ICMS

Symplectic Geometry and Transformation Groups

Jul 05, 2010 - Jul 09, 2010

15 South College Street, Edinburgh EH8 9AA

Organisers

Name	Institution
Cieliebak, Kai	Ludwig-Maximilians-Universität
Kedra, Jarek	University of Aberdeen & University of Szczecin
McDuff, Dusa	Barnard College, Columbia University
Polterovich, Leonid	Tel Aviv University & University of Chicago

Honorary Guest
Helmut Hofer, Institute for Advanced Study

This meeting will mark the 20th anniversary of the publication of the paper “The topology of symplectic maps” by Helmut Hofer, where he introduced a remarkable metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold. This metric is now known as the Hofer metric and is a classical notion of symplectic geometry. Hofer geometry has been extensively investigated for the last 20 years, but still there are many open problems.

Invited Speakers
Miguel Abreu
Augustin Banyaga
Misha Bialy
Yasha Eliashberg
Misha Entov
Boris Khesin
Francois Lalonde
Kaoru Ono
Yaron Ostrover
Pierre Py
Matthias Schwarz
Jean-Claude Sikorav
Claude Viterbo

Workshop funded by:
EPSRC
LMS
EMS
GMJT
ESF - CAST

PDF file of full report.  

Arrangements

Participation
Participation is by invitation only. This workshop is now full and applications are closed. The workshop will begin on Monday morning 5 July and finish on Friday afternoon 9 July 2010.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and, if so, how to apply for one. If you do require a visa, ICMS can provide a signed invitation letter.

Venue
The workshop will be held at 15 South College Street, Edinburgh. All lectures will be held in the Newhaven Lecture Theatre. To view this room and a list of the visual equipment available click here. In addition, two blackboards have recently been installed. Follow this link for a map showing the location of 15 South College Street. You may also find these maps useful: Map 1 & Map 2.

Wireless Access
The workshop venue, 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network.  For those without laptops, there will also be a couple of computers available for you to check your emails.

Travel
Information about travel to the UK and Edinburgh is available here. Lothian buses charge 1.20 GBP for a single journey and 3.00 GBP for a day ticket. Please note that the exact fare is required and no change is given.

A taxi directly from the airport will cost approximately 18.00 to 20.00 GBP to the city centre for a one-way journey.

From the airport, you can take the airport bus Airlink to the city centre. It will cost you 3.50 GBP for a single ticket, or 6.00 GBP return. More details on the airport bus Airlink is available here. The bus will take you to Waverley Bridge (next to Waverley Railway Station).

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge map. The second railway station is called Haymarket and is at the West End of the city centre.

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies that may occur on your trip.

Accommodation
If your invitation included the offer of accommodation, then ICMS will arrange this. En-suite rooms have been booked in two locations: The Kenneth MacKenzie Suite in the city centre, and Chancellor's Court in the Pollock Halls area.

Participants are also free to make their own arrangements and may claim back the cost, with receipts, up to a maximum of £55.00 a night for up to 6 nights. A list of Edinburgh accommodation of various sorts and prices is available here .

Meals and Refreshments
Morning and afternoon refreshments will be provided on each day of the workshop.

Buffet lunches will be provided in the Chapterhouse on the ground floor of 15 South College Street on Monday 5 July and Friday 9 July only.

The workshop dinner will take place on Thursday 8 July at 19.00 in The Magnum Restaurant, 1 Albany Street, Edinburgh, EH1 3PY.

The workshop grant will cover the cost of this catering.

Registration
Registration will take place 08.45 - 09.30 on Monday 5 July at 15 South College Street, Edinburgh. 

Financial Arrangements
Your individual financial arrangements will be laid out in your invitation and repeated in your final email, which will be sent shortly before the workshop.

If we have agreed to pay some of your travel costs, you will be informed by email. Reimbursement will take place after the workshop and will involve payment directly into your bank account. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with original receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world, their IBAN and SWIFT/BIC code. We cannot reimburse any item without a signed claim form and original receipts.

