1999 seminars

Peter Newstead, University of Liverpool
Coherent systems on algebraic curves

A coherent system on an algebraic variety (or scheme) consists of a coherent sheaf together with a subspace of its space of sections. For coherent systems on algebraic curves, one can introduce a concept of stability depending on a real parameter and construct corresponding moduli spaces. As the parameter varies, the stability condition can change only at a discrete (and in fact finite) set of values. For rank 2 bundles with one section, the situation has been thoroughly analysed by Thaddeus and used to give a proof of the Verlinde conjectures and other useful information on the moduli spaces of bundles on curves. In this talk I will be describing joint work (still in progress) with S. Bradlow, O. Garcia-Prada and V. Munoz to analyse at least some aspects of the general case.

Pol Vanhaecke, Université de Poitiers
Integrable systems and projective embeddings

Abelian varieties and their quotients can be embedded in projective space by using Painlevé analysis. This will be explained and applied to compute an explicit equation for the moduli space of rank two bundles with fixed determinant on a non-hyperelliptic Riemann surface of genus 3.

Ana Cannas da Silva, Instituto Superior Técnico
Fold-Forms on Four-Folds

A differential problem is said to satisfy the h-principle if any formal solution (i.e., a solution for the associated algebraic problem) is homotopic to a genuine (i.e., differential) solution. Gromov built the machinery of the h-principle on the work of Whitney, Nash, Kuiper, Smale, Hirsch, Poénaru, Phillips, etc. Without assuming familiarity with the h-principle, I will explain its application to prove the existence, on any orientable 4-manifold, of a closed 2-form with only fold type singularities. A formal solution necessary to conclude the argument is a stable complex structure, guaranteed by a result of Hirzebruch and Hopf.

Peter Gothen, Universidade do Porto
Homological algebra of quiver bundles

The category of representations of a finite quiver in the category of sheaves of modules on a ringed space is abelian. We show that this category has enough injectives by constructing an explicit injective resolution. From this resolution we deduce a long exact sequence relating the Ext groups in these two categories. We also show that under some hypotheses, the Ext groups are isomorphic to certain hypercohomology groups.

Joana Nunes da Costa, Universidade de Coimbra
Jacobi manifolds and Lie algebroids

A Jacobi manifold is a differentiable manifold equipped with a bivector and a vector field verifying some compatibility conditions. When the vector field vanishes identically the Jacobi manifold is a Poisson manifold, but there are other significant examples of Jacobi structures that are not Poisson, such as contact manifolds. We will exhibit some relations between Jacobi structures and Lie algebroids.

Carlos Florentino, Instituto Superior Técnico
Classical and non-abelian theta functions and the heat equation on Lie groups

All classical (abelian) theta functions, first studied by Jacobi and Riemann, satisfy the heat equation on a torus. When this torus is the Jacobian of a Riemann surface X, these theta functions have special properties reflecting the geometry of the surface. Since the non-abelian analogue of the Jacobian is the moduli space of vector bundles over X, non-abelian theta functions are defined as holomorphic sections on this moduli space. However, the analytic theory for these functions is still under investigation. We will show that, in some cases, all abelian and non-abelian theta functions can be obtained by a certain extension of the coherent state transform on a Lie group, which naturally induces the inner product on the space of theta functions.

Michèle Vergne, Ecole Polytechnique
Arrangement of hyperplanes, and number of points in convex polytopes

It is important to solve linear systems $Ax=b$, where the matrix $A$ has integral coefficients, and $x$ is a vector with integral positive coefficients. The number of solutions is a function $P_A(b)$ of $b$, called the partition function: if\[A = \begin{bmatrix}1 & 1 & 1\end{bmatrix}\] the equation reads $x_1+x_2+x_3=b$, and the number of solutions is $P_A(b)=\frac{1}{5!}(b+1)(b+2)(b+3)$ so represents the number of ways to partition a positive integer $b$ in 3 parts, adding up to $b$.

Many examples of such equations arise in statistics, algebraic geometry, representation theory, etc. The function $P_A(b)$ is “locally quasi polynomial”, a result due to Ehrhart, and which is based on the relation between the number $P_A(b)$ and the Riemann-Roch formula on a toric variety. I will present here a more efficient approach based on the cohomology of the complement of hyperplanes. Due to a multidimensional residue formula, this number $P_A(b)$ may be calculated as an integral over a cycle $C$ in the complement.

