Detailed Report
The symposium activities in 199798 centered around four main workshops, and included several lecture courses, seminar series, lecture series by visitors, some short meetings and an LMS Spitalfields Day. A closing workshop was held in July 2000 at the end of the three year grant period which gave the opportunity to cover the new developments resulting from Kontsevich's work in deformation theory.
There were three visitors for the whole year (Véronique ChloupArnould, Laurent Lazzarini, Joachim Weber) financed by other sources, a number of long term visitors (including Shigeru Mukai, Nagoya, who came for five months from October 1997 to April 1998), and 225 registered particpants for the longer activities as well as many who came to talks from nearby institutions.
In addition to the EPSRC grant of £ 80,000 visitors were also supported by
The mathematical topics covered a wide range of subjects centered around symplectic topology and geometry, as well as its many applications and related areas such as Kähler and algebraic geometry or the use of symmetry in mechanics, and several joint activities with these groups were organised (programmes are appended). Over recent years symplectic geometry has been a very active and lively subject with many new developments as well as new links to other subjects in mathematics as well as physics. This was reflected in the various workshops and lecture series at Warwick.
Many visitors came from other British universities, and reciprocally, many of the Symposium visitors, as well as the Symposium organisers, gave seminars and short lecture series at other British universities on symplectic geometry and related fields. A joint seminar between Manchester, Sheffield and Warwick on Bracket Geometries is one of the collaborations to come out of the contacts made in this Symposium year (it has just received funding from the LMS).
A large number of PhD students and postdocs particpated in the workshops and seminars. These are indicated by a ^{*} in the participant list (Appendix D).
More details of these workshops including lists of talks are given in Appendix B.
OPENING WORKSHOP
112 September 1997
(52 participants, 31 lectures)
The main purpose of the first workshop was to
provide an introduction to several different topics
which would be a focus of the Warwick Symposium 97/98.
One of the highlights was a lecture series by Leonid
Polterovich (Tel Aviv) about symplectic rigidity
ranging from symplectic packing problems, via Hofer's geometry,
length spectra, and classical mechanics, to Lagrangian knots,
and providing many interesting links between these
seemingly different topics. Ron Stern (Irvine) gave a
lecture series on symplectic 4manifolds with many interesting
applications of SeibergWitten invariants and surgery
along knots. Viterbo gave a series of lectures
on applications of Floer homology.
Victor Ginzburg (Santa Cruz) and Yael Karshon (Jerusalem),
gave a series of lectures on moment maps and symplectic reduction.
Simone Gutt (Brussels/Metz) and Joseph Wolf (Berkeley)
gave series of lectures on (deformation resp. direct limit)
quantization. And there were several additional lectures,
notably one by Ginzburg about his existence theorem
for a compact hypersurface without closed characteristics,
and two by JeanClaude Sikorav (Toulouse)
about an application of the theory of Jholomorphic curves.
WORKSHOP ON MOMENT MAPS AND QUANTIZATION
818 December 1997
(47 participants, 27 lectures)
The main focus of the Workshop was on quantization, with survey lectures
by Moshe Flato (Dijon) and Daniel Sternheimer (Dijon), and two series of lectures by
Boris Fedosov (Potsdam) on his geometrical approach and Alexander Karabegov (Dubna) on the
deformation quantization of Kähler manifolds. Simone Gutt (Brussels/Metz) lectured on
equivalence of star products, and the material she presented has been
combined with the Quantization seminars of Rawnsley to
produce a survey article on Deligne's approach to star products which
will appear in the Journal of Geometry and Physics.
A subtheme was the extension of results from symplectic to Poisson manifolds. Talks on the structure and quantization of Poisson manifolds were given by Santos Asin (Warwick) and Véronique ChloupArnould (Warwick/Metz). Johannes Huebschmann (Lille) and Kirill Mackenzie (Sheffield) lectured on geometrical aspects of Poisson manifolds.
Moment maps figured in the lectures on moduli spaces, a noncommutative version in Fedosov's second lecture, and in the lectures of James Montaldi (Nice) and Tatsuru Takakura (Tokyo).
The opening day of the Workshop was concluded with a seminar for a general audience by Dimitri Anosov (Steklov, Moscow) on some geometrical aspects of flows on surfaces.
In addition there were a number of lectures with a strong algebrogeometric content, namely by Hitchin (Oxford) and Pidstrigatch (Warwick) on the geometry of moduli spaces, by Mukai (Nagoya) about theta divisors, by McDuff (Stony Brook) about moduli spaces of stable maps, and by Ono (Hokkaido Univ) about symplectic fillings.
During the workshop Kenji Matsuki (Purdue) began his lecture course about the Mori program. (See Section 2.2 below.)
WORKSHOP ON SYMPLECTIC TOPOLOGY
23 March  3 April 1998
(78 participants, 36 lectures).
This workshop included an
LMS Spitalfields Day on Symplectic Topology
on 1 April 1998.
This workshop covered a wide spectrum of topics such as contact geometry (Giroux, Ohta), Legendrian knots (Chekanov, Fraser), Floer homology  both for 3manifolds and in symplectic geometry (Froyshov, Fukaya, Lazzarini, Ono, Schwarz, Weber), Hofer's geometry of the group of symplectomorphisms (Lalonde, Milinkovich), SeibergWitten invariants (Hutchings, Kanda, Lisca, Salamon, Stipsicz), circle actions (Tolman), integrable systems (Uhlenbeck), the topology of the symplectomorphism group (McDuff), ChernSimons theory (Tyurin), K3surfaces (Mukai), invariants of 4manifolds (Pidstrigatch, Stipsicz), and Donaldson's new theory of symplectic submanifolds and topological Lefschetz fibrations (Auroux).