Programme

 Monday 5 July

8.45 – 9.30	Registration and coffee in the Chapterhouse, Ground Floor, 15 South College Street
	Chair: Kaoru Ono (Hokkaido University)
9.30 – 10.30	Yasha Eliashberg (Stanford University) More about Legendrian surgery
10.30 – 11.30	Miguel Abreu (Instituto Superior Técnico) Contact homology of good toric contact manifolds
11.30 – 12.00	Informal discussions
12.00 – 14.00	Lunch provided in the Chapterhouse, Ground Floor
	Chair: Matthias Schwarz (Universität Leipzig)
14.00 – 15.00	Francois Lalonde (Université de Montréal) Homological Lagrangian monodromy
15.00 – 15.30	Tea/Coffee
15.30 – 16.30	Yasha Savelyev (University of Massachusetts Amherst) On Hofer geometry and quantum classes

 

Tuesday 6 July

	Chair: Miguel Abreu (Instituto Superior Técnico)
9.00 – 10.00	Augustin Banyaga (Pennsylvania State University) The Hofer topology on the group of all symplectic diffeomorphisms
10.00 – 10.30	Tea/Coffee
10.30 – 11.30	Misha Bialy (Tel Aviv University) Integrable geodesic flows on surfaces
11.30 – 14.00	Lunch break
	Chair: Francois Lalonde (Université de Montréal)
14.00 – 15.00	Kaoru Ono (Hokkaido University) (Tentative) Bulk deformations in Lagrangian Floer theory
15.00 – 15.30	Tea/Coffee
15.30 – 16.30	Claude Viterbo (Centre de Mathématiques Ecole Polytechnique) Non convex Mather theory

 

Wednesday 7 July

	Chair: Yasha Eliashberg (Stanford University)
9.00 – 10.00	Boris Khesin (University of Toronto) Dynamics on diffeomorphism groups: Burgers vs. Euler
10.00 – 10.30	Tea/Coffee
10.30 – 11.30	Helmut Hofer (Institute for Advanced Study) - EMS Lecture Polyfolds and SFT
	Free afternoon

 

Thursday 8 July

	Chair: Augustin Banyaga (Pennsylvania State University)
9.00 – 10.00	Matthias Schwarz (Universität Leipzig) Floer homology for cotangent bundles
10.00 –10.30	Tea/Coffee
10.30 – 11.30	Pierre Py (University of Chicago) Quasi-morphisms and Hofer's metric
11.30 – 14.00	Lunch break
	Chair: Misha Entov (Technion - Israel Institute of Technology)
14.00 – 15.00	Yaron Ostrover (Institute for Advanced Study) On the uniqueness of Hofer's geometry
15.00 – 15.30	Tea/Coffee
15.30 – 16.30	Yiming Long (Nankai University) Multiple closed geodesics on compact simply connected manifolds
18.00 –	Workshop Dinner - The Magnum Restaurant, 1 Albany Street, Edinburgh

 

Friday 9 July

	Chair: Claude Viterbo (Centre de Mathématiques Ecole Polytechnique)
9.00 – 10.00	Misha Entov (Technion - Israel Institute of Technology) C0-functional theory of Poisson brackets
10.00 – 10.30	Tea/Coffee
10.30 – 11.30	Jean-Claude Sikorav (École Normale Supérieure de Lyon) Hofer distance of Lagrangian submanifolds in a cotangent bundle
11.30 – 12.00	Informal discussions
12.00 – 14.00	Lunch provided in the Chapterhouse, Ground Floor
	Chair: TBA
14.00 – 15.00	Sheila Sandon (Université de Nantes) An integer valued bi-invariant metric on the group of contactomorphisms of R2n x S1
15.00 – 15.30	Tea/Coffee
15.30 – 16.30	Open problem session

 

 

Presentations:

