Poisson Geometry and Mathematical Physics
Org: Eckhard Meinrenken (Toronto)
- TOM BAIRD, University of Toronto, Toronto, Ontario, M5S 2E4, Canada
Moduli space of flat SU(2) bundles over nonorientable
I will present a computation of the cohomology groups of the moduli
space of flat SU(2) bundles over a closed nonorientable surface
S, which we identify via the holonomy map with X/SU(2),
where X : = Hom( p1 (S), SU(2) ). The
strategy will be to determine the equivariant cohomology ring
H*SU(2)(X), and then pass to H* ( X/SU(2) )
via a pair of long exact sequences and localization.
- OLEG BOGOYAVLENSKI, Queen's University, Kingston, Ontario, K7L 3N6
Invariant foliations with dynamical systems for the Poisson
brackets of hydrodynamic type
An invariant foliation Fm with an induced non-degenerate
metric áv,w ñ of constant curvature K is discovered
for any degenerate Poisson bracket of hydrodynamic type on a manifold
Mn with (2,0)-tensor gij(u) of rank m < n. An invariant
dynamical system V on Mn is introduced that is tangent to the
leaves of the foliation Fm. The dynamical system V is
applied for constructing the scalar and tensor invariants of the
Poisson bracket. Invariant (n-m)-dimensional nilpotent Lie algebras
Au are found that are embedded into the cotangent spaces
O. I. Bogoyavlenskij,
Schouten tensor and bi-Hamiltonian systems of hydrodynamic
J. Math. Phys. 47, 2006.
Invariant foliations for the Poisson brackets of hydrodynamic
Phys. Letters A (2006).
- HENRIQUE BURSZTYN, IMPA
Quasi-Poisson geometry and Dirac structures
In this talk, I will explain how to define hamiltonian spaces with
D/G-valued moment maps (where (D,G) is a group pair integrating a
Manin pair) in terms of Dirac structures, and prove that this approach
is equivalent to the original one of Alekseev and Kosmann-Schwarzbach
based on quasi-Poisson geometry. I will explain how the two viewpoints
complement one another and how they shed light on the theory of
G-valued moment maps, in the sense of Alekseev, Malkin and
- SAM EVENS, University of Notre Dame, Notre Dame, IN 46556, USA
Poisson geometry of the Grothendieck resolution
We construct a Poisson structure on the Grothendieck resolution X of
a complex semisimple group G. The natural map m: X ® G
is Poisson with respect to a Poisson structure p on G such that
closures of conjugacy classes are Poisson subvarieties. pG was
first constructed by Alekseev and Malkin. We determine symplectic
leaves on the Grothendieck resolution, and show that m resolves
singularities of the Poisson structure p on G.
This talk is based on joint work with Jiang-Hua Lu.
- SHAY FUCHS, University of Toronto, Canada
Additivity of spin-c Quantization Under Cutting
We describe a cutting construction for a compact oriented Riemannian
manifold M, endowed with an S1-equivariant spinc structure.
This produces two other equivariant spinc manifolds (the "cut
spaces"), denoted by Mcut+ and Mcut-.
The spinc structures on M, Mcut+ and Mcut-
(together with a connection on their determinant line bundles) enable
us to define virtual representations of S1, called the
"spin-c quantization" of the manifold.
We claim that the representation that corresponds to M is the sum of
the representations that correspond to those of the cut spaces, and we
outline the main steps in the proof.
- MARCO GUALTIERI, MIT
Holomorphic Poisson D-branes
I will define the notion of D-brane on a holomorphic Poisson manifold
and give some methods as well as consequences of their construction.
- MEGUMI HARADA, McMaster University
Orbifold cohomology of hypertoric varieties
Hypertoric varieties are hyperkähler analogues of toric varieties,
and are constructed as abelian hyperkähler quotients of a
quaternionic affine space. Just as symplectic toric orbifolds are
determined by labelled polytopes, orbifold hypertoric varieties are
intimately related to the combinatorics of hyperplane arrangements.
