




Symplectic Geometry / Géométrie symplectique (Lisa Jeffrey, Organizer)
 MAXIM BRAVERMAN, Northeastern University, Boston, Massachusetts 02466, USA
Quantization of noncompact Hamiltoninan Gspaces

In the first part of the talk I will present a new regularized version
of an index of a Dirac operator on a complete Riemannian manifold
endowed with an action of a compact Lie group G. I show that this
index is an invariant of a noncompact cobordism of the type
considered by V. Guillemin, V. Ginzburg and Y. Karshon. As a corollary
I obtain new proofs of the AtiyahSegalSinger equivariant index
theorem and of the GuilleminSternberg ``quantization commutes with
reduction'' conjecture.
In the second part of the talk I'll apply the above theorems to the
problem of geometric quantization of noncompact manifold. In
particular, I will construct a quantization of a noncompact symplectic
Gmanifold. This quantization is a virtual representation of G, in
which each irreducible representation appears with a finite
multiplicity.
 ANA CANNAS DA SILVA, Instituto Superior Técnico, Lisbon,
Av. Rovisco Pais, 1049001 Lisboa, Portugal
Index of contact diffeomorphisms and folded
symplectic structures

A folded symplectic structure is a closed 2form which is
nondegenerate except on a hypersurface, called the folding
hypersurface, and whose restriction to that hypersurface has maximal
rank. The folded tangent bundle is the Lie algebroid defined as the
free module over smooth functions whose sections are the vector fields
which are tangent to the folding hypersurface.
The mapping torus of a contact diffeomorphism admits a folded
symplectic structure. A contact diffeomorphism has an index defined
using the folded structure on its mapping torus. Epstein and Melrose
proved a formula for this index in the case of a cotangent sphere
bundle of a compact manifold, which goes back to a question of Atiyah
and Weinstein. This talk will report on the geometry of this index in
terms of the folded tangent bundle.
 VIKTOR GINZBURG, University of California at Santa Cruz, Department of Mathematics,
Santa Cruz, California 95064, USA
Existence of periodic orbits for some classes of Hamiltonian systems

We discuss the existence question for periodic orbits of Hamiltonian
systems describing the motion of a charge on a Riemannian manifold and
also the systems arising in semilocal generalizations of
WeinsteinMoser theorem. These classes of systems are as natural and
fundamental as conservative mechanical systems. However, known
symplectic topology methods developed to find periodic orbits on a
fixed energy level usually do not directly apply to the systems in
question because the energy level may fail to have contact type. We
state some new existence theorems, including results recently obtained
by Ely Kerman, for periodic orbits of such Hamiltonian systems and try
to explain the basic reason why periodic orbits exist for many systems
of this type.
 REBECCA GOLDIN, Maryland
A generalization of a theorem of GoreskyKottwitzMacPherson I

Let M be a compact, connected symplectic manifold with a Hamiltonian
action of a torus T^{n}. Define the oneskeleton of M to be the
union, over all codimension 1 subtori S Ì T^{n}, of the
fixedpoint sets M^{S}. Tolman and Weitsman proved that the
equivariant cohomology of M is computable from the oneskeleton
alone. Goresky, Kottwitz, and MacPherson showed that, if the
oneskeleton of M has (real) dimension at most 2, then the
equivariant cohomology can be computed from a combinatorial model of
the oneskeleton. We generalize this GKM theorem to obtain a
combinatorial computation of the equivariant cohomology of M when the
oneskeleton has higher dimension.
 VICTOR GUILLEMIN, Department of Mathematics, MIT, Cambridge, Massachusetts 02139, USA
Morse theory on graphs, part I

Let G be a finite dvalent graph and G an ndimensional
torus. An ``action'' of G on G is defined by a map which
assigns to each oriented edge, e, of G, a onedimensional
representation of G (or, alternatively, a weight, a_{e}, in the
weight lattice of G. For the assignment, e ® a_{e}, to be a
schematic description of a ``Gaction'', these weights have to
satisfy certain compatibility conditions: the GKM axioms). To this
action we attach an equivariant cohomology ring,
H_{G}(G) = H(G,a). By definition this ring contains the
equivariant cohomology ring of a point, S(g^{*}) = H_{G}(pt), as a
subring, and in this talk we will use graphical versions of standard
Morse theoretical techniques to analyze the structure of H_{G}(G)
as an S(g^{*})module.
 TARA HOLM, MIT
A generalization of a theorem of GoreskyKottwitzMacPherson II

This talk continues the material covered in `A generalization of
a theorem of GKM I'.
 JACQUES HURTUBISE, McGill University, Montreal, Quebec H3A 2K6
Integrable systems, loop groups and surfaces

Many of the most commonly studied integrable systems of mathematical
physics are best understood in terms of loop algebras and loop groups,
with appropriate Lie Poisson structures coming into play. It turns out
that these admit separations of variables in which various ruled
surfaces admitting symplectic forms come into play.
 LISA JEFFREY, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
Intersection pairings on singular quotients

This talk describes joint work with Y.H. Kiem, F. Kirwan and J. Woolf,
in which we extend earlier results on intersection pairints on smooth
symplectic quotients to the case of singular symplectic quotients.
 YOUNGHOON KIEM, Yale University
Intersection homology of symplectic torus quotients

Let M be a Hamiltonian Tspace with proper moment map m. In
general, the symplectic reduction M\mathrel//T = m^{1}(0)/T is a
compact pseudomanifold. We study the intersection (co)homology of the
reduced space in terms of the equivariant cohomology
H^{*}_{T}(m^{1}(0)) which is far better understood.
 MIKHAIL KOGAN, MIT
Schubert varieties and GelfandCetlin polytop

