THE GRADUATE STUDENT SECTION

WHAT IS. . .

Symplectic GeometryTara S. Holm

Communicated by Cesar E. Silva

Symplecticstructures arefloppier thanholomorphicfunctions or

metrics.

In Euclidean geome-try in a vector spaceover ℝ, lengths andangles are the funda-mental measurements,and objects are rigid.In symplectic geome-try, a two-dimensionalarea measurement isthe key ingredient, andthe complex numbersare the natural scalars.It turns out that sym-

plectic structures are much floppier than holomorphicfunctions in complex geometry or metrics in Riemanniangeometry.

The word “symplectic” is a calque introduced byHermann Weyl in his textbook on the classical groups.That is, it is a root-by-root translation of the word“complex” from the Latin roots

com – plexus,meaning “together – braided,” into the Greek roots withthe same meaning,

𝜎𝜐𝜇 – 𝜋𝜆𝜀𝜅𝜏𝜄𝜅ó𝜍.Weyl suggested this word to describe the Lie group thatpreserves a nondegenerate skew-symmetric bilinear form.Prior to this, that Lie group was called the “line complexgroup” or the “Abelian linear group” (after Abel, whostudied the group).

A differential 2-form𝜔 on (real) manifold𝑀 is a gadgetthat at any point 𝑝 ∈ 𝑀 eats two tangent vectors andspits out a real number in a skew-symmetric, bilinearway. That is, it gives a family of skew-symmetric bilinearfunctions

𝜔𝑝 ∶ 𝑇𝑝𝑀×𝑇𝑝𝑀 → ℝ

Tara S. Holm is professor of mathematics at Cornell Universityand Notices consultant. She is grateful for the support of the Si-mons Foundation.Her e-mail address is tsh@math.cornell.edu.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1450

depending smoothly on the point 𝑝 ∈ 𝑀. A 2-form𝜔 ∈ Ω2(𝑀) is symplectic if it is both closed (its exteriorderivative satisfies 𝑑𝜔 = 0) and nondegenerate (eachfunction 𝜔𝑝 is nondegenerate). Nondegeneracy is equiva-lent to the statement that for each nonzero tangent vector𝑣 ∈ 𝑇𝑝𝑀, there is a symplectic buddy: a vector 𝑤 ∈ 𝑇𝑝𝑀so that 𝜔𝑝(𝑣,𝑤) = 1. A symplectic manifold is a (real)manifold 𝑀 equipped with a symplectic form 𝜔.

Nondegeneracy has important consequences. Purely interms of linear algebra, at any point 𝑝 ∈ 𝑀wemay choosea basis of 𝑇𝑝𝑀 that is compatible with 𝜔𝑝, using a skew-symmetric analogue of the Gram-Schmidt procedure. Westart by choosing any nonzero vector𝑣1 and then finding asymplectic buddy𝑤1. These must be linearly independentby skew-symmetry. We then peel off the two-dimensionalsubspace that 𝑣1 and 𝑤1 span and continue recursively,eventually arriving at a basis

𝑣1,𝑤1,… ,𝑣𝑑,𝑤𝑑,which contains an even number of basis vectors. Sosymplectic manifolds are even-dimensional. This alsoallows us to think of each tangent space as a complexvector space where each 𝑣𝑖 and 𝑤𝑖 span a complexcoordinate subspace. Moreover, the top wedge power,𝜔𝑑 ∈ Ω2𝑑(𝑀), is nowhere-vanishing, since at each tangentspace,

𝜔𝑑(𝑣1,… ,𝑤𝑑) ≠ 0.In other words,𝜔𝑑 is a volume form, and𝑀 is necessarilyorientable.

Symplectic geometry enjoys connections to algebraiccombinatorics, algebraic geometry, dynamics, mathemati-cal physics, and representation theory. The key examplesunderlying these connections include:(1) 𝑀 = 𝑆2 = ℂ𝑃1 with 𝜔𝑝(𝑣,𝑤) = signed area of the

parallelogram spanned by 𝑣 and 𝑤;(2) 𝑀 any Riemann surface as in Figure 1 with the area

for 𝜔 as in (1);

⋯Figure 1. The area form on a Riemannian surfacedefines its symplectic geometry.

