I'll explain a Poisson-geometric proof of the Kashiwara-Vergne conjecture for quadratic Lie algebras, based on the equivariant Moser trick. This talk is based on a joint work with E. Meinrenken, preprint math.RT/0209346.

I will discuss the relationship between two notions of equivalence in Poisson geometry: one is gauge equivalence, that appears as the Poisson counterpart of Morita equivalence of star-product algebras via Kontsevich's formality correspondence; the other is Xu's Morita equivalence for integrable Poisson manifolds, that is a refinement of Weinstein's notion of dual pairs. As an application, I will show how to construct complete invariants of gauge and Morita equivalence for topologically stable Poisson structures on compact oriented surfaces.

Let M be a compact symplectic manifold with a Hamiltonian T action and moment map F. For H a subtorus of T, denote by M^H the fixed point set of the H action on T. The images of F(M) and F(M^H) for all one-dimensional subtori of T form a polytope carved into chambers. It is not at all trivial to count the number of these chambers. I will present an invariant which distinguishes the chambers in the case of SU(n) coadjoint orbits. The general story is still unknown. This is joint work with T. Holm.

The Gel'fand-Cetlin canonical basis for a finite-dimensional representation V_l of U(n,C) can be constructed by successive decompositions of the representation by a chain of subgroups

For G=U(n,H), the compact symplectic group (also described as the quaternionic unitary group), however, the decompositions are not multiplicity-free. However, in recent years, Molev et al. have found a Gel'fand-Cetlin type basis for representations of the symplectic group, using essentially new ideas, including the Yangian Y(2), an infinite-dimensional quantum group, and a subalgebra called the twisted Yangian Y^-(2). In this talk I will explain the symplectic and Poisson geometry underlying the canonical basis for finite-dimensional irreducible representations of U(n,H). In particular, I will construct an integrable system on the symplectic reductions of the coadjoint orbits of U(n,H) by U(n-1,H) and explain its correspondence with Molev et al.'s work.

Let M be a compact, connected symplectic manifold with a Hamiltonian action of a compact n-dimensional torus Tⁿ. Suppose that M is equipped with an anti-symplectic involution s compatible with the T-action. The real locus of M is the fixed point set M^s of s. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. We extend another classical result to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. We will mention several salient examples. This is joint work with Rebecca Goldin (George Mason University).

At the level of loop algebras, there is an equivalence between the integrable systems defined using r-matrices and the Hitchin systems for rigid holomorphic bundles over a Riemann surface. The latter generalise to cases when the bundle is not rigid. On the level of loop groups, however, there does not seem to be a suitable generalisation. An exception is provided by the dynamical r-matrices of Felder, Etingof, Varchenko et al., which turn out to correspond to G-bundles over an elliptic curve. We develop the geometry of this moduli space, which allows a fairly exhaustive elucidation of the Poisson geometry of the dynamical r-matrix. (joint with E. Markman)

Compact symplectic 4-manifolds which admit Hamiltonian torus actions must be blowups of CP² or of S² ×S². We show that the converse is false: we prove, using holomorphic machinery, that certain blow-ups of CP² do not admit torus actions or even circle actions.

This is joint work with Liat Kessler.

In this talk I will describe joint work with Kai Cieliebak and Viktor Ginzburg in which we use methods from symplectic topology to strengthen previous existence results for periodic orbits of Hamiltonian flows. More precisely, we show that for sufficiently small neighborhoods of compact symplectic submanifolds the symplectic homology of Floer and Hofer is nontrivial. This implies the existence of periodic orbits on a dense set of level sets near symplectic extrema.

According to the famous theorem of A. Weil the product of so-called Weil's symbols {f,g} over all the points of an algebraic curve G is equal to 1. Here f, g are non-zero meromorhpic functions on G. It turns out that one can obtain a very simple proof of this theorem just by looking at the Newton polygon of the equation of the image of the curve G under the meromorphic map f,g: G® (C^*)². Parshin generalized Weil's theorem to the multidimensional case and defined so-called Parshin's symbols of (n+1) meromorphic functions on a n-dimensional variety. Pashin's construction is pure algebraic. I will present a new topological explanation of the Parshin theory and a multidimensional generalization of the classical Vieta's formula for the product of all the roots of a polynomial.