Notable was a series of three lectures by Kenji Fukaya (Kyoto) about Categorical Mirror Symmetry, exploring fascinating links between symplectic and complex geometry. Although highly speculative, his lectures were full of original and inspiring ideas.
Another notable feature of the workshop was the presence of so many young mathematicians who already had made significant contributions to the subject. Among these were Dennis Auroux (IHES), Kim Froyshov (Oslo, now Harvard), Michael Hutchings (Harvard, now Stanford), Yutaka Kanda (Hokkaido Univ), Laurent Lazzarini (Warwick, now ETH Zürich), Matthias Schwarz (Stanford, now Chgicago), Joachim Weber (Warwick, now Stony Brook).
The Spitalfields Day consisted of four colloquial style lectures for a general mathematical audience by McDuff, Tolman, Hutchings, and Uhlenbeck. It was very well attended by British mathematicians, and gave an excellent introduction to a number of exciting new developments in symplectic topology and related subjects. The day closed with a social event (party with live blues band).
WORKSHOP ON SYMMETRIC HAMILTONIAN SYSTEMS
1617 July 1998
(organised by Mark Roberts)
(25 participants, 12 lectures).
This 2day workshop was the third in a series run under the auspices of an
LMS Scheme 3 project on ``Bifurcations and Symmetry'' coordinated by David
Chillingworth (Southampton). The aim was to bring together mathematicians
and physicists working on symmetric Hamiltonian systems from a wide range
of points of view. This meeting was held as part of the concluding workshop
on Symplectic Geometry (see below) to maximize the potential for
interaction between bifurcation theorists and symplectic geometers. Both
areas were well represented among the talks, as were applications to
specific physical systems.
Talks with a strong geometrical flavour included those by Anthony Bloch (Michigan) on integrable systems, Andrew Lewis (Warwick) on mechanical systems on homogeneous spaces and Sergey Pekarsky (Caltech) on kinematic connections for systems on Kähler manifolds. Symplectic geometry also featured prominently in presentations by Eugene Lerman (Illinois) and Mark Roberts (Warwick), both of whom described recent results on bifurcations of relative equilibria. This was also the theme of talks by Debra Lewis (UCSC) and Andre Vanderbauwhede (Gent). Gianne Derks (Surrey) described her work on the effects of dissipation on the stability of relative equilibria. Applications were represented by talks on systems of point vortices on the sphere (James Montaldi, Nice), on the dynamics of atomic systems (Dmitrii Sadovskii, U. du Littoral), on relative equilibria of rotating fluid masses (Esmeralda Sousa Dias, Lisbon), and on relative equilibria of molecules (Igor Kozin, Warwick).
Funding for the workshop was provided by EPSRC, LMS and the Warwick MRC.
WORKSHOP ON SYMPLECTIC GEOMETRY
1324 July 1998
(101 participants, 37 lectures).
The concluding workshop of the Symposium covered again
a wide spectrum of subjects within symplectic topology
and its many related areas.
Victor Ginzburg (Santa Cruz) and Yael Karshon (Jerusalem) gave two lectures about their joint work on abstract moment maps. Related to this was Sue Tolman's lecture about cricle reduced spaces. Mark Gross (Warwick) and Jon Wolfson (East Lansing) gave lectures about special Lagrangian fibrations. Yuli Rudyak (Heidelberg) and Misha Farber (Tel Aviv) gave lectures about LjusternikSchnirelman theory. Karl Friedrich Siburg (Freiburg), Yiming Long (Tianjin), and Misha Bialy (Tel Aviv) gave interesting lectures about Hamiltonian dynamics and its relation to modern symplectic invariants. There were lectures about Floer homology by Vicente Munoz (Malaga), YongGeun Oh (Madison), and Kaoru Ono (Hokkaido Univ), who explained his joint work with Fukaya, Kontsevich, Ohta, and Oh about new obstructions to the construction of Floer homology for Lagrangian intersections. There was a lecture by Andras Stipsicz (budapest) about symplectic Lefschetz fibrations and its Chern numbers.
Zoltan Szabo (Princeton) gave a series of three lectures about his very recent joint work with Peter Ozsvath in which they give a full proof of the symplectic Thom conjecture using the SeibergWitten invariants, and building on the previous work by Kronheimer and Mrowka.
Yasha Eliashberg (Stanford) gave two fascinating lectures about his joint work with Hofer and Givental about contact homology. This is a new invariant of contact manifolds which takes the form of a topological quantum field theory.
WORKSHOP ON QUANTISATION
2428 July 2000
(29 participants, 23 lectures).
This additional workshop of the Symposium was organised to cover an area
which had developed rapidly after the Symposium year was over, namely the
work of Maxim Kontsevich on the Formality Theorem and its consequences
for quantisation of Poisson manifolds. It was organised in parallel with
a meeting on Geometry and Analysis on Path Spaces, a satellite meeting
of the International Congress of Mathematical Physics 2000. There was
much interaction between the two meetings, with several sessions in
common.
