1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

2 INTRODUCTION

by beginning with Moser's ar gument for Darboux's theorem and finishing with a

sketch of the proof of Gromov's nonsqueezing theorem via the theory of holomorphic

curves and the concept of symplectic capacities.

Hofer shows in his course how the theory of holomorphic curves can be used

to study problems in Hamiltonian dynamics on contact manifolds. In recent years,

there have been a number of exciting counterexamples to the classic Seifert conjec-

ture, which states that a nowhere vanishing smooth vector field on S^3 must have

a periodic orbit. It is currently conjectured that it is even possible to construct a

volume-preserving counterexample. On the other hand, periodic orbit phenomena

do occur in the field of Hamiltonian dynamics under certain convexity assumptions.

Weinstein conjectured that for a special class of Hamiltonian vector fields, known

as Reeb vector fields on contact manifolds, periodic trajectories should always exist.

In his course Hofer presents his own proof of the Weinstein conjecture for all contact

structures on the 3-sphere, as well as on some other closed contact 3-manifolds.

In 1994, Seiberg and Witten caused a coup in smooth 4-dimensional topol-

ogy when they introduced the "Seiberg-Witten" equations. By an appropriate

counting of the solutions to these equations, one obtains new invariants of smooth

4-manifolds that are, in a certain sense, easier to handle than the instanton in-

variants introduced earlier by S. Donaldson. In his course, Taubes describes his

own fundamental theorem that on a symplectic 4-manifold with bi > 1, these

SW-invariants coincide with the "Gromov invariant", Gr, which is calculated by
counting pseudo holomorphic curves. This result provides a bridge between the
smooth and symplectic categories in dimension 4. Starting on a very basic level of
differential geometry,· Taubes's lectures develop the theory of the Seiberg-Witten

and the Gromov invariants and show how the statement SW = Gr can be worked

out.
The lectures of Salamon are devoted to symplectic Floer homology theory.
Floer married Gromov's theory of pseudo holomorphic curves and the Conley-
Zehnder variational technique in order to prove Arnold's conjecture, which states
that, in the nondegenerate case, the number of fixed points of a Hamiltonian dif-
feomorphism should be bounded below by the sum of the Betti numbers of the
underlying symplectic manifold. Several mathematicians contributed to the proof
of this conjecture. Currently it has been proved for all symplectic manifolds, but
only in terms of rational Betti numbers. In his lectures, Salamon outlines the proof
of the conjecture under certain restrictions. However, he discusses all major com-
ponents of the proof for the general case, including Novikov rings and the theory
of stable curves.
The theory of stable holomorphic curves is a starting point for Givental's course,
which is devoted to an interaction between symplectic geometry and theoretical
physics. The Gromov-Witten invariants defined by counting holomorphic curves in
symplectic manifolds can be organized into a fundamental system of solutions of a
set of linear differential equations, called Witten-Dijkgraaf-Verlinde-Verlinde equa-
tions, which arises in physics. Givental's lectures give a glimpse of a fascinating
new area involving an interplay of deep ideas from symplectic geometry, singular-
ity theory, and physics. The current development in this area was concentrated
around the mirror symmetry conjecture, where a recent breakthrough is due to
Givental. Givental's last lecture provides his point of view on the mirror symmetry
phenomenon.