We construct a flat degeneration of the flag manifold to the toric variety Y associated to the Gel\'fand-Cetlin polytope. Every Schubert variety X_w degenerates to a reduced union of toric subvarieties of Y, generalizing results of Gonciulea and Lakshmibai. The faces of the Gel\'fand-Cetlin polytope corresponding to the components of the degeneration of X_w are given by rc-graphs. We also explain how this degeneration is related to a construction of cycles representing equivariant Schubert classes in the flag manifold. This construction uses Gel\'fand-Cetlin action coordinates and the cycles are glued from pieces indexes by rc-graphs. This is joint work with Ezra Miller.

(joint work with Martin Pinsonnault)

We compute the rational homotopy type of the space of symplectic embeddings of the standard ball B⁴(c) Ì R⁴ into 4-dimensional rational symplectic manifolds of the form M_l = (S² ×S²,(1+l)w₀ Åw₀) where w₀ is the area form on the sphere with total area 1 and l belongs to the interval [0,1]. We show that, when l is zero, this space retracts to the space of symplectic frames, for any value of c. However, for any given l > 0, the rational homotopy type of that space changes as c crosses the critical parameter c_crit=l, which is the difference of areas between the two S² factors. We prove moreover that the full homotopy type of that space change only at that value, i.e. the restriction map between these spaces is a homotopy equivalence as long as these values of c remain either below or above that critical value. The same methods apply as well to other rational 4-manifolds like CP² or the topologically non-trivial S²-fibration over S².

This is work in progress. We discuss the definition and construction of contact fiber bundles. Applications include the contact versions of minimal coupling and of the cross-section theorem. In turn these are used to classify contact 5-manifolds with SU(2) invariant contact structures and to construct new examples of K-contact manifolds.

The main goal of the talk will be to present recent results of mine analogous to a theorem of B. Kim which describes the quantum cohomology ring of the generalized flag manifold G/B in terms of the integrals of motion of a certain completely integrable Hamiltonian system of Toda lattice type.

It is well-known that an orbifold M, all of whose stabilizer group actions is effective (an ``effective'' or ``reduced'' orbifold) can be presented as M = P/K, where P is a smooth manifold and K is a compact Lie group. For noneffective orbifolds, the corresponding result is unknown. We use the language of groupoids to understand the extra structure arising from the ineffective parts of the stabilizer groups, and show that a presentation exists in two cases. We will also discuss the difficulties with showing presentability in the general case. This work is joint with Andre Henriques.

In this talk I will show how the Tolman-Weitsman theorem enables us to compute the cohomology ring of symplectic quotients at the zero level set of the moment map when a circle acts on the symplectic manifold. This is done when the original manifold is a product of two-dimentional spheres or more generally when it is a product of manifolds such that the cohomology ring of each of them is generated by a degree two class.

We consider a proper action of the symmetry group G of a Poisson manifold P. The orbit space S=P/G is a differential space locally diffeomorphic to a subset of the Cartesian space. The ring of smooth functions on S is a Poisson algebra isomorphic to the algebra of smooth G-invariant functions on P.

We describe the structure of S directly in terms of derivations of the Poisson algebra of S. Orbits of the family of derivations that generate local one-parameter groups of local diffeomorphisms of S give rise to a stratification of S by Poisson manifolds. Orbits of the family of inner derivations define a singular symplectic foliation of S.

We extend our analysis of singular reduction of symmetries to subcartesian Poisson spaces.

We give an Euler MacLaurin formula with remainder for the sum of the values of a smooth function over the lattice points in a simple integral polytope.

(joint work with Yael Karshon and Shlomo Sternberg)

Standard geometric quantization procedure requires choosing a real or complex polarization and the quantum physics is independent of the choice if there exists a projectively flat connection on the vector bundle of Hilbert spaces over the space of polarizations. In this talk, I will discuss a family of quantizations determined by lattices in a symplectic vector space and study the problem of projective flatness.

We will discuss S¹-gerbes over a Lie groupoid (or more precisely, a stack), and describe their Dixmier-Douady class in terms of connection-like data. As an example, we present, for a compact simple Lie group G, an infite dimensional modle of S¹-gerbe whose Dixmier-Douady class corresponds to the canonical generator of the equvariant cohomology H_G³(G).

(joint work with Kai Behrend and Bin Zhang.)

The combinatorial construction of generators for the equivariant cohomology of GKM Hamiltonian spaces is essentially a multivariable interpolation problem on the moment polytope, and the result is given, in general, by a sum over a subset of paths. In the particular case of flag varieties, the same generators can also be computed using divided differences, and the result is then given as a sum over a subset of subwords of a reduced word. I will explain how these two results are related.