## Holomorphic curves and symplectic topology (WISM547)

### 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