga of toric geometry. In this talk, I'll describe recent results of myself and others relating Okounkov bodies on more general varieties with flat degenerations to toric varieties, integrable systems, and representation theory.
12:00-13:00 Mauricio Velasco, When is every nonnegative quadric a sum of squares?
Resumo. If X is a subvariety of projective space we ask the following question: When is every nonnegative quadric on X a sum of squares of linear forms in X? We identify a numerical invariant which is an obstruction for equality and show that it vanishes iff X is a variety of minimal degree. These result generalize a well known Theorem of Hilbert characterizing the pairs (d,k) for which every nonnegative homogeneous polynomial of degree 2d in k variables is a sum of squares and give us many new instance of the equality of much interest in optimization. These results are joint work with Gregoriy Blekherman (Georgia Tech) and Gregory Smith (Queen's University).

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