Presentation Details
Abreu, Miguel
Contact homology of good toric contact manifolds
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In this talk I will introduce toric contact manifolds and present their classification due to Banyaga-Molino, Boyer-Galicki and Lerman (2003). I will then show that any good toric contact manifold has well defined cylindrical contact homology and describe how it can be combinatorially computed from the associated moment cone. As a particular application, I will compute the cylindrical contact homology of a family of good toric contact structures arising in the work of Gauntlett-Martelli-Sparks-Waldram (2004), a group of mathematical-physicists, showing that it contains infinitely many new inequivalent contact structures on $S^2 x S^3$. This is joint work with Leonardo Macarini.
Banyaga, Augustin
The Hofer topology on the group of all symplectic diffeomorphisms
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We construct a Hofer-like metric on the group Symp(M,omega) of symplectic diffeomorphisms of a compact symplectic manifold and show that the resulting metric topology is intrinsic and the group of Hamiltonian diffeomorphisms is a closed subgroup. We get this way a new metric on the group of Hamiltonian diffeomorphisms.
Bialy, Misha
Integrable geodesic flows on surfaces
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It is a very classical problem going back to Liouville, Darboux and Poincare to describe those Riemannian metrics on surfaces which have integrable geodesic flows. I'd like to discuss new ideas and methods for this problem
Eliashberg, Yasha
More about Legendrian surgery
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I will discuss new joint work with F. Bourgeois and T. Ekholm; results which extend our paper about effects of Legendrian surgery on symplectic and contact invariants.
Entov, Michael
C^0-functional theory of Poisson brackets
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The Poisson bracket of two smooth functions on a symplectic manifold depends on their first derivatives. Nevertheless, it turns out that the Poisson bracket (as well as certain expressions involving iterated Poisson brackets of two functions) is rather sensitive to C^0-small perturbations of the functions. I will survey the C^0 functional theory of the Poisson brackets and an important role played in it by the Hofer metric and symplectic quasi-states. I will also discuss a relation between this theory and dynamics which has been found in our recent joint work with L.Buhovsky and L.Polterovich.
Hofer, Helmut
Polyfolds and SFT
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In this talk I describe some of the ideas of the polyfold approach to SFT.
Khesin, Boris
Dynamics on diffeomorphism groups: Burgers vs. Euler
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We describe a relation between curvatures of the group of volume-preserving diffeomorphisms (responsible for Lagrangian instability of ideal fluids) and the generation of shocks for potential solutions of the inviscid Burgers equation (important in mass transport). For this we characterize focal points of the group of volume-preserving diffeomorphisms, regarded as a submanifold in all diffeomorphisms and the corresponding conjugate points along geodesics in the Wasserstein space of densities. We also present the Hamiltonian framework for the corresponding mass transport problem as an infinite-dimensional Hamiltonian reduction on diffeomorphism groups, and describe its subriemannian version on a manifold with a bracket-generating distribution.
Lalonde, Francois
Homological Lagrangian monodromy
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Consider a Lagrangian submanifold in a symplectic manifold and its group of monodromies defined as the time-one maps of all Hamiltonian isotopies starting and ending at L. Using a relative form of the Seidel morphism, we will show that this group, at the homological level, is trivial when L is weakly exact.
Long, Yiming
Multiple closed geodesics on compact simply connected manifolds
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The problem of closed geodesics is a traditional and important topic in dynamical systems and differential geometry. There is a long standing conjecture that there exist infinitely many distinct closed geodesics on every compact Riemannian manifold. The current interest on this problem is on compact simply connected manifolds, specially spheres. So far not much is known on the multiplicity of closed geodesics on such manifolds when their dimensions are at least 3. Recently, Dr. Huagui Duan and myself proved the following Theorem: There exist at least two distinct closed geodesics on every compact simply connected Riemannian (or Finsler) manifold whose dimension is at least 2. In this lecture, I shall give a survey on the study of the problem of closed geodesics and explain some ideas in the proof of the above theorem.
Ono, Kaoru
Bulk deformations in Lagrangian Floer theory
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Based on the joint work with K. Fukaya, Y.-G. Oh, H. Ohta, I plan to explain bulk deformations in Lagrangian Floer theory and its applications. If time allows, I would also like to include implications for Lagrangian torus fibers in compact toric manifolds.
Ostrover, Yaron
On the uniqueness of Hofer's geometry
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In this talk we address the question whether Hofer's metric is unique among the Finsler-type bi-invariant metrics on the group of Hamiltonian diffeomorphisms. The talk is based on a recent joint work with Lev Buhovsky.
Py, Pierre
Quasi-morphisms and Hofer's metric
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In this talk we will survey some of the recent constructions of quasi-morphisms on groups of Hamiltonian diffeomorphisms (Floer theoretic as well as topological ones). In particular we will discuss some open problems related to Hofer's metric: continuity of Calabi quasi-morphisms on surfaces of positive genus, existence of $C^{0}$-robust bounds on Hofer's metric, and description of the mixed action-Maslov quasi-morphism. Along the way, we will survey some of the recent works of M. Entov and L. Polterovich, M. Rosenberg, F. Zapolsky, as well as of the speaker.
Sandon, Sheila
An integer valued bi-invariant metric on the group of contactomorphisms of R^{2n} x S^1
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I will describe a generalization to the contact case of the Viterbo metric on the Hamiltonian group of R^{2n}, and discuss how it is related to the theory of orderability and contact non-squeezing of Eliashberg, Kim and Polterovich.
Savelyev, Yasha
On Hofer geometry and quantum classes
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We describe quantum characteristic classes and their natural applications in Hofer geometry. One of the main classical tools will be Bott periodicity for the unitary group.
Schwarz, Matthias
Floer homology for cotangent bundles
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Floer homology on cotangent bundles for Hamiltonians equivalent to the geodesic flow is isomorphic to loop space homology. We show how this isomorphism extends to further structures such as loop product and cup product by chain level constructions.
Sikorav, Jean-Claude
Hofer distance of Lagrangian submanifolds in a cotangent bundle
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We study the Hofer distance on the space of Lagrangian submanifolds of $T^*M$ which are Hamiltonian istotopic to the zero section, where $M$ is a closed manifold. We prove a Hamilton-Jacobi differential inequality for a suitable generating function of the isotopy, which we use to prove some lower bounds on the distance.
Viterbo, Claude
Non convex Mather theory
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Using recent results and methods of symplectic homogenization, we shall show how one can construct invariant measures for Hamiltonian flows in non-convex situations.

