## Highlights - Symplectic topology is the study of the global phenomenon of symplectic geometry\. In contrast the local structure of a symplectic manifold is, by Darboux’s theorem, always equivalent to the standard structure on Euclidean space\. Hence there cannot be any local invariants in symplectic geometry\. This should be contrasted with Riemannian geometry where the curvature provides such local invariants\. These local invariants severely restrict the group of isometries and give rise to an infinite dimensional variety of nonequivalent Riemannian metrics\. (Dusa McDuff 9) - Here are some vaguely worded but typical questions: \(1\) Which manifolds support a symplectic structure? What symplectic invariants are there to distinguish one from another? \(ii\) Are there any special distinguishing features of Hamiltonian flows on arbitrary compact energy surfaces in Euclidean space? For example, must they always have a periodic orbit? \(iii\) Must a symplectomorphism always have ‘a lot’ of fixed points? \(iv\) What can be said about the shape of a symplectic image of a ball? Can it be long and thin, or must it always be in some sense round? \(v\) Is there a geometric way to understand the fact that a symplectic structure makes 2-dimensional measurements? \(A symplectic structure is a special kind of 2-form, but what geometric meaning does that have?\) (Dusa McDuff 9) - It establishes Darboux’s theorem and Moser’s stability theorem, and also contains an introduction to contact geometry, the odd-dimensional analogue of symplectic geometry\. (Dusa McDuff 13) - Consider the problem of minimizing the action integral (Dusa McDuff 20) - Consider a system whose configurations are described by points x in Fuclidean space R® which move along trajectories z\(t\)\. As we shall see in Lemma 1\.1 below, the assumption that these paths minimize some action functional gives rise to a system of n second-order differential equations called the Euler-Lagrange equations of the variational problem\. (Dusa McDuff 20) - We will then show how these equations can be transformed into a Hamiltonian system of 2n first-order equations\. (Dusa McDuff 20) - The function L is called the Lagrangian of this variational problem\. (Dusa McDuff 21) - A minimal path x : [to,t1] = R™ is a solution of the Euler— Lagrange equations (Dusa McDuff 21) - As alwaysin a variational problem, the Euler-Lagrange equations correspond to critical points of the functional under consideration rather than to actual minima or maxima\. (Dusa McDuff 21) - every symplectomorphism preserves volume\. (Dusa McDuff 39) - It has long been suspected that the group of symplectomorphisms is significantly smaller than that of volume-preserving diffeomorphisms, (Dusa McDuff 39) - However, there was po result which pinpointed a difference until Gromov proved his celebrated nonsqueezing theorem in 1985\. This says that a standard symplectic ball cannot be symplectically embedded into a thin cylinder\. (Dusa McDuff 39) - The minimum number of critical points of a function on a compact manifold M is a topological invariant of that manifold\. The minimum is taken over all smooth functions f : M — R and we denote it by Crit\(M\)\. It is obvious that Crit\(M\) > 2 since every nonconstant function must have a distinct maximum and minimum\. (Dusa McDuff 43) - The latter implies that the number of fixed (Dusa McDuff 43) - points of a map ¢ : M — M which is homotopic to the identity is greater than or equal to the number of zeros of a vector field on M \(when these are counted with appropriate multiplicities\) which is the alternating sum of the Betti numbers\. (Dusa McDuff 44) - The fact that the Arnold conjecture gives sharper estimates for the number of fixed points than the Lefschetz fixed point theorem is a global version of the observation that a function on a manifold must have more critical points than a vector field \(or 1-form\) must have zeros\. (Dusa McDuff 44) - The latter statement has a direct interpretation in symplectic geometry since the differential of a function is an exact Lagrangian submanifold of the cotangent bundle and the critical points are the intersections with the zero section\. (Dusa McDuff 44)