The highlights were talks on formality of the Hochschild complex by Arnal, Jones, A. Voronov; aspects of deformation quantisation by Bordemann, Gutt, Hudson, Maeda, Sternheimer, Waldmann; geometric aspects of quantisation by Hall, Landsman, Merkulov, Sergeev, T. Voronov.
Samples of visitors' comments on the Workshops are included
in Appendix E.)
The following three MSc courses were part of the Symposium:
Course of 6 lectures on Geometric Invariant Theory by Shigeru Mukai (Nagoya),
MarchApril 1998 (organised by Miles Reid).
Prof. S. Mukai from Nagoya (one of the world's top algebraic geometers) visited for 5 months from Oct 1997 to Mar 1998 and (in addition to many workshop and seminar talks) gave a short lecture course on Introduction to GIT (geometric invariant theory)
Topics were:
Course of 4 lectures on Gauge theory, Symplectic Geometry and Slightly
Deformed Algebraic Geometry by Prof. A. Tyurin (Steklov Inst.)
Prof. A. Tyurin from Moscow visited for 2 months during March  June 1998 and gave a short lecture course on his recent research
Topics were:
Lectures on A_{n}quiver algebras, intersection theory,
and Floer homology by Paul Seidel (IAS, Princeton),
1625 June 1998.
Lectures on Floer homology in Oxford by Dietmar Salamon (Warwick):
Lectures on Mp^{c}structures in Oxford by John Rawnsley (Warwick):
2122, 2829 May 1998.
Geometry and.... During the year the geometry group (Mark Gross, John Jones, Mario Micallef, Victor Pidstrigatch, John Rawnsley, Miles Reid, Dietmar Salamon) organised a seminar ``Geometry and ...'' (now ``Geometry Plus''), aiming to cover all the areas of crossfertilisation between the different branches of geometry, physics, algebra, number theory and so on. On almost every Monday of the 3 terms we had two sessions, one aiming at introducing technical topics to graduate students, the second more research oriented.
Floer homology seminar. During the spring and summer terms of 1998 Dietmar Salamon organised a seminar on Jholomorphic curves and quantum cohomology. In the spring term Dietmar Salamon gave a series of lectures on Gromov compactness and stable maps leading up to the construction of the GromovWitten invariants and a proof of the general Arnold conjecture along the lines of his joint work with Hofer. In the summer term Mark Gross gave a series of lectures entitled ``Counting curves in the quintic'' in which he explained the work of Yau et al about Givental's proof of the Mirror conjecture.
Quantization seminar. The quantization seminar met weekly from October 1997 and covered Deligne's approach to the classification of star products (Rawnsley, 4 seminars), Mp^{c}structures and Fedosov's index theorem (Rawnsley, 3 seminars) and Kontsevich's quantization of Poisson manifolds (Gutt & Rawnsley, 4 seminars). Maeda spoke on quantum diffeomorphisms, Asin on geometrical methods of constructing star products on Poisson manifolds, Terizakis on polarizations with singularities and their effect on geometric quantization, Loi on quantization of Kähler manifolds, ChloupArnould on results of Weinstein & Xu, and Cooper on Lagrangian submanifolds in deformation quantization.
Mechanics and Symmetry seminar.
In terms 1 and 2 Mark Roberts organized a seminar programme devoted to geometrical aspects of the theory of symmetric mechanical systems. Andrew Lewis gave a series of talks on Lagrangian reduction. This was followed by a series in which George Patrick (on a year long visit from Saskatchewan) described his work on `drift' dynamics near relative equilibria with nongeneric momenta. Mark Roberts then gave a number of seminars on the MarleGuilleminSternberg normal form for symplectic group actions and its applications to the reduction of symmetric Hamiltonian systems near relative equilibria. Other talks included a survey of properties of coadjoint orbits by John Rawnsley, and a talk on relative equilibria of atomic systems by Dmitrii Sadovskii (U. du Littoral).
Appendices
Some of the most interesting new applications of symplectic Floer
homology are due to Paul Seidel. In his thesis [127] he used
Floer homology to find symplectomorphisms, on a large class of
symplectic 4manifolds, which are smoothly, but not symplectically,
isotopic to the identity. The main ingredient in his proof is an exact
sequence in symplectic Floer homology, which is reminiscent of Floer's
exact sequence for the Floer homology of homology3spheres. Seidel
uses his exact sequence to compute the Floer homology groups of
generalised Dehn twists. Generalised Dehn twists also play a crucial
role in his joint work with Kovanov [46] about the symplectic
monodromy in A_{n} singularities. They prove, in particular, that the
braid group B_{n+1} embeds into the group of components of the
symplectomorphism group of the Milnor fibre. Floer homology is again an
essential ingredient to distinguish symplectic isotopy classes. There
are interesting relatins to the Burau representation of the braid group.
Seidel lectured about this work at Warwick in June.
Fukaya, Kontsevich, Oh, Ohta, and Ono discovered an interesting new
obstruction theory for the construction of Floer homology groups for
Lagrangian intersections (cf. [30]). This is a major project and
the work is still in progress. Ono lectured about these ideas in the
March and July workshops.
