1 July - 31 December 1994

**Organisers**: S Donaldson (*Oxford*), D McDuff (*Stony Brook*), D Salamon (*Warwick*), C Thomas (*Cambridge*)

Surfaces known as symplectic manifolds arise in many branches of pure and applied mathematics, for example they provide the natural framework in which to discuss classical mechanics. There has been dramatic progress in recent years in understanding their properties drawing on results from many disciplines, including differential geometry and topology, global analysis and the theory of partial differential equations. The programme will bring together mathematicians from all these fields together with theoretical physicists with the aim of further developing the theory for example by exploiting the parallelism between Yang-Mills theory in four dimensions (which was originally constructed by physicists the strong nuclear force between elementary particles) and symplectic geometry in spaces of arbitrary even dimension.

The following are expected to participate in the programme for all or part of the periods indicated:

**Scalia-Tomba** (Stockholm) **Scalia-Tomba** (Stockholm) **Arnold** (Moscow) **Kronheimer** (Oxford) **Audin** (Strasbourg) **Lu** (Arizona) ** Bates** (Berkeley) **MacKay** (Warwick) ** Baily **(Tel Aviv) **McDuff** (Stony Brook) ** Braam** (Oxford) **Oh** (Wisconsin) ** Donaldson** (Oxford) **Ono** (Bonn/Toyko) **Eells** (Cambridge) **Pansu** (Paris) **Eliashberg **(Stanford) ** Parker **(Michigan State) **Geiges** (Cambridge) **Polterovich **(Tel Aviv) ** Ginzburg** (Stanford) ** Rabinowitz** (Wisconsin) **Giroux **(Lyon) ** Salamon **(Warwick) ** Givental** (Berkeley) **Thomas** (Cambridge) ** Gompf **(Texas) ** Traynor** (Stanford) **Gonzalo** (Madrid) ** Van **(Bonn) ** Hofer **(Zurich) ** Weinstein **(Berkeley) ** Kirwan** (Oxford) ** Wolfson** (Michigan State) ** Kotschik** (Basel) **Xu** (UC Davis)

There will be a regular seminar programme throughout the period. As well as a short topical meetings, there will be the following workshops and conferences:

17--30 July:

**Instructional conference on 3--Dimensional Contact and 4-Dimensional Symplectic Manifolds**Principle speakers: Y Eliashberg, R Gompf, H Hofer, D McDuff.

25 Sept--8 Oct:

**Workshop on Floer Homology and Symplectic Invariants.**Organisers: DA Salamon and CB Thomas.

Financial support for this workshop has been requested from the EC and applications to participate from young mathematicians from Community Universities will be particularly welcomed. This workshop will be very wide ranging in that we shall try and cover as many of the recent developments in Floer theory as possible, together with more analytic topics such as capacities.

30 Oct--12 Nov:

**Workshop on Applications of Symplectic Geometry.**Organisers: L Jeffrey and others.

The topics to be covered include moduli spaces and the moment map, cohomology and localisation formulae, group representations, integrable systems.

4--17 December:

**Conference on Symplectic Manifolds, Instantons and Algebraic Geometry.**Organisers: SK Donaldson and others.

Besides the relations between instanton invariants and the topology of symplectic 4-manifolds and complex surgaces, this conference is also intended to pull together and exploit the ideas developed in the first three special meetings. Continuing from the workshops, for example, there will be further talks on symplectic homology, mirror symmetry, quantum cohomology, and links with theoretical physics.