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Symplectic Geometry Tara S THE GRADUATE STUDENT SECTION WHAT IS... Symplectic Geometry Tara S. Holm Communicated by Cesar E. Silva In Euclidean geome- depending smoothly on the point 푝 ∈ 푀.A 2-form try in a vector space 휔 ∈ Ω2(푀) is symplectic if it is both closed (its exterior Symplectic over ℝ, lengths and derivative satisfies 푑휔 = 0) and nondegenerate (each structures are angles are the funda- function 휔푝 is nondegenerate). Nondegeneracy is equiva- mental measurements, lent to the statement that for each nonzero tangent vector floppier than and objects are rigid. 푣 ∈ 푇푝푀, there is a symplectic buddy: a vector 푤 ∈ 푇푝푀 In symplectic geome- so that 휔푝(푣, 푤) = 1.A symplectic manifold is a (real) holomorphic try, a two-dimensional manifold 푀 equipped with a symplectic form 휔. area measurement is Nondegeneracy has important consequences. Purely in functions or the key ingredient, and terms of linear algebra, at any point 푝 ∈ 푀 we may choose the complex numbers a basis of 푇푝푀 that is compatible with 휔푝, using a skew- metrics. are the natural scalars. symmetric analogue of the Gram-Schmidt procedure. We It turns out that sym- start by choosing any nonzero vector 푣1 and then finding a plectic structures are much floppier than holomorphic symplectic buddy 푤1. These must be linearly independent functions in complex geometry or metrics in Riemannian by skew-symmetry. We then peel off the two-dimensional geometry. subspace that 푣1 and 푤1 span and continue recursively, The word “symplectic” is a calque introduced by eventually arriving at a basis Hermann Weyl in his textbook on the classical groups. 푣1, 푤1, … , 푣푑, 푤푑, That is, it is a root-by-root translation of the word “complex” from the Latin roots which contains an even number of basis vectors. So symplectic manifolds are even-dimensional. This also com – plexus, allows us to think of each tangent space as a complex meaning “together – braided,” into the Greek roots with vector space where each 푣푖 and 푤푖 span a complex the same meaning, coordinate subspace. Moreover, the top wedge power, 휎휐휇 – 휋휆휀휅휏휄휅ó휍. 휔푑 ∈ Ω2푑(푀), is nowhere-vanishing, since at each tangent Weyl suggested this word to describe the Lie group that space, 푑 preserves a nondegenerate skew-symmetric bilinear form. 휔 (푣1, … , 푤푑) ≠ 0. Prior to this, that Lie group was called the “line complex In other words, 휔푑 is a volume form, and 푀 is necessarily group” or the “Abelian linear group” (after Abel, who orientable. studied the group). Symplectic geometry enjoys connections to algebraic A differential 2-form 휔 on (real) manifold 푀 is a gadget combinatorics, algebraic geometry, dynamics, mathemati- that at any point 푝 ∈ 푀 eats two tangent vectors and cal physics, and representation theory. The key examples spits out a real number in a skew-symmetric, bilinear underlying these connections include: way. That is, it gives a family of skew-symmetric bilinear 2 1 (1) 푀 = 푆 = ℂ푃 with 휔푝(푣, 푤) = signed area of the functions parallelogram spanned by 푣 and 푤; 휔푝 ∶ 푇푝푀 × 푇푝푀 → ℝ (2) 푀 any Riemann surface as in Figure 1 with the area for 휔 as in (1); Tara S. Holm is professor of mathematics at Cornell University and Notices consultant. She is grateful for the support of the Si- mons Foundation. Her e-mail address is [email protected]. For permission to reprint this article, please contact: ⋯ [email protected]. Figure 1. The area form on a Riemannian surface DOI: http://dx.doi.org/10.1090/noti1450 defines its symplectic geometry. 1252 Notices of the AMS Volume 63, Number 11 THE GRADUATE STUDENT SECTION 2푑 (3) 푀 = ℝ with 휔푠푡푑 = ∑ 푑푥푖 ∧ 푑푦푖; the second type. Let us briefly examine results of each (4) 푀 = 푇∗푋, the cotangent bundle of any manifold kind. 푋, thought of as a phase space, with 푝-coordinates If a symplectic manifold exhibits a certain flavor of coming from 푋 being positions, 푞-coordinates in symmetries in the form of a Lie group action, then it the cotangent directions representing momentum admits a momentum map. This gives rise to conserved directions, and 휔 = ∑ 푑푝푖 ∧ 푑푞푖; quantities such as angular momentum, whence the name. (5) 푀 any smooth complex projective variety with 휔 The first example of a momentum map is the height induced from the Fubini-Study form (this includes function on a 2-sphere (Figure 2). smooth normal toric varieties, for example); (6) 푀 = 풪휆 a coadjoint orbit of a compact connected semisimple Lie group 퐺, equipped with the Kostant- Kirillov-Souriau form 휔. For the group 퐺 = 푆푈(푛), this class of examples includes complex projective 푛−1 푛 space ℂ푃 , Grassmannians G 푟푘(ℂ ), the full flag variety ℱℓ(ℂ푛), and all other partial flag varieties. Plenty of orientable manifolds do not admit a symplectic structure. For example, even-dimensional spheres that are at least four-dimensional are not symplectic. The reason is 1 2 that on a compact manifold, Stokes’ theorem guarantees Figure 2. The momentum map for 푆 acting on 푆 