Colóquio de Geometria e Aritmética
Rio de Janeiro

Programa: 2018.1

29/3, IMPA, Auditorio 1.

10:30-11:30 Alex Abreu (UFF), Mapa de Abel universal.
Resumo. Nesta palestra mostraremos como resolver o mapa de Abel universal Mg,nJg, onde Jg é a compactificação de Esteves da Jacobiana universal. Para isso utilizaremos a geometria tropical, em particular resolveremos o problema análogo para o moduli de curvas tropicais.
12:00-13:00 Dhruv Ranganathan (MIT), Curves, maps, and singularities in genus one.
Resumo. I will outline a new framework based on tropical and logarithmic methods to study genus one curve singularities and discuss its relationship with the geometry of moduli spaces. I will focus on two applications of these ideas. First, they allow one to explicitly factorize the rational maps among log canonical models of the moduli space of n-pointed elliptic curves. Second, they reveal a modular interpretation for Vakil and Zinger's famous desingularization of the space of elliptic curves in projective space, as well as a short and conceptual proof of that result. In fact, the same methods yield logarithmically smooth compactifications of the space of elliptic curves in toric varieties. If time permits, I will discuss applications to some questions in classical enumerative geometry. This is based on joint work with Keli Santos-Parker and Jonathan Wise, building on prior work of Speyer, Smyth, Viscardi, Vakil, and Zinger.

27/4, UFRJ. Sala C116 Bloco C

10:30-11:30 Alex Massarenti, On the birational geometry of moduli spaces of points on the line
12:00-13:00 Henrique Bursztyn, Morita equivalence in algebra and geometry
Resumo. Morita equivalence, native to ring theory, is an equivalence relation among unital algebras which compares them by means of their categories of modules. This notion of equivalence plays a key role e.g. in noncommutative geometry. There are also versions of Morita equivalence in classical geometry, notably for Poisson manifolds. I will discuss these notions of Morita equivalence in algebra and geometry, and explain how one can find a concrete link betweem them through "deformation quantization".

25/5, IMPA, sala 232.

10:30-11:30 Alessia Mandini (PUC), Hyperpolygons and parabolic Higgs bundles
Resumo. Hyperpolygons spaces are a family of (finite dimensional, non-compact) hyperkaehler spaces, that can be obtained from coadjoint orbits by hyperkaehler reduction. Jointly with L. Godinho, we show that these space are diffeomorphic (in fact, symplectomorphic) to certain families of parabolic Higgs bundles. In this talk I will describe this relation and use it to analyse the fixed points locus of a natural involution on the moduli space of parabolic Higgs bundles. I will show that each connected components of the fixed point locus of this involution is identified with a moduli spaces of polygons in Minkowski 3-space.
This is based on joint works with Leonor Godinho and with Indranil Biswas, Carlos Florentino and Leonor Godinho
12:00-13:00 Kevin Destagnol (MPI Bonn), Prime and squarefree values of polynomials in moderately many variables
Resumo. The classical Schinzel's hypothesis and its quantitative version, the Bateman-Horn's conjecture, states that a system of polynomials in one variable takes infinitely many simultaneously prime values under some necessary assumptions. We will present in this talk a proof of a generalization of these conjectures to the case of a integer form in many variables. In particular, we will establish that a polynomial in moderately many variables takes infinitely many prime (and also squarefree) values under some necessary assumptions. The proof relies on the Birch's circle method but can be achieved in 50% fewer variables than in the classical Birch setting. Moreover this result can be applied to study the Hasse principle and weak approximation for some normic equations.

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