By developing hyperkähler analogues of symplectic techniques
developed by Goldin, Holm, and Knutson, we give an explicit
combinatorial description of the Chen-Ruan orbifold cohomology of an
orbifold hypertoric variety in terms of the combinatorial data of a
rational cooriented weighted hyperplane arrangement. Time permitting,
we detail several explicit examples, including some computations of
orbifold Betti numbers (and Euler characteristics).
This is joint work with R. Goldin.
- TARA HOLM, Cornell University, Department of Mathematics, Ithaca, NY
Integral cohomology of symplectic quotients
I will describe some work in progress on computing the integral
cohomology of symplectic reductions. Under a hypothesis on the
isotropy groups, we may prove that the Kirwan map from equivariant
cohomology of the total space to the ordinary cohomology of the
reduced space, both with integer coefficients, is a surjection. We
will apply this result to compute the integral cohomology of certain
This talk is based on joint work with Susan Tolman.
- GREG LANDWEBER, University of Oregon, Eugene, OR 97403, USA
Equivariant formality in K-theory
This talk will introduce the notion of equivariant formality in
K-theory. For Borel equivariant cohomology theories, equivariant
formality is the statement that the Leray-Serre sequence for the
fibration M ® MG ® BG collapses at the E2 stage, giving an
isomorphism HG(M) @ H(M) ÄHG(pt) as modules over
HG(pt). In the equivariant bundle construction of K-theory, we
do not have such a fibration, so we introduce a different definition,
that K(M) @ KG(M) ÄR(G) Z, tensoring down
rather than tensoring up.
We will prove that compact Hamiltonian G-spaces are always
equivariantly formal in K-theory, using as our main tool the Kunneth
spectral sequence, and showing that the higher R(G)-torsion in
KG(M) vanishes. It follows that the forgetful map KG(M) ®K(M) is surjective, and as a corollary, we will show that every
complex line bundle over M admits a lift of the G-action.
This talk consists of joint work with Megumi Harada.
- EUGENE LERMAN, Illinois, Urbana-Champaign
Is it useful to think of orbifolds as stacks?
I have been told by a number of people that one should think of
orbifolds as Deligne-Mumford stacks. I have also been warned that
"it is difficult to explain what stacks are and it is even more
difficult to explain why it's the right way to think about
orbifolds." I will report on what happened when I tried to come to
grips with orbifolds as stacks.
- YI LIN, University of Toronto, BA 6172, 40 St. George St., Toronto,
Ontario, M5S 2E4
Hamiltonian actions on generalized complex manifolds and the
We first review the definition of Hamiltonian actions on generalized
complex manifolds and present some non-trivial examples. Given a
Hamiltonian action of a compact Lie group on a generalized complex
manifold which satisfies the [`(¶)] ¶-lemma, we show
that there is an equivariant version of the [`(¶)]¶-lemma which is a direct generalization of the equivariant
dG d-lemma in symplectic geometry.
- JOHAN MARTENS, MPI Bonn / University of Toronto
Euler characteristics of GIT quotients
We will discuss work in process determining Euler characteristics over
GIT quotients (symplectic reductions) as residues of expressions in
equivariant K-theory. We will indicate the relationship with the
Jeffrey-Kirwan theorem in cohomology.
- MARKUS PFLAUM, Goethe-University, FB Mathematik, Robert-Mayer-Str. 10,
60054 Frankfurt/Main, Germany
Cyclic cohomology of deformation quantizations over orbifolds
In the talk, the cyclic homology of deformation quantizations of the
convolution algebra over a proper etale groupoid G will be studied.
It is shown that cyclic homology recovers the (additive) structure of
the orbifold cohomology of the orbit space X=G0/G. As a
consequence, the space of traces on the deformed convolution algebra
has dimension equal to the number of sectors of the underlying
orbifold. Using these results, I then elaborate on the application in
the algebraic index theory of orbifolds.
Joint with N. Neumaier, H. Posthuma and X. Tang.
