


Symplectic Geometry / Géométrie symplectique (Org: Lisa Jeffrey and/et Eckard Meinrenken)
 ANTON ALEKSEEV, Section of Mathematics, University of Geneva, Geneva, Switzerland
Poisson Geometry and the KashiwaraVergne conjecture
[PDF] 
I'll explain a Poissongeometric proof of the KashiwaraVergne
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.
 HENRIQUE BURSZTYN, University of Toronto, Toronto, Ontario
Notions of equivalence for Poisson manifolds
[PDF] 
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 starproduct 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.
 REBECCA GOLDIN, George Mason University, Fairfax, Virginia 22039, USA
Counting chambers of the moment polytope
[PDF] 
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 onedimensional 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.
 MEGUMI HARADA, University of CaliforniaBerkeley, California, USA
The symplectic geometry of the Gel'fandCetlin basis
for representations of the symplectic group
[PDF] 
The Gel'fandCetlin canonical basis for a finitedimensional
representation V_{l} of U(n,C) can be constructed by
successive decompositions of the representation by a chain of subgroups
U(1,C) Ì U(2,C) Ì ¼U(n1,C) Ì U(n,C). 

A key point in the analysis is that the decomposition of an irreducible
representation of U(k,C) under the subgroup
U(k1,C) is multiplicityfree. Guillemin and Sternberg
constructed in the 1980s the Gel'fandCetlin integrable system on the
coadjoint orbits O_{l} of U(n,C), which is
the symplectic geometric version, via geometric quantization, of the
Gel'fandCetlin canonical basis for V_{l}.
For G=U(n,H), the compact symplectic group (also described
as the quaternionic unitary group), however, the decompositions
are not multiplicityfree. However, in recent years, Molev et al. have
found a Gel'fandCetlin type basis for representations of the
symplectic group, using essentially new ideas, including the Yangian
Y(2), an infinitedimensional 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
finitedimensional 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(n1,H) and explain its correspondence with Molev et
al.'s work.
 TARA HOLM, University of California, Berkeley, California 94720, USA
Symplectic quotients and real loci
[PDF] 
Let M be a compact, connected symplectic manifold with a Hamiltonian
action of a compact ndimensional torus T^{n}. Suppose that M is
equipped with an antisymplectic involution s compatible with
the Taction. 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).
 JACQUES HURTUBISE, Centre de Recherches Mathématiques
Dynamical rmatrices and bundles on elliptic curves
[PDF] 
At the level of loop algebras, there is an equivalence between the
integrable systems defined using rmatrices 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 rmatrices of Felder,
Etingof, Varchenko et al., which turn out to correspond to
Gbundles 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 rmatrix. (joint with E. Markman)
 YAEL KARSHON, University of Toronto and The Hebrew University of Jerusalem
Blowups of CP^{2} without torus actions
[PDF] 
Compact symplectic 4manifolds which admit Hamiltonian torus actions
must be blowups of CP^{2} or of S^{2} ×S^{2}. We
show that the converse is false: we prove, using holomorphic
machinery, that certain blowups of CP^{2} do not
admit torus actions or even circle actions.
This is joint work with Liat Kessler.
 ELY KERMAN, SUNYStony Brook, Stony Brook, New York, USA
Symplectic homology and periodic orbits near symplectic extrema
[PDF] 
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.
 ASKOLD KHOVANSKII, Toronto
Newton polyhedra and Parshin's symbols
[PDF] 
According to the famous theorem of A. Weil the product of socalled
Weil's symbols {f,g} over all the points of an algebraic curve
G is equal to 1. Here f, g are nonzero 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 socalled Parshin's
symbols of (n+1) meromorphic functions on a ndimensional 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.
 MISHA KOGAN, Northeastern University, Boston, Massachusetts 02115, USA
Degenerating Schubert varieties to unions of toric varieties
associated to rcgraphs
[PDF] 
We construct a flat degeneration of the flag manifold to the toric
variety Y associated to the Gel\'fandCetlin 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\'fandCetlin polytope corresponding to the
components of the degeneration of X_{w} are given by rcgraphs. 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\'fandCetlin action coordinates and the
cycles are glued from pieces indexes by rcgraphs. This is joint
work with Ezra Miller.
 FRANÇOIS LALONDE, QuébecMontréal
Critical values for the moduli space of symplectic balls
in a rational 4manifold
[PDF] 
(joint work with Martin Pinsonnault)
We compute the rational homotopy type of the space of symplectic
embeddings of the standard ball B^{4}(c) Ì R^{4} into
4dimensional rational symplectic manifolds of the form M_{l} = (S^{2} ×S^{2},(1+l)w_{0} Åw_{0}) where
w_{0} 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 S^{2} 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 4manifolds
like CP^{2} or the topologically nontrivial S^{2}fibration over
S^{2}.
 EUGENE LERMAN, University of IllinoisUrbanaChampaign, USA
Contact fiber bundles
[PDF] 
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 crosssection theorem. In turn these are
used to classify contact 5manifolds with SU(2) invariant contact
structures and to construct new examples of Kcontact manifolds.
 LIVIU MARE, University of Toronto, Toronto, Ontario M5S 3G3
Quantum cohomology of flag manifolds and Toda lattices
[PDF] 
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.
 DAVID METZLER, Florida
Presentation of noneffective orbifolds
[PDF] 
It is wellknown 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.
 RAMIN MOHAMMADALIKHANI, McGill University, CRM
Cohomology ring of symplectic reductions by circle actions
[PDF] 
In this talk I will show how the TolmanWeitsman 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 twodimentional
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.
 JEDRZEJ SNIATYCKI, University of Calgary, Calgary, Alberta T2N 1N4
Singular reduction of Poisson spaces
[PDF] 
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 Ginvariant 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 oneparameter 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.
 JONATHAN WEITSMAN, University of California, Santa Cruz, California 95064, USA
Euler MacLaurin formulas for simple polytopes
[PDF] 
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)
 SIYE WU, Department of Mathematics, University of Colorado, Boulder
Colorado 803090396, USA
Lattice polarization and projective flatness
[PDF] 
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.
 PING XU, Pennsylvania State, USA
Equivariant gerbes over compact simple Lie groups
[PDF] 
We will discuss S^{1}gerbes over a Lie groupoid (or more precisely, a
stack), and describe their DixmierDouady class in terms of
connectionlike data. As an example, we present, for a compact simple
Lie group G, an infite dimensional modle of S^{1}gerbe whose
DixmierDouady class corresponds to the canonical generator of the
equvariant cohomology H_{G}^{3}(G).
(joint work with Kai Behrend and Bin Zhang.)
 CATALIN ZARA, Department of Mathematics, Yale University, New Haven,
Connecticut, USA
Morse interpolation and divided differences
[PDF] 
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.

