Colóquio de Geometria e Aritmética
Rio de Janeiro


Programa: 2016.2

30/9, IMPA. Sala 236

10:30-11:30 Cristhian Garay (UFF), The fundamental theorem of tropical differential algebraic geometry (joint work with F. Aroca and Z. Toghani)
Resumo. Let I be an ideal of the ring of Laurent polynomials in n variables with coefficients in a real-valued field (K,v). The fundamental theorem of tropical algebraic geometry states the equality trop(V(I))=V(trop(I)) between the tropicalization trop(V(I)) of the closed subscheme V(I) and the tropical variety V(trop(I)) associated to the tropicalization of the ideal trop(I).
In this talk we discuss an analogous result for a differential ideal G of the ring of differential polynomials in n variables with coefficients in the DVR K[[t]], where K is an uncountable algebraically closed field of characteristic zero. We define the tropicalization trop(Sol(G)) of the set of solutions Sol(G) of G, and the set of solutions Sol(trop(G)) associated to the tropicalization of the ideal trop(G). There is a tropicalization morphism trop which goes from Sol(G) to Sol(trop(G)).
We show the equality trop(Sol(G))=Sol(trop(G)), answering a question recently raised by D. Grigoriev.
Link to the article in the ArXiv : arXiv:1510.01000
12:00-13:00 Dan Avritzer (UFMG), Linear Systems of quadrics containing a linear space.
Resumo. Consider a complete intersection X of two quadrics Q and Q' in P^5. Assume X is nonsingular and choose a line L in X and a 3-dimensional linear subspace M disjoint from L. Consider the projection p from P^5 to M with center L and restrict p to X; this gives a birational map from X to M whose inverse is not defined at the genus two quintic in M given by a quadric and two cubics.
In this talk, I will explain how to extend this construction by considering a linear system of k quadrics in P^N containing a linear space. I will also apply this to the Fano variety of lines in X.
(This is joint work with C. Peskine)

21/10, UFRJ. Sala C 119

10:30-11:30 Atoshi Chowdhury (IMPA), Two approaches to compactifying the relative Picard scheme over degenerations of varieties
Resumo. Consider a family X of d-dimensional varieties, in which the central fiber X_0 is a singular variety. The relative Picard scheme for this family may fail to be compact: that is, given a line bundle on X\X_0, there may be no way to extend it to a line bundle on X -- or there may be infinitely many ways to do so.
For families of curves (d=1), there are two closely related approaches to resolving this problem: one can extend a line bundle on X\X_0 using either a line bundle on a modification of X_0, or a sheaf (not necessarily invertible) on X_0 itself. In each approach, one then obtains a compactified Picard scheme by imposing a stability condition on the line bundles or sheaves used. It is well known that the two approaches produce isomorphic compactifications.
I'll discuss work in progress (joint with Eduardo Esteves) on extending this picture to families of higher-dimensional varieties (d>1). In particular, I'll describe partial generalizations of each approach to higher dimension, maps relating them to each other, and results indicating that the two approaches should be equivalent in the case of families of surfaces (d=2).
12:00-13:00 Diego Marques (UnB), Sobre Alguns Problemas de Kurt Mahler
Resumo. Em 1976, o matemático alemão Kurt Mahler publicou um livro cujo capítulo 3 era completamente devotado ao comportamento aritmetico de funções transcendentes. Nesse capítulo, ele enunciou três problemas. Nessa palestra falaremos sobre a história desses tipos de problemas e sobre a nossa solução para dois deles. Além disso, será apresentada uma breve introdução a teoria dos numeros transcendentes. Alguns trabalhos apresentados aqui foram feitos em conjunto Carlos Gustavo Moreira e Josimar Ramirez.

25/11, UFF. Campus do Gragoatá, Bloco H, sala 407 (como de chegar)

10:30-11:30 Vinicius Gripp Barros Ramos (IMPA), Symplectic embeddings, number theory and billiards
Resumo. The study of symplectic embeddings lies at the core of symplectic topology and its flexbility and ridigity properties has been shown to be very interesting and difficult to predict. In this talk, I will explain a theorem of McDuff-Schlenk relating the Fibonacci numbers with the sharp symplectic embeddings of four-dimensional ellipsoids into balls. I will also talk about a recent result relating these sharp bounds to lengths of billiards in a disk.
12:00-13:00 Letterio Gatto (Torino), On the Equations of Plücker Quadrics
Resumo. Let G(r,n) be the complex Grassmann manifold parametrizing r-dimensional subspaces of Cn, understood as the locus of decomposable tensors in r-th exterior product of Cn. The goal is to sketch the proof of a formula, obtained with P. Salehyan, that characterizes the image of G(r,n) in H*(G(r,n),C) via a natural isomorphism of the r-th exterior product of Cn with H*(G(r,n),C), to be described in the talk. The formula, obtained within a familiar finite-dimensional context, asymptotically recovers the celebrated KP-hierarchy, a system of infinitely many quadratic PDEs eventually seen, in yet another way, as the Plücker equations of a Grassmannian parametrizing infinite-dimensional subspaces of C. Most of the computations will be explicitly performed in the case of G(2,n) to simplify the combinatorial issues, although exactly the same arguments work to describe the Plücker embedding of G(r,n), for all 2≤ r≤ n at once.

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