- MARTIN PINSONNAULT, Fields Institute & University of Toronto
Maximal Tori in the Hamiltonian groups of 4-manifolds
Let (M,w) be a symplectic 4-manifold. Let Symp be
its group of symplectomorphisms and denote by Ham its
subgroup of Hamiltonian diffeomorphisms. Let M be the set
of maximal tori in Ham and let T be the subset
of 2-dimensional tori. Both Symp and Ham act
by conjugation on M and T. We will explain
why the quotient space M / Symp is finite, and
describe what the finiteness of T / Ham would
imply for the homotopy type of Symp.
- ROMARIC PUJOL, University of Toronto
Poisson Groupoids, Duality and Classical Dynamical
I will briefly recall basic notions about Poisson groupoids and their
duality. As examples of these, I will present the class of
bidynamical Poisson groupoids, and show the relations with the
classical dynamical Yang-Baxter equation (CDYBE) and Lie
quasi-bialgebras. From this geometric point of view, we are able to
give explicit solutions of the CDYBE.
- REYER SJAMAAR, Cornell University, Ithaca, New York
Torsion and abelianization in equivariant cohomology
Let X be a topological space upon which a compact connected Lie
group G acts. It is well known that the equivariant cohomology
HG* (X,Q) is isomorphic to the subalgebra of Weyl group invariants
of the equivariant cohomology HT* (X,Q), where T is a maximal
torus of G. We establish a similar relationship for coefficient
rings more general than Q.
Our results rely on work of Grothendieck and Demazure concerning the
intersection theory of flag varieties and have applications to the
cohomology of homogeneous spaces and, potentially, symplectic
This is a report on joint work with Tara Holm.
- XIANG TANG, Washington University, St. Louis, MO, 63130, USA
Algebraic structures on Hochschild cohomology of an orbifold
In this talk, we study algebraic structures on the Hochschild
cohomology of the convolution algebra over a proper etale
We show that the Gerstenhaber bracket defines a twisted
Schouten-Nijenhuis bracket between multivector fields on the
corresponding inertia orbifold [^(X)] of X = G0/G. This leads
an interesting connection to symplectic reflection algebra.
We will define a de Rham model for the Chen-Ruan orbifold cohomology,
and explain its relations to the ring structure on the Hochschild
Joint work with G. Halbout, N. Neumaier, M. Pflaum, H. Posthuma and
- AISSA WADE, Penn State
Poisson fiber bundles
Poisson fiber bundles are natural generalizations of symplectic fiber
I will discuss an approach to Poisson fiber bundles which is based on
the theory of Dirac structures on manifolds. I will review various
constructions of coupling Dirac structures on manifolds.
- JONATHAN WEITSMAN, Santa Cruz
- GRAEME WILKIN, Johns Hopkins University
The Lojasiewicz inequality for Higgs bundles
An important technique in the study of symplectic (and hyperkähler)
quotients is to use the Morse theory of the norm-square of the moment
map. The Lojasiewicz inequality is a key estimate in the process of
proving that the gradient flow of a functional converges, which is
essential in order for Morse theory to work. This technique was first
used for certain infinite-dimensional problems by Leon Simon, and then
extended by Johan Rade to study the gradient flow of the Yang-Mills
functional in two and three dimensions. Here we extend Rade's version
of the Lojasiewicz inequality to functionals which are invariant under
a group action and also satisfy a certain ellipticity condition on the
Hessian. In particular, this can be applied to the norm square of the
hyperkähler moment map for Higgs bundles over a compact Riemann
- CHRISTOPHER WOODWARD, Rutgers New Brunswick
Gauged pseudoholomorphic maps on cylindrical end surfaces
Salamon, Mundet, and others introduced the notion of "vortex
equations" which simultaneously generalize Gromov's pseudoholomorphic
curves and the notion of flat connection on a surface. We study the
moduli space of solutions to the vortex equations on curves with
cylindrical end, and show how they fit into the framework of loop
group actions/group-valued moment maps developed by Alekseev,
Meinrenken, and the second author. This leads to invariants of a
Hamiltonian G-manifold taking values in the certain spaces of
invariant distributions on a group, which is analogous to the orbifold
Gromov-Witten invariants of Chen-Ruan.
This is joint with Eduardo Gonzalez.