1.
Discrete Torsion and Twisted
Orbifold Cohomology - Ruan
, Yongbin
One of the remarkable insights of orbifold string theory is an indication
of the existence of a new cohomology theory of orbifolds containing the
so-called twisted sectors as the contribution of singularities. Mathematically,
such an orbifold cohomology theory has been constructed by
Chen-Ruan [CR]. The author believes that there is a "stringy" geometry
and topology of orbifolds whose core is orbifold cohomology. One
aspect of this new geometry and topology is the twisted orbifold cohomology
and its relation to discrete torsion. Again, the twisting process
has its roots in physics. Physicists usually work over a global quotient
X = Y/G only, where G is a finite group acting smoothly...
3.
A NEW CONSTRUCTION OF
POISSON MANIFOLDS - Ibort
, A.; Martínez Torres
, D.
A new technique is presented for construction of Poisson
manifolds. This technique is inspired by surgery
ideas used to define Poisson structures on 3-manifolds
and Gompf's surgery construction for symplectic manifolds.
As an application of these ideas it is proved that
for all n ? d ? 4, d even, any finitely presentable group
is the fundamental group of a n-dimensional orientable
closed Poisson manifold of constant rank d. The unimodularity
of some of the Poisson structures thus constructed
is studied.
4.
Distinguishing the Chambers
of the Moment Polytope - Goldin
, R. F.; Holm
, T. S.; Jeffrey
, L. C.
Let M be a compact manifold with a Hamiltonian T
action and moment map ?. The restriction map in rational
equivariant cohomology from M to a level set ?-1(p)
is a surjection, and we denote the kernel by I(p). When T
has isolated fixed points, we show that I(p)
distinguishes
the chambers of the moment polytope for M. In particular,
counting the number of distinct ideals I(p) as (p)
varies over different chambers is equivalent to counting
the number of chambers.
5.
Moment Maps and Equivariant Szegö Kernels - Paoletti
, Roberto
Let M be a connected n-dimensional complex projective manifold and
consider an Hermitian ample holomorphic line bundle (L; hL) on M. Suppose
that the unique compatible covariant derivative ?L on L has curvature
-2?i? where ?
is a Kähler form. Let G be a compact connected Lie group
and ?: G x M ? M a holomorphic Hamiltonian action on (M; ?
). Let \frac g
be the Lie algebra of G, and denote by ? : M ? g* the moment map.
¶
Let us also assume that the action of G on M linearizes to a holomorphic
action on L; given that the action is Hamiltonian, the obstruction...
6.
The Verlinde formula as fixed point formulas - Alekseev
, A.; Meinrenken
, E.; Woodward
, C.
We express the index of the SpinC-Dirac operator on symplectic
quotients of a Hamiltonian loop group manifold in terms of fixed point data. As an application
we prove Verlinde formulas for the SpinC-quantization of moduli spaces of
flat bundles over surfaces.
7.
Minimal annuli with and without slits - Colding
, Thomas H.; Minicozzi
, William P.
In this paper we bound the oscillation of the unit normal of minimal annuli with and
without slits. Our estimates are independent of the ratio of the inner and outer radii.
Hence, we recover standard removable singularity results as the inner radius goes to zero.
The estimate for annuli with slits is important in proving a removable singularities theorem
for minimal limit laminations.
8.
Knots and Contact Geometry I: Torus Knots and the Figure Eight Knot - Etnyre
, John B.; Honda
, Ko
We classify Legendrian torus knots and Legendrian figure eight knots in the
tight contact structure on S3 up to Legendrian isotopy. A a corollary to
this we also obtain the classification of transversal torus knots and transversal figure eight knots
up to transversal isotopy.
9.
Grothendieck Groups of Poisson Vector Bundles - Ginzburg
, V.L.
A new invariant of Poisson manifolds, a Poisson K-ring, is
introduced. Evidence is given that this invariant is more tractable
than such invariants as Poisson (co)homology. A version of this
invariant is also defined for arbitrary Lie algebroids. Basic
properties of the Poisson K-ring areproved and the
Poisson K-rings are calculated for a number of examples.
In particular, for the zero Poisson structure the K-ring
is the ordinary K0-ring of the manifold and
for the dual space to a Lie algebra the K-ring is the
ring of virtual representations of the Lie algebra.
It is also shown that the K-ring is an invariant of
Morita equivalence. Moreover, the K-ring is a...
11.
Geometric Invariants of the Hofer Norm - McDuff
, D.
