Colóquio de Geometria e Aritmética
Rio de Janeiro


Programa: 2014.1

28/03, UFF. Sala de seminários da pós-graduação, 7o. andar

10:30-11:30 Alex Massarenti (IMPA), Varieties of Sums of Powers
Resumo. In 1770 Edward Waring stated that every integer is a sum of at most 9 positive cubes. Later on Jacobi and others considered the problem of finding all the decompositions of a given number into sums of cubes. Since then many problems related to additive decompositions have been named after Waring. The setup we are interested in is that of homogeneous polynomials over an algebrically closed field. In particular we consider the decompositions of a general homogeneous polynomial of degree d as sums of d-powers of linear forms. Varieties of Sums of Powers parametrize these additive decompositions. The study of these varieties dates back to Sylvester and Hilbert. However, only few of them, for special degrees and number of variables, are concretely identified. We will discuss some biregular and birational aspects of these varieties. In particular, we will prove the rational connectedness of many Varieties of Sums of Powers in arbitrary degrees and number of variables.
12:00-13:00 Benjamin Collas (Münster), Stack inertia of moduli spaces of curves and Grothendieck-Teichmüller theory
Resumo. In Arithmetic Geometry of moduli spaces of curves, one studies the action of the absolute Galois group of rationals on the associated algebraic fundamental groups. The purpose of Grothendieck-Teichmüller theory is to provide a framework to achieve "explicit" results for this action. In this talk, we first present how a Grothendieck-Teichmüller group is constructed in the context of mapping class groups of surfaces, and how it relates to this consideration. We then explain how this theory is applied in order to characterize the Galois action on a certain type of elements: the stack inertia -- or (pro)torsion -- elements. Our approach is mainly group theoretic and relies on properties of mapping class groups of surfaces, braid groups and group cohomology theory.

25/04, IMPA. Sala: Auditório 3

10:30-11:30 Ethan Cotterill (UFF), Dimension counts for singular rational curves
Resumo. Rational curves are essential tools for classifying algebraic varieties. Establishing dimension bounds for families of embedded rational curves that admit singularities of a particular type arises arises naturally as part of this classification. Singularities, in turn, are classified by their value semigroups. Unibranch singularities are associated to numerical semigroups, i.e. sub-semigroups of the natural numbers. These fit naturally into a tree, and each is associated with a particular weight, from which a bound on the dimension of the corresponding stratum in the Grassmannian may be derived. Understanding how weights grow as a function of (arithmetic) genus g, i.e. within the tree, is thus fundamental. We establish that for genus g \leq 8, the dimension of unibranch singularities is as one would naively expect, but that expectations fail as soon as g=9. Multibranch singularities are far more complicated; in this case, we give a general classification strategy and again show, using semigroups, that dimension grows as expected relative to g when g \leq 5. This is joint work with Lia Fusaro Abrantes and Renato Vidal Martins.
11:30-14:00 Lunch break
14:00-15:00 Anthony Várilly-Alvarado (Rice), Note a mudança do horario! Explicit Arithmetic on Algebraic Surfaces
Resumo. The geometric complexity of a variety is a good proxy for its arithmetic complexity. Using the classification of algebraic surfaces as a guide for geometric complexity, we will discuss explicit techniques for computing cohomological obstructions to the existence and distribution of rational points on algebraic surfaces, with a view toward identifying a boundary between arithmetically "well-behaved" varieties, like rational surfaces, and arithmetically "wild" varieties, like surfaces of general type.
15:00-15:30 Café

30/05, UFRJ. Sala: C 116

10:30-11:30 Reimundo Heluani (IMPA), From Lie algebras to modular forms and elliptic curves
Resumo. Ever since Frobenius it has been noticed that characters of representations are special functions. Every known family of orthogonal polynomials, Bessel functions, trigonometric functions, spherical functions and such appear this way. A new phenomenon started to unveil in the late 70's and 80's when it was discovered that characters of modules of certain infinite dimensional Lie algebras and groups where modular invariant. At the time there were plenty of examples both coming from physics and representation theory, but no good explanation of this phenomenon. It was in the mid 90's that Zhu settled this question putting in a rigorous framework some ideas from string theory: these characters are naturally defined (flat) sections of certain bundles on the moduli space of elliptic curves. This immediately shows the modular invariance since a coarse version of this moduli space is simply H (the upper half plane) divided by SL(2,Z) (the modular group).
In later years many examples have arisen involving super Lie algebras instead of simply Lie algebras. In all these examples some extended version of modularity is found. We show that under certain conditions the characters of modules for these Lie algebras are Jacobi modular forms (that is invariant for the group SL(2,Z) \ltimes Z^2) by naturally constructing them as sections of some flat bundles on the moduli space of elliptic supercurves. The reduced part of this moduli space parametrizes an ellptic curve and a line bundle over it, so it is simply the universal elliptic curve (or rather its Jacobian), but some very involved subtleties arise when working on families over supercommutative rings.
This is joint work in Jethro Van Ekeren.
12:00-13:00 Michela Artebani (Concepción), About the Berglund-Hübsch mirror construction
Resumo. In the nineties the physicists Berglund and Hübsch defined a duality between (finite quotients of) Calabi-Yau hypersurfaces of weighted projective spaces defined by polynomials of the following type
W(x_1,\dots,x_n)=\sum_{i=1}^n \prod_{j=1}^n x_j^{a_{ij}}.
Recently Chiodo and Ruan proved that such dual Calabi-Yau's satisfy the requirement of classical topological mirror symmetry, that is their Hodge numbers are exchanged. In this talk we will explain how the Berglund-Hübsch construction can be formulated in terms of toric geometry and we will show that it actually consists of a duality between polytopes, in the spirit of Batyrev mirror symmetry.
This is joint work with Paola Comparin.

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