View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CiteSeerX WHAT IS SYMPLECTIC GEOMETRY? DUSA MCDUFF Abstract. In this talk we explain the elements of symplectic geometry, and sketch the proof of one of its foundational results — Gromov’s nonsqueezing theorem — using J-holomorphic curves. 1. First notions Symplectic geometry is an even dimensional geometry. It lives on even dimensional spaces, and measures the sizes of 2-dimensional objects rather than the 1-dimensional lengths and angles that are familiar from Euclidean and Riemannian geometry. It is naturally associated with the field of complex rather than real numbers. However, it is not as rigid as complex geometry: one of its most intriguing aspects is its curious mixture of rigidity (structure) and flabbiness (lack of structure). In this talk I will try to describe some of the new kinds of structure that emerge. First of all, what is a symplectic structure? The concept arose in the study of classical mechanical systems, such as a planet orbiting the sun, an oscillating pendulum or a falling apple. The trajectory of such a system is determined if one knows its position and velocity (speed and direction of motion) at any one time. Thus for an object of unit mass moving in a given straight line one needs two pieces of information, the position q and velocity (or more correctly momentum) p :=q ˙. This pair of real numbers 2 (x1, x2) := (p, q) gives a point in the plane R . In this case the symplectic structure ω is an area form (written dp ∧ dq) in the plane. Thus it measures the area of each open region S in the plane, where we think of this region as oriented, i.e. we choose a direction in which to traverse its boundary ∂S. This means that the area is signed, i.e. as in Figure 1.1 it can be positive or negative depending on the orientation. By Stokes’ theorem, this is equivalent to measuring the integral of the action p dq round the boundary ∂S. This might seem a rather arbitrary measurement. However, mathematicians in the nineteenth century proved that it is preserved under time evolution. In other words, if a set of particles have positions and velocities in the region S1 at the time t1 then at any later time t2 their positions and velocities will form a region S2 with the same area. Area also has an interpretation in modern particle (i.e. quantum) physics. Heisenberg’s Uncertainty Principle says that we can no longer know both position and velocity to Date: March 31, 2009, revised June 23 2009. Talk given at the European Women in Mathematics conference of Sept 2007 in Cambridge, UK.. Partially supported by NSF grant DMS 0604769. 1 2 DUSA MCDUFF Figure 1.1. The area of the region S1 is positive, while that of S2 is negative. an arbitrary degree of accuracy. Thus we should not think of a particle as occupying a single point of the plane, but rather lying in a region of the plane. The Bohr-Sommerfeld quantization principle says that the area of this region is quantized, i.e. it has to be an integral multiple of a number called Planck’s constant. Thus one can think of the symplectic area as a measure of the entanglement of position and velocity. An object moving in the plane has two position coordinates q1, q2 and correspond- ingly two velocity coordinates p1 =q ˙1, p2 =q ˙2 that measure its speed in each direction. So it is described by a point 4 (x1, x2, x3, x4) = (p1, q1, p2, q2) ∈ R 4 in the 4-dimensional space R . The symplectic form ω now measures the (signed) area 4 of 2-dimensional surfaces S in R by adding the areas of the projections of S to the (x1, x2)-plane and the (x3, x4)-plane. Thus, as is illustrated in Figure 1.2, ω(S) = area(pr12(S)) + area(pr34(S)). Notice that ω(S) can be zero: for example S might be a little rectangle in the x1, x3 directions which projects to a line under both pr12 and pr34. More technically, ω is a differential 2-form written as ω = dx1 ∧ dx2 + dx3 ∧ dx4, R and we evaluate the area ω(S) = S ω by integrating this form over the surface S.A similar definition is made for particles moving in n-dimensions. The symplectic area form ω is again the sum of contributions from each of the n pairs of directions: (1.1) ω0 = dx1 ∧ dx2 + dx3 ∧ dx4 + ··· + dx2n−1 ∧ dx2n. We call this form ω0 because it is the standard symplectic form on Euclidean space. The letter ω is used to designate any symplectic form. To be even more technical, one can define a symplectic form ω on any even dimen- sional smooth (i.e. infinitely differentiable) manifold M as a closed, nondegenerate 2-form, where the nondegeneracy condition is that for each nonzero tangent direction v there is another direction w such that the area ω(v, w) of the little (infinitesimal) parallelogram spanned by these vectors is nonzero. (For a geometric interpretation of WHAT IS SYMPLECTIC GEOMETRY? 