by that [휔] ≠ 0 ∈ 퐻2(푀; ℝ). In other words, compact rotation. symplectic manifolds must exhibit nonzero topology in In this case, the conserved quantity is angular momen- degree 2 cohomology. The only sphere with this property tum, and the height function is the simplest example of is 푆2. a perfect Morse function on 푆2. When the Lie group is a Example (3) gains particular importance because of the product of circles 푇 = 푆1 × ⋯ × 푆1, we say that the mani- nineteenth century work of Jean Gaston Darboux on the fold is a Hamiltonian 푇-space, and the momentum map structure of differential forms. A consequence of his work is denoted Φ ∶ 푀 → ℝ푛. In 1982 Atiyah and independently is Guillemin and Sternberg proved the Convexity Theorem Darboux’s Theorem. Let 푀 be a two-dimensional symplec- (see Figure 3): tic manifold with symplectic form 휔. Then for every point Convexity Theorem. If 푀 is a compact Hamiltonian 푇- 푝 ∈ 푀, there exists a coordinate chart 푈 about 푝 with space, then Φ(푀) is a convex polytope. It is the convex hull coordinates 푥 , … , 푥 , 푦 , … , 푦 so that on this chart, 1 푑 1 푑 of the images Φ(푀푇) of the 푇-fixed points. 푑 휔 = ∑ 푑푥푖 ∧ 푑푦푖 = 휔푠푡푑. 푖=1 This makes pre- cise the notion that symplectic geome- Tools from the try is floppy. In Figure 3. Atiyah and Guillemin-Sternberg proved that Riemannian geome- 1970s and 1980s set if a symplectic manifold has certain symmetries, the try there are local the stage for image of its momentum map is a convex polytope. invariants such as curvature that dis- dramatic progress This provides a deep connection between symplectic tinguish metrics. and algebraic geometry on the one hand and discrete Darboux’s theorem in symplectic geometry and combinatorics on the other. Atiyah’s proof says that symplectic demonstrates that the momentum map provides Morse forms are all lo- geometry and functions on 푀 (in the sense of Bott), bringing the power cally identical. What of differential topology to bear on global topological remains, then, are topology. questions about 푀. Momentum maps are also used to global topological construct symplectic quotients. Lisa Jeffrey will discuss questions such as, What is the cohomology ring of a the cohomology of symplectic quotients in her 2017 particular symplectic manifold? and more subtle sym- Noether Lecture at the Joint Mathematics Meetings, and plectic questions such as, How large can the Darboux many more researchers will delve into these topics during charts be for a particular symplectic manifold? the special session Jeffrey and I are organizing. Two tools developed in the 1970s and 1980s set the Through example (2), we see that two-dimensional sym- stage for dramatic progress in symplectic geometry and plectic geometry boils down to area-preserving geometry. topology. Marsden and Weinstein, Atiyah, and Guillemin Because symplectic forms induce volume forms, a natural and Sternberg established properties of the momentum question in higher dimensions is whether symplectic ge- map, resolving questions of the first type. Gromov intro- ometry is as floppy as volume-preserving geometry: can a duced pseudoholomorphic curves to probe questions of manifold be stretched and squeezed in any which way so December 2016 Notices of the AMS 1253 THE GRADUATE STUDENT SECTION long as volume is pre- the common underlying tool. The constructions rely on served? Using pseudoholo- subtle arguments in complex analysis and Fredholm Gromov proved morphic curves, Gromov theory. These techniques are essential to current work that symplectic dismissed this possibility, in symplectic topology and mirror symmetry, and they proving that symplectic provide an important alternative perspective on invariants maps are more maps are more rigid than of four-dimensional manifolds. volume-preserving ones. In Further details on momentum maps may be found in rigid than ℝ2푑 we let 퐵2푑(푟) denote [CdS], and [McD-S] gives an account of pseudoholomorphic the ball of radius 푟. In curves. volume- 1985 Gromov proved the preserving nonsqueezing theorem (see References Figure 4): [CdS] A. Cannas da Silva, Lectures on Symplectic Geometry, ones. Lecture Notes in Mathematics, 1764, Springer-Verlag, Nonsqueezing Theorem. Berlin, 2001. MR 1853077 There is an embedding [McD-S] D. McDuff and D. Salamon, Introduction to Symplec- 퐵2푑(푅) ↪ 퐵2(푟) × ℝ2푑−2 tic Topology, Oxford Mathematical Monographs, Oxford University Press, New York, 1995. MR 1373431 preserving 휔푠푡푑 if and only if 푅 ≤ 푟. One direction is straightforward: if 푅 ≤ 푟, then 퐵2푑(푅) ⊆ 퐵2(푟) × ℝ2푑−2. To find an obstruction to ABOUT THE AUTHOR the existence of such a map, Gromov used a pseudo- ABOUT THE AUTHOR In addition to mathematics, holomorphic curve in 퐵2(푟) × ℝ2푑−2 and the symplectic In addition to mathematics, Tara Holm enjoys gardening, embedding to produce a minimal surface in 퐵2푑(푅), Tara Holm enjoys gardening, cooking with her family, and forcing 푅 ≤ 푟.
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