In [29] Fukaya interprets (noncommutative) mirror symmetry as a
duality between symplectic and complex geometry, following
StromingerYauZaslov. The (complex) mirror of a symplectic manifold is
interpreted as a moduli space of Lagrangian submanifolds with flat line
bundles, up to Hamiltonian isotopy. Floer homology becomes a sheaf over
the mirror. Another ingredient of this approach is Floer homology for
Lagrangian foliations (rather than submanifolds). The details have been
worked out rigorously in the case of the torus. In more general contexts
this approach is still speculative but provides a rich source of new
ideas. In the March workshop Fukaya gave an inspiring series of lectures
about his new approach.
In [73] Schwarz gives a detailed proof for the existence of a
natural ring isomorphism between Floer homology and quantum cohomology.
This is based on his previous joint work with Piunikhin and
Salamon [121].
Some of the most interesting recent work about YangMills Floer homology
of 3manifolds is due to Froyshov. In [28] he uses the
dimension of a certain equivariant Floer homology group to establish an
integer invariant YÆ h(Y) of homology 3spheres which respects
connected sums:

Gaio and Salamon worked on pseudoholomorphic curves
in symplectic quotients and their relations to solutions of a certain
deformed equation in the ambient symplectic manifold
which couples equivariant maps to connections via the curvature
and the moment map (cf. [32] and [31]).
Potential applications should include the relation between
the GromovWitten invariants of symplectic quotients in different
chambers, and a geometric proof of the relation between
the quantum cohomology of the Grassmannian and the Verlinde algebra.
Salamon and Weber worked on the relation between
the Floer homology of the cotangent bundle
for a classical Hamiltonian (kinetic plus potential energy)
with the cohomology of the loop space (cf [69]
and [90]). The technique of proof involves an adiabatic limit
in which the perturbed pseudoholomorphic curves
in the cotangent bundle degenerate to
solutions of the heat equation, i.e. gradient flow lines
of the classical action. This is related to Viterbo's
work in [89]. Weber lectured about his results in
the March workshop.
In [50] Lazzarini studied
pseudoholomorphic discs with Lagrangian
boundary conditions and his results enabled him to prove
the Arnold conjecture for strongly negative
Lagrangian submanifolds.
He lectured about these results in the March workshop.
A codimension zero submanifold U Ã M with corners
determines naturally a singular Lagrangian subvariety
L_{U} Ã T^{*}M. In [42,43]
Kasturirangan and Oh constructed Floer homology
groups for the Lagrangian pair (L_{U},L_{0}) where
L_{0} is a Hamiltonian geformation of the zero section,
related this to the Conley index,
and obtained a refinement of the Arnold conjecture
for Lagrangian intersections in cotangent bundles.
Oh lectured about these results in the July workshop.
In [88,89] Viterbo constructed a natural
pushforward map for Floer homology groups of open manifolds,
induced by a certain codimension zero inclusions.
Applications include:
1. a proof of the Weinstein conjecture
in cotangent bundles of simply connected manifolds,
2. the theorem that subcritical Stein manifolds
do not admit exact Lagrangian embeddings,
3. for exact Lagrangian submanifolds L Ã T^{*}S^{n}
the projection LÆ S^{n} has nonzero degree.
Viterbo lectured about these results in the opening workshop.
Another application is the theorem that the real part
of a Fano variety cannot be hyperbolic.
Viterbo lectured about this in the July workshop.
Recently Ozsvath and Szabo [120] found a proof of the symplectic Thom conjecture, which asserts that symplectic submanifolds (of symplectic 4manifolds) minimise the genus among all submanifolds representing the same homology class. This had previously been proved by KronheimerMrowka, and independently by MorganSzaboTaubes. The new work of Ozsvath and Szabo is based on a product formula for the SeibergWitten invariants in [117]. Szabo gave a series of lectures about this work in the July workshop.
In [104] Fintushel and Stern established a relation
between the SeibergWitten invariants of a smooth
4manifold X with that of the manifold X_{T,K}
obtained by forming the connected sum,
along a suitable torus T Ã X,
with the product M_{K}×S^{1}, where
M_{K} is obtained by performing
0surgery on a knot K Ã S^{3}.
The relation has the form

In [77] Stipsicz
found interesting applications of the SeibergWitten
invariants to the geography of smooth 4manifolds.
He constructed a sequence of new simply connected
symplectic 4manifolds C_{n} for which the
numbers c_{1}(C_{n})^{2}/c_{h}(C_{n}) converging to 9.
In particular, this gives rise to infinitely
many simply connected symplectic 4manifolds
with positive signature. In [78]
He establishes the rationality of complex
curves in simply connected Kähler surfaces
with b^{+} > 1. In [79] he characterises
the minimality of symplectic Lefschetz fibration.
At various times Stipsicz gave lectures
at the Warwick Symposium about his work.
Stefan Bauer discovered a refinement of the SeibergWitten
invariants [5] which enabled him to distinguish the
diffeomorphism type of certain connected sums of elliptic
surfaces which are homeomorphic and whose SeibergWitten
invariants vanish. He lectured about his work in the March
workshop.
In [109,110] Hutchings and Lee establish an interesting
link between Reidemeister torsion and circle valued
Morse theory. They prove that the topological
torsion of a manifold M  with twisted coefficients, determined
by a cohomology class a ‘ H^{1}(M,\mathbb Z) 
is equal to the product of the torsion of the MorseNovikov
complex with a zetafunction, obtained by counting closed
orbits. Both the MorseNovikov complex and the zeta function
are determined by the gradient flow of a closed 1form representing
the class a. This result has interesting consequences
concerning the relation of the SeibergWitten invariants of
a 3manifold with Reidemeister torsion.
