Colóquio de Geometria e Aritmética
Rio de Janeiro
Descrição: Colóquio rotativo entre IMPA, UFRJ e UFF. Acontecerá toda última sexta feira do mês, contando com duas palestras, uma de 10:30 as 11:30 e outra de 12:00 as 13:00. As palestras serão ministradas por pesquisadores locais e do exterior das áreas de geometria algébrica e aritmética.
O objetivo principal é estimular a troca de ideias e a colaboração entre as instituições do Rio de Janeiro. A pausa para o café e o almoço após o colóquio darão oportunidade para discussões entre os participantes e interação com os palestrantes.
Esperamos uma grande participação dos pesquisadores da área. Estudantes de mestrado e doutorado são encorajados a participar.
24/8, UFF. Campus do Gragoatá, Bloco H, sala 407.
Misha Verbitsky (IMPA),
Transcendental Hodge algebra|
Resumo. Let M be a projective manifold. Transcendental Hodge lattice of weight p is the smallest rational Hodge substructure in Hp(M) containing Hp,0(M). Transcendental Hodge lattice is a birational invariant of M. Yu. Zarhin computed the transcendental Hodge lattice for a K3 surface, and proved that it is an irreducible representation of a unitary or orthogonal group over a number field. I will prove that the direct sum of all transcendental Hodge lattices for any projective manifold is an algebra, and compute it (using Zarhin's theorem) explicitly for all hyperkahler manifolds in terms of
irreducible representations of unitary or orthogonal groups over number fields.
Omid Amini (ENS, Paris),
Trees in algebraic geometry|
Resumo. The aim of the talk is to show through examples of recent results how combinatorial objects called trees naturally arise and play a role in understanding the geometry of algebraic varieties and their asymptotics. The results themselves are motivated by arithmetic geometry and mathematical physics.
28/9, IMPA, sala 236.
Luciane Quoos (UFRJ),
Resumo. Let Fq be the finite field with q elements. A polynomials f(x) in Fq[x] is said to be a permutation polynomial (PP for short) if it is a bijection as a function in Fq. In this talk it will be presented some criterions to decide when a polynomial over Fq is a PP and some families that can be obtained using these criterions. The notion of the Carlitz Rank of a PP will be introduced and a bound involving its Carlitz Rank, the degree of a certain polynomial and the cardinality of the finite field will be given.
Kostiantyn Iusenko (USP),
Stable representations of posets.|
Resumo. Representations of finite dimensional algebras can be approached combinatorially via representations of posets (due to L.A. Nazarova
and A.V. Roiter) and representations of quivers (due to P. Gabriel). The problem of classifying representations of "most" algebras is wild in a sense
that it is as difficult as the problem of classifying representations of free algebras. Nevertheless, one can use geometrical approach by considering the spaces whose points correspond naturally to isomorphism classes of representations. Using standard GIT methods A. King defined the moduli spaces of quiver representations. In this talk we will discuss certain aspects related to study of moduli space of poset representations. We will see that the Euler quadratic form associated with a poset plays significant role here: for calculation of dimension of moduli space and for canonical choice of stability (which is certain analogue of Schofield's characterization of Schurian roots for quiver). Also we plan to discuss the behavior of Coxeter transformations on stable representations.
26/10, UFRJ, sala C116.
Alex Massarenti (UFF),
On Mori chamber and stable base locus decompositions|
Resumo. The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We will investigate the differences between these decompositions for moduli spaces of complete collineations. Furthermore, we will provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we will construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our criterion is sharp.
Edgar Costa (MIT),
Resumo. In this talk, we will focus on how one can deduce some geometric invariants of an abelian variety or a K3 surface by studying their Frobenius polynomials.
In the case of an abelian variety, we show how to obtain the decomposition of the endomorphism algebra, the corresponding dimensions, and centers.
Similarly, by studying the variation of the geometric Picard rank, we obtain a sufficient criterion for the existence of infinitely many rational curves on a K3 surface of even geometric Picard rank.
30/11, UFF. Campus do Gragoatá, Bloco H, sala 407.
Nivaldo Medeiros (UFF)
Carolina Araujo (IMPA)
Cecília Salgado (UFRJ)
Eduardo Esteves (IMPA)
Marco Pacini (UFF)
Oliver Lorscheid (IMPA)
Apoio: CNPq, Capes, Faperj
Contato: colga at impa.br