3 Figure 1.2. The symplectic area ω(S) is the sum of the area of its pro- jection pr12(S) to the plane (x1, x2) = (p1, q1) given by the velocity and position in the first direction together with the area of the correspond- ing projection pr34(S) for the two coordinates in the second direction. I have drawn the first 3 coordinates; the fourth is left to your imagina- tion. Figure 1.3. The fact that ω is closed means that the symplectic area of a surface S with boundary does not change as S moves, provided that the boundary is fixed. Thus in (I) the surfaces S and S0 have the same area. Diagram (II) illustrates the nondegeneracy condition: for any direction v at least one of the family of 2-planes spanned by v and a varying other direction w has non zero area. these conditions on ω see Figure 1.3.) A manifold is said to be symplectic or to have a symplectic structure if it is provided with a symplectic form. 4 DUSA MCDUFF The first important theorem in symplectic geometry is that locally1 all symplectic forms are the same. Darboux’s Theorem: Given a symplectic form ω on a manifold M and any point on M one can always find coordinates (x1, . , x2n) defined in an open neighborhood U of this point such that in this coordinate system ω is given on the whole open set U by formula (1.1). This is very different from the situation in the usual (Riemannian) geometry where one can make many local measurements (for example involving curvature) that distin- guish among different structures. Darboux’s theorem says that all symplectic structures are locally indistinguishable. Of course, as mathematicians have been discovering in the past 20 years, there are many very interesting global invariants that distinguish different symplectic structures. But most of these are quite difficult to define, often in- volving deep analytic concepts such as the Seiberg–Witten equations or J-holomorphic curves. A symplectic form ω has an important invariant, called its cohomology class [ω]. This class is determined by the areas ω(S) of all closed2 surfaces S in M. In fact, for compact M the class [ω] is determined by a finite number of these areas ω(Si) and so contains only a finite amount of information. Cf. Figure 1.3 where we pointed out that the area ω(S) does not change if we move S around. There is a similar flabbiness in the symplectic structure itself. A fundamental theo- rem due to Moser says that one cannot change the symplectic form in any important way by deforming it, provided that the cohomology class is unchanged. More precisely, if ωt, t ∈ [0, 1], is a smooth path of symplectic forms such that [ω0] = [ωt] for all t, then all these forms are “the same” in the sense that one can make them coincide by moving the points of M appropriately.3 The important point here is that we cannot find new structures by deforming the old ones, provided that we fix the integrals of our forms over all closed surfaces. This result is known as Moser’s Stability Theorem, and is an indication of robustness of structure.4 2. Symplectomorphisms Another consequence of the lack of local features that distinguish between different symplectic structures is that there are many ways to move the points of the underlying space M without changing the symplectic structure ω. Such a movement is called a symplectomorphism. This means first that 1This means “on suitably small open sets”. 2A closed surface is something like the surface of a sphere or donut; it has no edges and no holes. 3In technical language we say that these forms are all diffeomorphic. 4For precise statements, many proofs, and a list of references on all the topics mentioned here see [9]. There are also other more elementary books such as Cannas [2]. For simplicity, we shall only work here in dimensions 2 and 4. But all the results have higher dimensional analogs. WHAT IS SYMPLECTIC GEOMETRY? 5 • φ is a diffeomorphism, that is, it is a bijective (one to one and onto) and smooth (infinitely differentiable) map φ : M → M, giving rise to the movement x 7→ φ(x) of the points x of the space M; and second that • it preserves symplectic area, i.e.