Participants

Name	Institution
Abreu, Miguel	Instituto Superior Técnico
Albers, Peter	Purdue University
Banyaga, Augustin	Pennsylvania State University
Batoreo, Marta	UC Santa Cruz
Bialy, Misha	Tel Aviv University
Bielawski, Roger	University of Leeds
Brandenbursky, Michael	Technion-Israel Institute of Technology
Cieliebak, Kai	Ludwig-Maximilians-Universität
Diogo, Luis	Stanford University
Eliashberg, Yasha	Stanford University
Entov, Michael	Technion - Israel Institute of Technology
Evans, Jonny	University of Cambridge
Fuchs, Urs	Purdue University
Gal, Swiatoslaw	Uniwersytet Wrocławski
Hein, Doris	UC Santa Cruz
Hofer, Helmut	Institute for Advanced Study
Humiliere, Vincent	Institut Mathématique de Jussieu
Iriyeh, Hiroshi	Tokyo Denki University
Karshon, Yael	University of Toronto
Katic, Jelena	University of Belgrade
Kedra, Jarek	University of Aberdeen & University of Szczecin
Khesin, Boris	University of Toronto
Lalonde, Francois	Université de Montréal
Lanzat, Sergei	Technion - Israel Institute of Technology
Laudenbach, Francois	Université de Nantes
Long, Yiming	Nankai University
McDuff, Dusa	Barnard College, Columbia University
Milinkovic, Darko	University of Belgrade
Ono, Kaoru	Hokkaido University
Ostrover, Yaron	Institute for Advanced Study
Polterovich, Leonid	Tel Aviv University & University of Chicago
Py, Pierre	University of Chicago
Ranicki, Andrew	University of Edinburgh
Reis, Rui	University of Aberdeen
Sandon, Sheila	Université de Nantes
Savelyev, Yasha	University of Massachusetts Amherst
Schlenk, Felix	Université de Neuchâtel
Schwarz, Matthias	Universität Leipzig
Sepe, Daniele	University of Edinburgh
Siburg, Karl Friedrich	Technische Universität Dortmund
Sikorav, Jean-Claude	École Normale Supérieure de Lyon
Smith, Ivan	University of Cambridge
Spacil, Oldrich	University of Aberdeen
Spaeth, Peter	KIAS
Usher, Michael	University of Georgia
Viterbo, Claude	Ecole Normale Supérieure
Waldron, Jack	University of Oxford
Wysocki, Kris	Pennsylvania State University
Zapolsky, Frol	Max Planck Institute for Mathematics in the Sciences
Ziltener, Fabian	University of Toronto