This is joint work with Velleda Baldoni-Silva and Andras Szenes.

Henrique Bursztyn, MSRI and Université Libre de Bruxelles
Dirac Geometry

Dirac manifolds were introduced by Weinstein and Courant to provide a geometric framework for the study of mechanical systems with constraints. Examples of Dirac structures on manifolds include (pre-)symplectic forms, Poisson structures and foliations. In this talk, I will describe the basic features of the geometry of Dirac manifolds, pointing out their role in Poisson geometry and explaining their connection with the world of Lie algebroids and groupoids.

David Martínez, Universidad Carlos III de Madrid
Odd dimensional almost complex geometry

Approximately holomorphic techniques were introduced by Donaldson to prove the existence of symplectic submanifolds in compact symplectic manifolds. Further generalizations led to different constructions as Lefschetz pencils (Donaldson), normal forms for maps to CP2 (Auroux), embeddings to CPN (Muñoz, Presas, Sols). Here we will present a joint work (Ibort, Martínez-Torres, Presas) that extends some of the known results for symplectic manifolds to its odd dimensional counterparts.

Ettore Aldrovandi, Florida State University
Oncharacteristic classes related to uniformization

An (analytic) approach to the uniformization of a compact Riemann surface of genus greater than one is to look at metrics of constant negative curvature. Such metrics (more precisely, the condition that characterizes them) admit a generating functional, extending the classical Liouville's one. This functional is naturally interpreted as the square of the metrized holomorphic tangent bundle in a suitably defined hermitian-holomorphic Deligne cohomology group. For a pair of line bundles on the Riemann surface, it generalizes Deligne's tame symbol. If time permits, we will also outline relations with group cohomology for Kleinian groups and volume calculations in hyperbolic 3-space.

Rui Loja Fernandes, Instituto Superior Técnico
Linearizing actions and Poisson brackets

A classical result of H. Cartan states that if a compact Lie group acts by analytic transformations and has a fixed point then in suitably chosen local analytic coordinates around the fixed point the action is linear. Bochner extended this result to smooth compact Lie group actions. Guillemin and Sternberg proved that any semisimple Lie algebra action by analytic transformations can be analytically linearized, and gave a counter-example in the smooth case. We will show that any compact Lie algebra action by smooth transformations can be smoothly linearized, and we apply this result to prove the Levi decomposition of Poisson brackets. This is joint work with Philippe Monnier.

Michèle Audin, IRMA Strasbourg
A redução simpléctica e o problema do satélite

Consideramos um satélite sobre uma órbita circular à volta da Terra. O problema é de saber se o sistema é integrável (se o movimento for bastante regular, quase-periódico). É possível mostrar que não é utilizando o critério galoisiano de Morales e Ramis, apesar de os grupos de Galois serem grandes demais.

Stefan Mueller-Stach, University of Essen
Motivic Geometry

We give a rather elementary definition of motivic cohomology and present in the sequel several new structural results. Motivic Cohomology is related to algebraic K-theory in the same way as singular cohomology is related to topological K-theory. New methods lead to strong interactions with integrable systems and number theory.

Grigory Mikhalkin, University of Utah
Toric surfaces, Gromov-Witten invariants and tropical algebraic geometry

The talk presents a new formula for the Gromov-Witten invariants of arbitrary genus in the projective plane as well as for the related enumerative invariants in other toric surfaces. The answer is given in terms of certain lattice paths in the relevant Newton polygon. The length of the paths turns out to be responsible for the genus of the holomorphic curves in the count. The formula is obtained by working in terms of the so-called tropical algebraic geometry. This version of algebraic geometry is simpler than its classical counterpart in many aspects. In particular, complex algebraic varieties themselves become piecewise-linear objects in the real space. The transition from the classical geometry is provided by consideration of the "large complex limit" (which is also known as "dequantization" or "patchworking" in some other areas of Mathematics).

Denis Auroux, Ecole Polytechnique
Branched coverings, Luttinger surgery and non-isotopy

In contrast with the spectacular progress made on the symplectic isotopy problem for smooth curves in the complex projective plane, various instances of non-isotopy phenomena for symplectic curves in complex surfaces have been discovered over the recent years. Using branched coverings as our main tool, we show that these examples can be reinterpreted in terms of surgery operations along Lagrangian tori in symplectic 4-manifolds. Various examples will be discussed, relating the study of symplectic curves in complex surfaces to that of symplectic 4-manifolds and their topological invariants.