O microsuporte do feixe das soluções holomorfas de um $\cal D$-Módulo coerente coincide com a sua variedade característica, como foi provado por Kashiwara e Schapira no fim dos anos oitenta. Neste seminário tratarei outros feixes de soluções microlocais, enquadrando, por exemplo, a condição clássica de hiperbolicidade fraca.

Provamos que se $F:M^{2n}\to N^{2n}$ é uma imersão minimal duma variedade compacta de dimensão real $2n$ e $N$ é uma variedade Kahler-Einstein de dimensão complexa $2n$ com primeira classe de Chern não positiva, então, sob certas condições na curvatura de $M$, ângulos de Kähler iguais ou pluriminimalidade de $F$, e não existência de direcções complexas, conclui-se que $F$ é Lagrangiana. O caso $n=2$ é aquele em que melhores conclusões se podem obter.

Nós desenvolvemos uma técnica baseada em métodos da teoria de representações dos grupos de Lie compactos e nos espaços osculadores de suas órbitas que nos permite estudar diversas classes de variedades geometricamente interessantes, com algumas aplicações importantes incluindo a recíproca de um teorema de Bott e Samelson e novos resultados sobre variedades "taut" bem como provas mais simples de outros resultados de classificação (trabalho em conjunto com G. Thorbergsson).

The general setup for a categorical construction of state-sum invariants of PL $4$-manifolds given by the speaker contains all known categorical/combinatorial constructions as especial cases and has the potential of being more general. In order to find more examples of this construction, one would like to understand whether there is a link with the famous smooth invariants of $4$-manifolds defined by Donaldson and by Seiberg and Witten. For this reason one first ought to understand the differential geometry behind the categorical construction. As a first tiny step in this direction Roger Picken and the speaker studied the holonomy of gerbes with connections and proved some interesting structural theorems about such holonomies. This talk will first sketch the results about the contruction above in order to make clear the motivation for the interest in gerbes with connections, and then will explain the results about the aforementioned holonomies.

A theorem of Delzant states that any symplectic manifold $(M,\omega)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\mathbb{T}^n = \mathbb{R}^{n}/2pi\mathbb{Z}^n$, is a smooth projective toric variety completely determined (as a Hamiltonian $\mathbb{T}^n$-space) by the image of the moment map $\phi:M\to\mathbb{R}^n$, a convex polytope $P=\phi(M)\subset\mathbb{R}^n$. In this talk I will show, using symplectic (action-angle) coordinates on $P\times \mathbb{T}^n$, how all $om$-compatible toric complex structures on $M$ can be effectively parametrized by smooth functions on $P$. I will also discuss some topics suited for application of this symplectic coordinates approach to Kähler toric geometry, namely: explicit construction of extremal Kähler metrics, spectral properties of toric manifolds and combinatorics of polytopes.

One tool for a local study of singular analytic spaces is to consider collections of sections of these spaces with linear spaces of various codimension. In the case of surfaces, the study of hyperplane sections at singular points leads to the study of limits of tangent hyperplanes. We give various characterisations of these hyperplanes and of their sections by means of equisingularity of curves.

Lie algebroids are generalizations of classical Lie algebras, and of Lie algebras of vector fields. They appear in various branches of mathematics (foliations, Poisson geometry, equivariant geometry, non-commutative geometry etc.), providing a stimulating interplay. In this talk I will discuss certain cohomological aspects of algebroids, as well as their relevance in some particular cases.

Designers of secure software systems need to monitor and quantify event clustering in order to minimize information leakage to probes by an attacker, perhaps through introduction of obscuring procedures in a restricted memory device such as a smartcard. An ideal situation would be to have scheduling that to an attacker resembles closely a random sequence of events. The basic “random” model for stochastic events is the Poisson process; for events on a line this results in an exponential distribution of intervals between events. Here we discuss the differential geometry of manifolds of gamma distributions, which contain exponential distributions as a special case; this gives a means of quantifying departures from randomness.

A regular Poisson manifold can be described as a foliated space supporting a tangentially symplectic form. Examples of foliations will be presented in the talk that are not induced by any Poisson structure although the basic obstructions vanish. On the other hand, tangentially symplectic structures are solutions of a certain partial differential relation that can be shown to satisfy the $h$-principle when some natural condition on the underlying foliated space is fulfilled. The proof of this statement can be sketched if time permits.

The interaction between geometry in the adjacent dimensions $2$, $3$ and $4$ is a theme which runs through a great deal of the work of mathematicians over the past few years. Investigations are following three intertwining threads: DG, SG and AG - Differential Geometry, Symplectic Geometry and Algebraic Geometry. The picture we will find pulls together various ideas which have been touched on in the mathematical literature but striking constructions which emerge do not seem to be well-known. We will try to fill this gap.