This note discusses some geometrically defined seminorms on the group Ham
(M,?)
of Hamiltonian diffeomorphisms of a closed symplectic manifold
(M,?),
giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm.
As a consequence we show that if an element in Ham
(M,?)
is sufficiently close to identity in the C2-topology then it may be joined to the
identity by a path whose Hofer length is minimal among all paths, not just among paths in the same homotopy
class relative to endpoints. Thus, true geodesics always exist for the Hofer norm. The main step in
the proof is to show that a "weighted" version of...
12.
New Smooth counterexamples to the Hamiltonian Seifert conjecture - Kerman
, Ely
We construct a new aperiodic symplectic plug and hence new smooth counterexamples to
the Hamiltonian Seifert conjecture in ?2n
for n ? 3. In other words, we describe an alternative procedure, to those of V.L. Ginzburg
[Gi1, Gi2] and M. Herman [Her], for producing smooth Hamiltonian flows, on symplectic manifolds of
dimension at least six, which have compact regular level sets that contain no periodic orbits.
The plug described here is a modification of those built by Ginzburg. In particular, we use
a different "trap" which makes the necessary embeddings of this plug much easier to construct.
13.
Strict Quantization of Solvable Symmetric Spaces - Bieliavsky
, Pierre
This work is a contribution to the area of Strict Quantization (in the sense of
Rieffel) in the presence of curvature and non-Abelian group actions. More precisely,
we use geometry to obtain explicit oscillatory integral formulae for strongly
invariant strict deformation quantizations of a class of solvable symplectic symmetric spaces.
Each of these quantizations gives rise to a field of (pre)-C*-algebras whose fibers
are function algebras which are closed under the deformed product. The symmetry group of the
symmetric space acts on each fiber by C*-algebra automorphisms.
14.
Invariants of Legendrian Knots and Coherent Orientations - Etnyre
, John B.; Ng
, Lenhard L.; Sabloff
, Joshua M.
We provide a translation between Chekanov's combinatorial theory for
invariants of Legendrian knows in the standard contact ?3
and Eliashberg and Hofer's contact homology. We use this translation to
transport the idea of "coherent orientations" from the contact homology world to
Chekanov's combinatorial setting. As a result, we obtain a lifting of Chekanov's
differential graded algebra invariant to an algebra over ?[t,t-1] with a full
? grading.
15.
A h-principle for open relations invariant under foliated isotopies - Bertelson
, Mélanie
This paper presents a natural extension to foliated spaces of the
following result due to Gromov: the h-principle for open, invariant differential relations is valid
on open manifolds. The definition of openness for foliated spaces adopted here involves a
certain type of Morse functions. Consequences concerning the problem of existence of
regular Poisson structures, the original motivation for this work, are presented.
16.
Correction: The Verlinde formula as fixed point formulas - Alekseev
, A.; Meinrenken
, E.; Woodward
, C.
Due to a publisher error, the correct version of sections 5.4, 5.5, and 5.6 was not printed.
The correct version is includedhere.
18.
A classification of topologically stable Poisson structures on a compact oriented structure - Radko
, Olga
Poisson structures vanishing linearly on a set of smooth closed disjoint
curves are generic in the set of all Poisson structures on a
compact connected oriented surface. W construct a complete set
of invriants classifying these structures up to an
orient-preserving Poisson isomorphism. We show that there is a
set of non-trivial infinitesimal deformations which generate
the second Poisson cohomology and such that each of the deformations
changes exactly one of the classifying invarients. As an
example, we consider Poisson structures on the sphere which vanish
linearly on a set of smooth closed disjoint curves.
19.
The symplectic vortex equations and invariants of
Hamiltonian group actions - Cieliebak
, Kai; Gaio
, A. Rita; Mundet i Riera
, Ignasi; Salamon
, Dietmar A.
In this paper we define invariants of Hamiltonian group actions
for central regular values of the moment map. The key hypotheses
are that the moment map is proper and that the ambient
manifold is symplectically aspherical. The invariants are based
on the symplectic vortex equations. Applications include an
existence theorem for relative periodic orbits, a computation for
circle actions on a complex vector space, and a theorem
about the relaton between the invariants introduced here and the
Seiberg-Witten invariants of a product of a Riemann surface
with a two-sphere.
20.
A remark about Donaldson's construction of
symplectic submanifolds - Auroux
, D.
We describe a simplification of Donaldson's arguments
for the construction of symplectic hypersurfaces [4] or
Lefschetz pencils [5] that makes it possible to avoid any
reference to Yomdin's work on the complexity of real
algebraic sets.
1.