1252 Notices of the AMS Volume 63, Number 11

THE GRADUATE STUDENT SECTION(3) 𝑀 = ℝ2𝑑 with 𝜔𝑠𝑡𝑑 = ∑𝑑𝑥𝑖 ∧𝑑𝑦𝑖;(4) 𝑀 = 𝑇∗𝑋, the cotangent bundle of any manifold

𝑋, thought of as a phase space, with 𝑝-coordinatescoming from 𝑋 being positions, 𝑞-coordinates inthe cotangent directions representing momentumdirections, and 𝜔 = ∑𝑑𝑝𝑖 ∧𝑑𝑞𝑖;

(5) 𝑀 any smooth complex projective variety with 𝜔induced from the Fubini-Study form (this includessmooth normal toric varieties, for example);

(6) 𝑀 = 𝒪𝜆 a coadjoint orbit of a compact connectedsemisimple Lie group𝐺, equipped with the Kostant-Kirillov-Souriau form 𝜔. For the group 𝐺 = 𝑆𝑈(𝑛),this class of examples includes complex projectivespace ℂ𝑃𝑛−1, Grassmannians G 𝑟𝑘(ℂ𝑛), the full flagvariety ℱℓ(ℂ𝑛), and all other partial flag varieties.

Plenty of orientable manifolds do not admit a symplecticstructure. For example, even-dimensional spheres that areat least four-dimensional are not symplectic. The reason isthat on a compact manifold, Stokes’ theorem guaranteesthat [𝜔] ≠ 0 ∈ 𝐻2(𝑀;ℝ). In other words, compactsymplectic manifolds must exhibit nonzero topology indegree 2 cohomology. The only sphere with this propertyis 𝑆2.

Example (3) gains particular importance because of thenineteenth century work of Jean Gaston Darboux on thestructure of differential forms. A consequence of his workisDarboux’s Theorem. Let𝑀 be a two-dimensional symplec-tic manifold with symplectic form 𝜔. Then for every point𝑝 ∈ 𝑀, there exists a coordinate chart 𝑈 about 𝑝 withcoordinates 𝑥1,… , 𝑥𝑑, 𝑦1,… ,𝑦𝑑 so that on this chart,

𝜔 =𝑑∑𝑖=1

𝑑𝑥𝑖 ∧𝑑𝑦𝑖 = 𝜔𝑠𝑡𝑑.

Tools from the1970s and 1980s set

the stage fordramatic progress

in symplecticgeometry and

topology.

This makes pre-cise the notion thatsymplectic geome-try is floppy. InRiemannian geome-try there are localinvariants such ascurvature that dis-tinguish metrics.Darboux’s theoremsays that symplecticforms are all lo-cally identical. Whatremains, then, areglobal topologicalquestions such as, What is the cohomology ring of aparticular symplectic manifold? and more subtle sym-plectic questions such as, How large can the Darbouxcharts be for a particular symplectic manifold?

Two tools developed in the 1970s and 1980s set thestage for dramatic progress in symplectic geometry andtopology. Marsden and Weinstein, Atiyah, and Guilleminand Sternberg established properties of the momentummap, resolving questions of the first type. Gromov intro-duced pseudoholomorphic curves to probe questions of

the second type. Let us briefly examine results of eachkind.

If a symplectic manifold exhibits a certain flavor ofsymmetries in the form of a Lie group action, then itadmits a momentum map. This gives rise to conservedquantities such as angular momentum, whence the name.The first example of a momentum map is the heightfunction on a 2-sphere (Figure 2).

Figure 2. The momentum map for 𝑆1 acting on 𝑆2 byrotation.

In this case, the conserved quantity is angular momen-tum, and the height function is the simplest example ofa perfect Morse function on 𝑆2. When the Lie group is aproduct of circles 𝑇 = 𝑆1 ×⋯×𝑆1, we say that the mani-fold is a Hamiltonian 𝑇-space, and the momentum mapis denoted Φ ∶ 𝑀 → ℝ𝑛. In 1982 Atiyah and independentlyGuillemin and Sternberg proved the Convexity Theorem(see Figure 3):Convexity Theorem. If 𝑀 is a compact Hamiltonian 𝑇-space, then Φ(𝑀) is a convex polytope. It is the convex hullof the images Φ(𝑀𝑇) of the 𝑇-fixed points.

Figure 3. Atiyah and Guillemin-Sternberg proved thatif a symplectic manifold has certain symmetries, theimage of its momentum map is a convex polytope.

This provides a deep connection between symplecticand algebraic geometry on the one hand and discretegeometry and combinatorics on the other. Atiyah’s proofdemonstrates that the momentum map provides Morsefunctions on 𝑀 (in the sense of Bott), bringing the powerof differential topology to bear on global topologicalquestions about 𝑀. Momentum maps are also used toconstruct symplectic quotients. Lisa Jeffrey will discussthe cohomology of symplectic quotients in her 2017Noether Lecture at the Joint Mathematics Meetings, andmanymore researchers will delve into these topics duringthe special session Jeffrey and I are organizing.

