Department of Mathematics

Fabian Ziltener

## Holomorphic curves and symplectic topology (WISM547)

### blokken 3 + 4, 2014/ 2015

### General information

**Target audience**: This course is aimed at master and Ph.D.-students in mathematics and theoretical physics.

**Hoorcollege**: Fridays 9:00-10:45, MIN 207, except for week 22, when the hoorcollege will be in MIN 204

**Werkcollege**: Fridays 11:00-12:45, usually in MIN 016, for exceptions see osiris (Click on Rooster.)

**Presentations**: Each student will be asked to present the solutions to two of the exercises of the assignments during the werkcollege.

**Tentamen**: There will be a written or oral final exam. Place and time: tba.

**Eindcijfer**: The presentations of the solutions of the exercises will count for 20 % of the final mark. The final exam will count for 80 % of this mark.
### Aims:

Provide an overview over some important results in symplectic topology, such as Gromov’s nonsqueezing theorem, and an introduction to the technique of (pseudo-)holomorphic curves.
### Course description

Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in Hamilton’s equation. It is related to the theory of dynamical systems and - via holomorphic curves – to algebraic geometry. Symplectic topology is a subfield of symplectic geometry, in which global properties of symplectic manifolds are studied. Some highlights of symplectic topology are:

M. Gromov’s celebrated nonsqueezing theorem, which states that a ball in R^{2n} of radius bigger than 1 does not symplectically embed into a symplectic cylinder of radius 1
obstructions to Lagrangian embeddings
the existence of an exotic symplectic structure on R^{2n}

I will explain and motivate the statements and main ideas of the proofs of these results. The proofs are based on holomorphic curves. Such a curve is a map from a Riemann surface to an almost complex manifold that solves the Cauchy-Riemann equation. Crucial ingredients of the proofs are bubbling, Fredholm, and transversality results for this equation.

I will indicate how by counting holomorphic curves, we obtain symplectic invariants, the Gromov-Witten invariants. If time permits, I will also discuss a famous conjecture by V. Arnol’d, which states a lower bound on the number of periodic orbits of a Hamiltonian system. Versions of this conjecture have been proven with the help of Floer homology. This homology is based on a version of the Cauchy-Riemann equation involving a Hamiltonian perturbation.
### Prerequisites

Basics of differential geometry (WISB342) and symplectic geometry (e.g. the mastermath course taught in Fall 2014), such as symplectic manifolds, the canonical symplectic form on a cotangent bundle, Lagrangian submanifolds, Hamiltonian diffeomorphisms, Liouville’s, Darboux’s, and Weinstein’s theorems. (Lecture notes containing the relevant material are available upon request. Please send me an e-mail at f dot ziltener at-sign uu dot nl.) Some knowledge of functional analysis (WISB315) and partial differential equations will also be helpful, in particular familiarity with Sobolev spaces and Fredholm operators.
### Literature

D. McDuff and D.A. Salamon, J-holomorphic Curves and Symplectic Topology, AMS Colloquium Publications, Volume 52, 2004.

D. McDuff and D.A. Salamon, Introduction to Symplectic Topology, 2. edition, Oxford Mathematical Monographs, The Clarendon Press, 1998.
### Learning goals

After completion of the course, the student

understands the statements and relevance of the highlights of symplectic topology discussed in this course
is able to discuss these results in examples
understands the main ideas behind the technique of holomorphic curves
is able to present mathematics in a concise and understandable way