GelfandCetlin completely integrable system provides a set of action
coordinates on every flag manifold. These coordinates map the flag
manifold onto the GelfandCetlin polytope. We investigate the
relationship between Schubert varieties and faces of the GelfandCetlin
polytope. In particular, for every Schubert variety we construct a
cycle in the same homology class, which is given by the preimage of
some faces of the polytope. These faces can be identified with already
discovered combinatorial objects (called RCgraphs). This approach
provides new geometric interpretations of some formulas for double
Schubert polynomials.
 EUGENE LERMAN, University of Illinois, Urbana, Illinois, USA
Contact cuts

We describe a contact analog of the symplectic cut construction
As an application we show that the group of contactomorphisms for a
particular overtwisted contact structure on the three sphere contains
countably many nonconjugate two tori.
 STEVEN LILLYWHITE, University of Toronto, Toronto, Ontario M5S 3G3
The topology of symplectic reductions of loop spaces

We compute the real homotopy type of a symplectic reduction of the loop
space of the 2sphere by SO(3). This symplectic reduction may be
identified with a moduli space of arclength parametrised closed curves
in Euclidean space. In this case, the Kirwan map turns out to be an
isomorphism. We discuss how to compute the equivariant cohomology of
loop spaces.
 EYAL MARKMAN, University of Massachusetts, Amherst, Massachusetts 01003, USA
Hyperkahler varieties; their reflections and monodromy

Reflections of holomorphic symplectic varieties
are certain symplectic birational (surgery) operations.
They play a central role in the study of K3 surfaces and higher
dimensional holomorphic symplectic varieties.
1) We construct holomorphicsymplectic reflections modeled after
dual pairs of Springer resolutions of nilpotent coadjoint orbits.
2) We derive a PicardLefschetz formula, describing the reflection
on the level of cohomology rings.
3) We apply these results to study the monodromy representation on the
cohomology ring of Hilbert scheme of points on a K3 surface (and
other moduli spaces of vector bundles on a K3).
 REYER SJAMAAR, Department of Mathematics,
Cornell University, Ithaca, New York 148534201, USA
Moment maps and Riemannian symmetric pairs

This is joint work with Luis O'Shea. Our main result is a
``real form'' of Kirwan's convexity theorem. We apply our result to
flag varieties of real semisimple groups and thus obtain eigenvalue
inequalities, which generalize inequalities found by Weyl, Ky Fan,
Kostant, Klyachko, and many others. For instance, we find an answer
to the following question: given the singular values of two
rectangular matrices (of the same size), what are the possible
singular values of their sum?
 JENNIFER SLIMOWITZ, Rice University, Houston, Texas, USA
Shortening the Hofer length of Hamiltonian circle actions

Consider a Hamiltonian circle action on a symplectic manifold (M,w). If the moment map of this action obtains its maximum at an
isolated fixed point at which one of the weights has absolute value
greater than one, then this circle action may be deformed to a loop in
Ham^{c}(M) with shorter Hofer length. Following ideas of Polterovich,
we produce shorter loops and describe an estimate for the amount by
which the original loop may be shortened.
 MARGARET SYMINGTON, TexasAustin
Singular Lagrangian fibrations in dimension four

We study the class of all closed symplectic 4manifolds M that
admit a smooth map to a two dimensional base that, over an open dense
subset U of the base, give the manifold the structure of a Lagrangian
fibration. The preimage of a point in the complement of U may be a
singular fiber of a restricted type. The set of allowable singular
fibers includes lower dimensional tori (as appear in a toric variety)
and nodal fibers (which appear in holomorphic fibrations and a
Lagrangian fibration of the K3 surface).
 SUE TOLMAN, University of Illinois at UrbanaChampaign, Urbana,
Illinois 618012975, USA
Symplectic packing, Gromov width, and Grassmanians

Joint work with Y. Karshon.
As an application of our project to classify complexity one spaces,
we find a symplectic packing of several Grassmanians by two
balls, and hence obtain an lower bound on the Gromov Width.
 LISA TRAYNOR, Bryn Mawr College
Invariants for legendrian tangles

A recent development has been the construction of legendrian knot
invariants from contact homology. This standard contact homology is
defined using holomorphic curves in the symplectization of a contact
manifold. For legendrian tangles, there is an alternative definition
for contact homology coming from the theory of generating functions. I
will compare and contrast these notions of contact homology.
 THOMAS TRELOAR, Department of Mathematics, University of Maryland, College Park,
Maryland 20742, USA
Manin pairs and the symplectic geometry of polygons

We obtain a symplectic structure on the moduli space of closed
ngons with fixed sidelengths in the 3sphere. This structure
arises from a quasihamiltonian reduction on the fusion product of
ncopies of S^{3}, where S^{3} is equipped with a quasipoisson
bivector induced from Manin pair (su_{2} Åsu_{2}, su_{2}). This is
a completely integrable system and the hamiltonian flows can be
calculated. More generally, the symplectic structure on the moduli
space of ngons in any constant curvature 3space can be given as a
reduction associated to a Manin pair (d,g) where g = su_{2}. These
structures relate to the structures obtained from the moduli space of
flat connections on a npunctured 2sphere.
 JONATHAN WEITSMAN, CaliforniaSanta Cruz
To be announced

 CATALIN ZARA, Department of Mathematics, MIT, Cambridge, Massachusetts 02139, USA
Morse theory on graphs, part II

Continuation of the previous talk.