Through example (2), we see that two-dimensional sym-plectic geometry boils down to area-preserving geometry.Because symplectic forms induce volume forms, a naturalquestion in higher dimensions is whether symplectic ge-ometry is as floppy as volume-preserving geometry: can amanifold be stretched and squeezed in any which way so

December 2016 Notices of the AMS 1253

THE GRADUATE STUDENT SECTION

Gromov provedthat symplecticmaps are more

rigid thanvolume-

preservingones.

long as volume is pre-served? Using pseudoholo-morphic curves, Gromovdismissed this possibility,proving that symplecticmaps are more rigid thanvolume-preserving ones. Inℝ2𝑑 we let 𝐵2𝑑(𝑟) denotethe ball of radius 𝑟. In1985 Gromov proved thenonsqueezing theorem (seeFigure 4):

Nonsqueezing Theorem.There is an embedding

𝐵2𝑑(𝑅) ↪ 𝐵2(𝑟) × ℝ2𝑑−2

preserving 𝜔𝑠𝑡𝑑 if and only if 𝑅 ≤ 𝑟.One direction is straightforward: if 𝑅 ≤ 𝑟, then

𝐵2𝑑(𝑅) ⊆ 𝐵2(𝑟) × ℝ2𝑑−2. To find an obstruction tothe existence of such a map, Gromov used a pseudo-holomorphic curve in 𝐵2(𝑟) × ℝ2𝑑−2 and the symplecticembedding to produce a minimal surface in 𝐵2𝑑(𝑅),forcing 𝑅 ≤ 𝑟.

On the other hand, a volume-preserving map existsfor any 𝑟 and 𝑅. Colloquially, you cannot squeeze asymplectic camel through the eye of a needle.

Gromov’s work has led to many rich theories ofsymplectic invariants with pseudoholomorphic curves

Figure 4. The Nonsqueezing Theorem gives geomet-ric meaning to the aphorism on the impossibility ofpassing a camel through the eye of a needle: A sym-plectic manifold cannot fit inside a space with a nar-row two-dimensional obstruction, no matter how bigthe target is in other dimensions.This cartoon originally appeared in the article “TheSymplectic Camel” by Ian Stewart—published in theSeptember 1987 issue of Nature—we thank him forhis permission to use it. Cosgrove is a well-knowncartoonist loosely affiliated with mathematicians. Hedid drawings for Manifold for a few years and mostrecently did a few sketches for the curious cookbookSimple Scoff:https://www2.warwick.ac.uk/newsandevents/pressreleases/50th_anniversary_cookery.

the common underlying tool. The constructions rely onsubtle arguments in complex analysis and Fredholmtheory. These techniques are essential to current workin symplectic topology and mirror symmetry, and theyprovide an important alternativeperspective on invariantsof four-dimensional manifolds.

Further details on momentum maps may be found in[CdS], and [McD-S] gives anaccountofpseudoholomorphiccurves.

References[CdS] A. Cannas da Silva, Lectures on Symplectic Geometry,

Lecture Notes in Mathematics, 1764, Springer-Verlag,Berlin, 2001. MR 1853077

[McD-S] D. McDuff and D. Salamon, Introduction to Symplec-tic Topology, Oxford Mathematical Monographs, OxfordUniversity Press, New York, 1995. MR 1373431

ABOUT THE AUTHOR

In addition to mathematics, Tara Holm enjoys gardening, cooking with her family, and exploring the Finger Lakes. This column was based in part on her AMS–MAA Invited Address at MAA MathFest 2016 held in August.

Tara S. Holm

ABOUT THE AUTHOR

In addition to mathematics, Tara Holm enjoys gardening, cooking with her family, and exploring the Finger Lakes. This column was based in part on her AMS–MAA Invited Ad-dress at MathFest 2016 held in August. Tara S. Holm

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A Decade Ago in the Notices: PseudoholomorphicCurves

“WHAT IS…Symplectic Geometry?” discusses Gromov’s“nonsqueezing theorem,” a key result in symplecticgeometry. Gromovproved the theoremusing thenotionof a pseudoholomorphic curve, which he introduced in1986.

Readers interested in these topics might also wishto read “WHAT IS…a Pseudoholomorphic Curve?” bySimon Donaldson, which appeared in the October 2005issue of the Notices. “The notion [of pseudoholomor-phic curve] has transformed the field of symplectictopology and has a bearing on many other areas suchas algebraic geometry, string theory and 4-manifoldtheory,” Donaldson writes. Starting with the basic no-tion of a plane curve, he gives a highly accessibleexplanation of what pseudoholomorphic curves are.He notes that they have been used as a tool in sym-plectic topology in two main ways: “First, as geometricprobes to explore symplectic manifolds: for examplein Gromov’s result, later extended by Taubes, on theuniqueness of the symplectic structure on the complexprojective plane…Second, as the source of numericalinvariants: Gromov–Witten invariants.”

Another related piece “WHAT IS…a Toric Variety?”by Ezra Miller appeared in the May 2008 issue ofthe Notices.

1254 Notices of the AMS Volume 63, Number 11