Colóquio de Geometria e Aritmética
Rio de Janeiro


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Programa: 2017.1

31/3, IMPA. Sala 224

10:30-11:30 Alessio Corti (Imperial College), Picard-Fuchs equations of minimal ramification.
Resumo. Ordinary differential equations (ODE) with polynomial coefficients arise naturally in algebraic geometry as Picard-Fuchs equations. I will define the notion of ramification of such an ODE, and ask for a classification of those of smallest ramification. I will show several examples.
12:00-13:00 Roozbeh Hazrat (Western Sydney University), Leavitt path algebras.
Resumo. From a directed graph one can generate an algebra which captures the movements along the graph. One such algebras are Leavitt path algebras.
Despite being introduced only 10 years ago, Leavitt path algebras have arisen in a variety of different contexts as diverse as analysis, symbolic dynamics, noncommutative geometry and representation theory. In fact, Leavitt path algebras are algebraic counterpart to graph C*-algebras, a theory which has become an area of intensive research globally. There are strikingly parallel similarities between these two theories. Even more surprisingly, one cannot (yet) obtain the results in one theory as a consequence of the other; the statements look the same, however the techniques to prove them are quite different (as the names suggest, one uses Algebra and other Analysis). These all suggest that there might be a bridge between Algebra and Analysis yet to be uncovered.
In this talk, we introduce Leavitt path algebras and try to classify them by means of (graded) Grothendieck groups. We will ask nice questions!

28/4, IMPA. Sala 236

10:30-11:30 Marcelo Escudeiro Hernandes (UEM, Maringá), The semiring and semimodule associated to an algebroid curve.
Resumo. We introduce the semiring Γ of values with respect to the tropical operations associated to an algebroid curve. As a set, Γ determines and is determined by the well known semigroup of values S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the results of Zariski and Waldi that characterize the topological type of the curve.
The principal ingredient is the concept of Standard Basis for the local ring of the curve that give us a computational method to compute the minimal system of generators of Γ. This idea can be apply in other situations, for example to compute the set of values (a semimodule) of an fractionary ideal of the local ring.
12:00-13:00 María Pe Pereira (Universidad Complutense de Madrid), Valuative criterion, arcs and adjacencies of plane curves.
Resumo. I would talk about a joint work with J. Fern\'andez de Bobadilla and P. Popescu- Pampu. In this work we make the link between the study of arc spaces and the classical theory of plane curves. We study two notions of adjacency between prime divisors lying above the origin O of the complex plane. These divisors are the exceptional components of a composition of blow ups over the origin of the plane. We say that E is adjacent to F if there is a deformation of germs of curves at O whose special and general members have strict transforms on a common model of E and F which intersect F and E respectively at smooth points of the exceptional divisor. We say that E is Nash-adjacent to F if the closure of the space of arcs associated to F is contained in the closure of the one associated to E. Nash-adjacency implies adjacency.
We will study the relations between these two types of adjacencies and see what we managed to do in order to describe them.

26/5, UFF. Campus do Gragoatá, Bloco H, sala 407 (como chegar)

10:30-11:30 Maral Mostafazadehfard (IMPA), Anti-canonical cover of secant varieties of a rational normal curve.
Resumo. (see link)
12:00-13:00 Angel Carocca (Universidad de la Frontera, Chile), A generalization of the Recillas's construction
Resumo. The well known Recillas trigonal construction shows that Prym varieties above trigonal curves are Jacobian varieties of tetragonal curves and, conversely, that all tetragonal Jacobians are Pryms.
In this work we show that a similar result applies in a more general situation: Let p be a prime number. Then, given an unramified double cover X_1 of a p-gonal curve Z with total ramification, there exist a 2^{p-1}-gonal curve Y and unramified double covers X_2, ..., X_k of Z, with k = \dfrac{2^{p-1}-1}{p} for p \geq 3 and k=2 for p=2, such that JY is isomorphic to the product of the P(X_i/Z)}.

23/6, UFRJ. Sala C116

10:30-11:30 Frauke Bleher (Iowa), Holomorphic differentials on curves.
Resumo. This talk is about joint work with Ted Chinburg and Aristides Kontogeorgis on the Galois module structure of the holomorphic differentials of the top curve of a finite Galois cover of curves in positive characteristic p. The case in which p divides the order of the group is more subtle, since then not all representations are semi-simple. We show that when a Sylow p subgroup of the Galois group is cyclic, the Galois module structure of the holomorphic differentials can be effectively determined by ramification data alone. I will illustrate the results with an example involving a modular curve mod p.
12:00-13:00 Ted Chinburg (University of Pennsylvania), Brauer group invariants of knots.
Resumo. In this talk I will describe some work with Matt Stover and Alan Reid on new invariants of knots which are elements of the Brauer group of curves. A basic technique in the subject is to study two dimensional representations of the fundamental group of a knot. The new invariants arise from studying natural quaternion Azumaya algebras over subsets of the SL_2 character variety of a knot. This leads to a connection between problems in knot theory and the Tate-Shafarevitch group of Jacobians of curves.

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