Hutchings lectured about these results in the March workshop.
Salamon is working on the relation between the SeibergWitten monopoles
on symplectic Lefschetz fibrations and holomorphic sections of
corresponding singular fibrations, in which the fibres have been
replaced by symmetric products of a Riemann surface. The 3dimensional
analogue relates the SeibergWitten Floer homology of a mapping torus
Y_{f}, associated to a diffeomorphism f:SÆS to the
symplectic Floer homology of the induced symplectomorphism of the
symmetric products of S [68]. This is a SeibergWitten
version of the AtiyahFloer conjecture. The expected results are
closely related to the work of Taubes and MengTaubes. Salamon lectured
about this work in the March workshop.
In [59] Ohta and Ono proved that symplectic fillings
of certain quotients S^{3}/G must have a negative
definite intersection form. This work is based on results by Froyshov.
Ono lectured about it in the December workshop
and Ohta in the March workshop.
In the March workshop Kanda lectured about
his extension of Taubes' relation between the SeibergWitten
and the Gromov invariants to certain noncompact
symplectic 4manifolds.
Many different aspects of quantisation were in development during the period of the Symposium. The recent work of Kontsevich on the Formality of the Hochschild complex and its consequences for deformation quantisation had not been digested during the main activities of the Symposium, but a small continuation workshop including this theme was held during the summer of 2000.
In the Symposium proper one of the main themes was deformation quantisation for symplectic manifolds. This had been revolutionised by Fedosov who bypassed all the difficulties of the traditional approach of constructing star products through gluing together local products. Instead Fedosov uses a geometrical method based on the construction of a flat connection in the Weyl bundle from a (curved) symplectic connection on the manifold.
Deligne's account of the classification of star products using classical deformation theory methods was simplified and extended by Gutt and Rawnsley in work largely done during seminars and workshops of the Symposium.
Ginzburg and Karshon [1] described the work of the Guillemin group on generalised moment maps which provide a simplified approach to symplectic reduction, just retaining the features needed to control topological properties of the reduced spaces and rebuild in some cases the original space from the reduction data.
Tolman [137] described classification results for circle actions in low dimensions, and methods for computing intersection cohomology of reduced spaces.
Geometrical properties of momentum maps were obtained by R. Sjamaar [136], A.R. Gaio and D.A. Salamon [32].
Jones and Rawnsley [111] found a restriction on the signature when a manifold admits a Hamiltonian circle action.
Donaldson proved the existence of symplectic submanifolds and topological Lefschetz fibrations for all symplectic manifolds. In the March workshop Auroux lectured about this new theory and his own related work in [94].
In [96] Biran proved sharp estimates for symplectic
packing problems in \mathbb CP^{2}. He then used Donaldson theory
of symplectic Lefschetz pencils study this problem
for general symplectic 4manifolds [7].
His approach is to fill as much as possible of the symplectic manifold
by a disc bundle over the symplectic submanifold representing
a multiple of the symplectic form. The complement
turns out to be a Lagrangian skeleton which carries
information about the underlying symplectic manifold.
Biran discussed these remarkable new ideas in
his lectures in the July workshop.
Polterovich [125] recently found new interesting
links between ergodic theory and symplectic topology.
He constructed contractible strictly ergodic Hamiltonian
loops on large classes of symplectic manifolds
and proved that the asymptotic Hofer norm of every
strictly ergodic loop must be zero.
He also posed a number of interesting open questions.
In the September workshop Polterovich gave a series of five lectures about symplectic rigidity starting with applications of pseudoholomorphic curves to symplectic packing problems (McDuffPolterovich [114]), to the Lagrangian knot problem (EliashbergPolterovich [100]), and to the deformation problem (persistence of exceptional divisors). He then discussed Hofer's metric on the group of Hamiltonian diffeomeorphisms, his joint work with Bialy about geodesics in this metric, the relation between length spectra in Hofer's metric, invariants of symplectic fibrations, and Gromov's Karea (cf. [122,123,124]), and the relation between Lagrangian knots and classical mechanics (cf. [95,101]).
In [119,126] Oprea and Rudyak proved that every
spherical symplectic 2nmanifold has LjusternikSchnirelman
category 2n+1. Hence they were able to prove
the Arnold conjecture for such manifolds
in full generality (the minimal number of fixed points of a Hamiltonian
symplectomorphism is equal to the minimal number of critical
points of a function).
In [47] LalondeMcDuffPolterovich proved
the flux conjecture for a large class of symplectic manifolds.
In [48] They established the topological rigidity
of Hamiltonian loops. Namely, if two loops of symplectomorphisms
in (M,w_{0}) and (M,w_{1}) are smoothly isotopic,
and one of them is symplectically isotopic to a
loop of Hamiltonian symplectomorphisms, then so is the other.
In other words, the Hamiltonian condition (zero Flux)
is a topological property. Their proof involves the
study of symplectic fibrations over the 2sphere,
and is based on the work of Seidel [128].
Lalonde and McDuff lectured about this work in
the July workshop.
Le Hong Van studied harmonic almost complex structures
on symplectic manifolds.
Schwarz found a continuous section of the action spectrum
over the universal cover of the group of Hamiltonian
symplectomorphisms [71]. This gives rise to a
new biinvariant metric on the group of Hamiltonian
symplectomorphisms. As an application Schwarz proved
the existence of infinitely many geometricall distinct
periodic orbits for certain Hamiltonian symplectomorphisms.
