Department of Mathematics
Fabian Ziltener
Seminar Orientation on Research in Geometry (WISM556)
dates: October 5 and 12, 2015
place: HFG 610, mathematics department
title: Some highlights of symplectic geometry
abstract: Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in Hamilton's equation. It is related to dynamical systems, algebraic geometry, and string theory.
I will start my lectures by explaining basic notions, such as symplectic forms and Hamilton's equation. After this I will discuss the following highlights of symplectic geometry:

the existence of an exotic symplectic structure on R^{2n},

a famous conjecture by V. Arnol'd, which states a lower bound on the number of periodic orbits of a Hamiltonian system,

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.
I will motivate and explain the statements and the ideas of proof of these results and the conjecture. (Full proofs would fill an entire lecture course.)