We give a construction to obtain canonically an isotropic average of given $C^1$-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein's submanifold averaging theorem and apply Moser's trick. We show some applications to the existence of invariant isotropic submanifolds under group actions, to images of isotropic submanifolds under moment maps, and to contact geometry.

I will try to give a global overview of the main mathematical aspects and results of General Relativity. No particular knowledge will be assumed from the audience except basic Riemannian geometry.

We will present the first results from Hamilton on Ricci Flow for three and four dimensional compact manifolds. These results give a classification of such manifolds with positive Ricci curvature (on 3 dimensions) and with positive curvature operator (on 4 dimensions).

There are some well-known polytope decomposition formulas that write the characteristic function of a polytope in terms of the characteristic functions of cones associated to its faces. These formulas provide us with ways of counting lattice points in a polytope. We give weighted versions of these decompositions and show some interactions with toric varieties and Euler Maclaurin formulas for simple lattice polytopes.

I will discuss some results classifying holomorphic vector bundles on certain algebraic surfaces in terms of linear algebra data associated to a quiver, and extensions of this result to some threefolds. The surface results go back to work of Atiyah and others on instantons, extended by Kronheimer and Nakajima. The threefold results are much more recent; they connect nicely to some recent physics work on supersymmetric quiver gauge theory.

There are many examples of surfaces of general type with geometric genus $p_g =0$, but a complete classification is still unknown. In this talk we survey some classical constructions of these surfaces, and describe an approach to obtain new examples.

Geometric structures on topological spaces

• Klein geometries.
• Geometric atlases for Ehresmann structures.
• Fundamental domains and identification spaces.
• Developing the universal covering space.
• Representions of the fundamental group.
• The hierarchy of Klein geometries.
• Pathological developing maps.

Affine structures

• Affine structures on the \(2\)-torus.
• Benzecri's theorems.
• Euler class of a surface group representations.
• The Milnor-Wood inequality.
• Flat bundles and geometric structures.

Deformation spaces: geometries on the space of geometries.

• Markings.
• Definition of the deformation space.
• The Ehresmann-Thurston holonomy theorem.
• Representation spaces.

Poisson geometry on Fricke spaces

• Symplectic geometry.
• Fricke's theorem on rank two free groups.
• Fenchel-Nielsen coordinates and their generalizations.
• Action of the mapping class group.
• Penner-Fock coordinates.

\(RP2\)-structures on surfaces

• Coxeter groups.
• Convex projective structures; Hitchin's conjecture.
• Choi's convex decomposition theorem.
• Regularity of the boundary, bulging deformations.
• Survey of recent progress: higher dimensions and other geometries.

Symmetric differentials are sections of the symmetric powers of the sheaf of differentials. Geometrically they describe multi-foliations, in the same way differential 1-forms (i.e symmetric differential of order 1) describe foliations. Information on the existence and asymptotic growth of symmetric differentials gives important information on the geometry of the projective variety, e.g. hyperbolicity. We discuss the non-existence of symmetric differentials on subvarieties of $ P^n$ of low codimension, the jumping phenomenon and applications toward the hyperbolicity of hypersurfaces of $ P^3$. The jumping phenomenon refers to the number of symmetric differentials along a family of projective varieties. This contrasts with the constancy of the plurigenera (the dimension of the space of sections of powers of the canonical bundle).

Eliashberg and Polterovich introduced a partial order on the universal cover of the identity component of the group of contact diffeomorphisms. We show the triviality of this partial order for some contact manifolds by studying contact squeezing properties of certain domains. Also we will discuss how to apply symplectic field theory to squeezing problems.

We use a partition of the critical set of the norm of the square of a moment map described by Paradan to give new weighted decomposition formulas for simple polytopes. As an application, Euler Maclaurin formulas for simple integral polytopes are given. This is joint work with Leonor Godinho.

I will discuss a reduction procedure for Courant algebroids, generalized complex, and generalized Kähler structures. In the context of generalized complex structures, this procedure interpolates between holomorphic reduction of complex manifolds and symplectic reduction. Our theory is based on extended notions of group action and moment map, which reflect the enhanced symmetry group of a Courant algebroid. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on $CP^2$. This is joint work with G. Cavalcanti and M. Gualtieri.

In this talk I will discuss a generalization of the Fredholm index for operators on groupoids. A (pseudo)differential operator on a Lie groupoid is a family of (pseudo)differential operators on $s$-fibers, satisfying invariance and smoothness conditions. The analytic index of such an operator is defined as a $K$-theory map, using the normal groupoid construction and $K$-theory for $C$*-algebras. For the pair groupoid, one recovers operators on manifolds and the Fredholm index. The main non-trivial examples to keep in mind are Melrose's $b$-groupoid approach to the index theorem on manifolds with boundary, and Connes and Skandalis index theorem for operators on foliated manifolds.