Another application is a proof of the fact that
the diameter if the group of Hamiltonian
diffeomorphisms of the 2torus (with respect to the
Hofer metric) is infinite [72].
Schwarz lectured about this work in the March workshop.
In [87] Viterbo discovered certain isoperimetric
inequalities for the displacement energy.
He found many interesting applications
concerning obstructions to Lagrangian embeddings,
periodic orbits of billiard problems,
and the closure of the symplectomorphism group.
Viterbo lectured about these results in
the March workshop.
One of the most exciting recent developments in this area is the contact homology discovered by Eliashberg, Givental, and Hofer (cf. [21]. This invariant takes the form of a topological quantum field theory, which assigns to every contact manifold a super Poisson algebra (roughly speaking, an algebra of functions on a symplectic (super) vector space generated by the periodic solutions of the Reeb flow). A symplectic manifold with contact boundary gives rise to a Lagrangian subalgebra, and gluing symplectic manifolds along a contact boundary corresponds to symplectic reduction. the correspondiong invariants for closed symplectic manifolds are the GromovWitten invariants. Eliashberg gave two lectures about these ideas in the July workshop.
Yuri Chekanov gave lectures about his invariants of
Legendrian knots (cf. [15]).
These invariants represent a special
case of the contact homology by EliashbergGiventalHofer
and they can be constructed with combinatorial techniques.
They give rise to examples of Legendrian knots which
are smoothly isotopic and have the same Bennequin invariant
and rotation number, but are not Legendrian isotopic.
This should be contrasted with the theorem of
Eliashberg and Fraser, which asserts that
topologically trivial Legendrian knots are trivial if and only
classified by their Bennequin invariant and rotation number
(cf. [102] and the March workshop).
Lisa Traynor worked on invariants of Legendrian tangles.
During his stay at Warwick Chekanov also worked on the proof of the
fourpoint conjecture by Arnold.
Emmanuel Giroux (in the March workshop) and his former student
Vincent Colin worked on contact structures on 3manifolds.
Colin constructed tight contact structures on many 3manifolds.
Mark Gross's most recent work on mirror symmetry [34] represents very substantial progress on the StromingerYauZaslow approach to mirror symmetry via special Lagrangian fibrations. Following his previous joint work with Pelham Wilson, which concentrated on verifying the topological aspects of the SYZ conjectures in one example, Gross has generalised Hitchin's results in the case of trivial connections, developing the crucial metric aspects of the subject by showing how to write down the complex and Kähler structures of the mirror in terms of the geometry of the initial manifold.
A. Tyurin [83] has developed a theory of special Lagrangian geometry and its relations with BohrSommerfeld quantisation.
During the year Mark Roberts (Warwick) worked on a number of aspects of the stability and bifurcation theory of relative equilibria of symmetric Hamiltonian systems with two EPSRC research assistants, Andrew Lewis and Igor Kozin, and two long term EPSRC visiting fellows, James Montaldi (Nice) and George Patrick (Saskatchewan). Work with Lewis included the development of Lagrangian and Hamiltonian reduction theories for mechanical systems defined on the tangent bundles of homogeneous spaces. Patrick and Roberts showed that the set of relative equilibria of a generic Hamiltonian system which is invariant under a free action of a compact group is stratified by the symmetry type of the generatormomentum pair. Lewis, Patrick and Roberts also initiated a project which aims to extend aspects of the stability theory of relative equilibria to systems for which the action of the symmetry group has noncompact isotropy subgroups. On the applied side Montaldi and Roberts continued work on the existence and stability of relative equilibria of systems of point vortices on the sphere (a joint project with Chjan Lim (Rensselaer PI)) while work with Kozin centred on the computation and interpretation of bifurcation diagrams for the relative equilibria of triatomic molecules.
^{*}indicates a PhD student or postdoc.
Abenda, Simonetta (Bologna, Italy)
Agnihotri, Sharad (Amsterdam, The Netherlands)
Akveld^{*}, Meike (ETHZurich, Switzerland)
Altinok^{*}, Selma (Warwick, UK)
Anjos, Silvia (Stony Brook, USA)
Anosov, Dimitri (SteklovMoscow, Russia)
Arezzo^{*}, Claudio (Warwick, UK)
Arnal, Didier (Metz, France)
Asin^{*}, Santos (Warwick, UK)
Audin, Michèle (Paris, France)
Auroux, Denis (Ecole Polytechnique, France)
Baguis^{*}, Pierre (Brussels, Belgium)
Barrett, John (Nottingham, UK)
Bartocci, Claudio (Genova, Italy)
Bauer, Stefan (Bielefeld, Germany)
Bertelson^{*}, Mélanie (Stanford, USA)
Bhupal^{*}, Mohan (MPIBonn, Germany)
Bialy, Misha (Tel Aviv, Israel)
Bieliavsky^{*}, Pierre (Bruxelles, Belgium)
Biran, Paul (Stanford, USA)
Bloch, Anthony (Michigan, USA)
Bloore, Fred, J. (Liverpool, UK)
Bonneau^{*}, Philippe (Bourgogne, France)
Bordemann, Martin (Freiburg, Germany)
Bourgeois, Frédéric (Bruxelles, Belgium)
Braverman, Maxim (Jerusalem, Israel)
Bridges, Thomas J. (Surrey, UK)
Burns, Dan (Michigan, USA)
Burstall, Francis (Bath, UK)
Cahen, Michel (Bruxelles, Belgium)
Caiber, Mirel (Warwick, UK)
Calderon, Francisco (Sevilla, Spain)
Cannas de Silva, Ana (UC Berkeley, USA)
CastanoBernard^{*}, Ricardo (Warwick, UK)
Castelvecchi, Davide (Stanford, USA)
Chaperon, Marc (ParisJussieu, France)
Chekanov, Yuri (Moscow, Russia)
Chiang, Mengjung (UrbanaChampaign, USA)
Chillingworth, David (Southampton, UK)
ChloupArnould^{*}, Veronique (Metz, France)
Ciocci, Maria Cristina (Gent, Belgium)
Ciriza, Eleonora (Roma, Italy)
Coelho, Zaq (Porto, Portugal)
Cohen, Ralph (Stanford, USA)
Colin, Vincent (Lyon, France)
Cooper^{*}, Paul G. (Warwick, UK)
Cruz^{*}, Ines (Porto, Portugal)
Cushman, Richard (Utrecht, The Netherlands)
Damian, Mihai (Toulouse, France)
Daskalopoulos, Georgios (Brown, USA)
Derks, Gianne (Surrey, UK)
Djordjevic, Goran S. (Nis, Yugoslavia)
Domitrz, Wojciech (Warsaw , Poland)
Dostoglou, Stamatis (MissouriColumbia, USA)
Eells, James (Warwick, UK)
Eliashberg, Yakov (Stanford, USA)
Farber, Michael (Tel Aviv, Israel)
Fedosov, Boris (Potsdam, Germany)
Fernandes, Emmanuel (Louvain, Belgium)
Fernandes, Rui Loja (Lisbon, Portugal)
Flaschka, Hermann (Arizona, USA)
Flato, Moshe (Bourgogne, France)
Fraser, Maia (Montreal, Canada)
Froyshov, Kim A. (Oslo, NORWAY)
Fukaya, Kenji (Kyoto, Japan)
Furter, Jacques (Brunel, UK)
Gaio^{*}, Rita (Warwick, UK)
Gammella^{*}, Angela (Metz, France)
Gekhtman, Michael (Williamsburg, USA)
Ginzburg, Viktor (UC Santa Cruz, USA)
Giroux, Emmanuel (ENSLyon, France)
Giunashvili, Zaqro (Academy of Sciences, Georgia)
Gothen, Peter (Porto, Portugal)
Gross, Mark (Warwick, UK)
Gutt, Simone (Bruxelles, Belgium)
Habermann, Katharina (MPI Bonn, Germany)
Halic, Mihai (Institut Fourier, France)
Hall, Brian (Notre Dame, USA)
Hannabuss, Keith C. (Oxford, UK)
He, Xinyu (Warwick, UK)
Herrera^{*}, Rafael (Oxford, UK)
Hickin, David (Warwick, UK)
Hind, Richard (MPI Bonn, Germany)
Hitchin, Nigel (Oxford, UK)
Horowitz^{*}, Joel (Bruxelles, Belgium)
Hoyle, Mark (MIT, USA)
Hrabak^{*}, Sean Paul (KCL, UK)
Hudson, Robin, L. (Nottingham Trent , UK)
Huebschmann, Johannes (Lille, France)
Hutchings^{*}, Michael (Harvard, USA)
Izadi, Elham (UG Athens, USA)
Jones, John D.S. (Warwick, UK)
Kanda, Yutaka (Hokkaido, Japan)
Karabegov, Alexander (Dubna, Russia)
Karshon, Yael (Hebrew University, Israel)
Kharlamov, V (Strasbourg, France)
Khudaverdyan, H (UMIST, UK)
Kim, Youngsun (Warwick, UK)
Kirwan, Frances (Oxford, UK)
Konno, Hiroshi (Tokyo, Japan)
Konno, Kazuhiro (isa, Italy)
Kovalev, Alexei (Edinburgh, UK)
Lalonde, Francois (Montréal, Canada)
Lamb^{*}, Jeroen (Warwick, UK)
Landsman, Nicolaas (Cambridge, UK)
Lang, Jens (München, Germany)
Laudenbach, Francois (Ecole Polytechnique, France)
Lazzarini^{*}, Laurent (Warwick, UK)
LeHong, Van (MPI Bonn, Germany)
Lei, Tan (Warwick, UK)
Lerman, Eugene (UrbanaChampaign, USA)
Lewis^{*}, Andrew (Warwick, UK)
Lewis, Debra (UC Santa Cruz, USA)
Lisca, Paolo (Pisa, Italy)
Livotto^{*}, Andrea Giulio (Warwick, UK)
Lizan, Véronique (Toulouse, France)
Loi^{*}, Andrea (Cagliari, Italy)
Long, Yiming (Nankai, PR CHINA)
McDuff, Dusa (SUNY at Stony Brook, USA)
Mackenzie, Kirill (Sheffield, UK)
Maeda, Yoshiaki (Keio, Japan)
Maeno, Toshiaki (Kyoto, Japan)
Mahassen, Nadim (Swansea, UK)
Marshall, Ian D. (Leeds, UK)
Matsuki, Kenji (Purdue, USA)
Matsushita, Daisuke (Kyoto, Japan)
Markus, Larry (Warwick, UK)
Matessi^{*}, Diego (Warwick, UK)
Merkulov, Sergei (Glasgow, UK)
Metzler, David (Rice, USA)
Micallef, Mario (Warwick, UK)
Milinkovic, Darko (WisconsinMadison, USA)
Missarov, Moukadas (Kazan, Russia)
Mitsumatsu, Yoshihiko (Chuo, Japan)
Miyaoka, Yoichi (MPI Bonn, Germany)
Mohnke, Klaus (Siegen, Germany)
Mohsen, Jean Paul (ENSLyon, France)
Montaldi, James (Nice, France)
Mukai, Shigeru (Nagoya, Japan)
Munn^{*}, Jonathan M. (Warwick, UK)
Munoz, Vicente (Malaga, Spain)
Nakagawa, Yasuhiro (Tohoku, Japan)
Nasir, Sazzad Mahmud (Cambridge, UK)
Norbury^{*}, Paul (Melbourne, Australia)
Oh, YongGeun (Wisconsin, USA)
Ohba, Kiyoshi (Ochanomizu, Japan)
Ohta, Hiroshi (Nagoya, Japan)
Omoda, Yasuhiro (Kyoto, Japan)
Ono, Kaoru (Hokkaido, Japan)
Osipova^{*}, Daria (Hull, UK)
Pansu, Pierre (ParisSud, France)
Paoletti, Roberto (Pavia, Italy)
Park, Doug (Princeton, USA)
Parthasarathy, K.R. (Nottingham Trent, UK)
Patrick, George W. (Saskatchewan, Canada)
Pauly, Christian (Nice, France)
Pekarsky, Sergey (CalTech, USA)
Pidstrigatch, Victor (Warwick, UK)
Pinsonnault, Martin (Montreal, Canada)
Pinto, Alberto (Porto, Portugal)
Polterovich, Leonid (Tel Aviv, Israel)
Prajapat, Jyotshana (TIFR, India)
Prokhorov, Yuri (Lille, France)
Reid, Miles (Warwick, UK)
Robbin, Joel (Wisconsin, USA)
Roberts, Mark (Warwick, UK)
Rogers, Alice (KCL, UK)
Rudyak, Yuli (Siegen, Germany)
Rugh, Hans Henrik (Warwick, UK)
Rumynin^{*}, Dmitriy (Warwick, UK)
Rybicki, Tomasz (Rzesow, Poland)
Sadovskii, Dmitrii (Littoral, France)
Santa Cruz^{*}, Sergio (Recife, Brazil)
Schlenk, Felix (ETH Zurich, Switzerland)
Schwarz, Matthias (Stanford, USA)
Seidel, Paul (IAS Princeton, USA)
Sergeev, Armen (Steklov Moscow, Russia)
Sevennec, Bruno (ENSLyon, France)
Siburg, Karl Friedrich (Freiburg, Germany)
Sikorav, Jean Claude (Toulouse, France)
Sitta, Angela (Brunel, UK)
Sjamaar, Reyer (Cornell, USA)
Sleewaegen^{*}, Pierre (Bruxelles, Belgium)
Smith, Ivan (Oxford, UK)
Sousa Dias, Esmeralda (Lisboa, Portugal)
Stavracou^{*}, Jenny (Brussels, Belgium)
Stern, Ron (UC Irvine, USA)
Sternheimer, Daniel (Bourgogne, France)
Stipsicz, András I. (Budapest, Hungary)
Strien, Sebastian van (Warwick, UK)
Swann, Andrew (Bath, UK)
Swift, S. Timothy (Southampton, UK)
Szabó, Zoltán (Princeton, USA)
Szendröi^{*}, Balázs (Cambridge, UK)
Takakura, Tatsuru (Chuo, Japan)
Tang, Chun Chung (Cambridge, UK)
Terizakis^{*}, George (Warwick, UK)
Thomas, Charles B. (Cambridge, UK)
Tokieda, Tadashi (UrbanaChampaign, USA)
Tokunaga, Kenichi (Kyoto, Japan)
Tolman, Susan (MIT, USA)
Traynor, Lisa (Bryn Mawr, USA)
Tyurin, Andrei (Steklov Moscow, Russia)
Uhlenbeck, Karen (UT Austin, USA)
Valero^{*}, Carlos (Oxford, UK)
Valli, Giorgio (Pavia, Italy)
Vanderbauwhede, Andre (Gent, Belgium)
Vassilakis, Theodore (Brown, USA)
Vidussi, Stefano (Pavia, Italy)
Viterbo, Claude (Orsay, France)
Voronov, A. (Michigan State, USA)
Voronov, Theodore (UMIST, UK)
Waldman^{*}, Stefan (Bruxelles, Belgium)
Wang, Bryan (Adelaide, Australia)
Weber, Joachim (Warwick, UK)
Wentworth, Richard A. (UC Irvine, USA)
Wilson, Pelham (Cambridge, UK)
Wolf, Joseph (UC Berkeley, USA)
Wolfson, Jon (Michigan State, USA)
Wood^{*}, David (Oxford, UK)
Wurzbacher, Tilmann (Strasbourg, France)
Yoshioka, Akira (Tokyo, Japan)
Zakalyukin, Vladimir M. (Moscow, Russia)
There were many more comments of a similar nature.
See front page.