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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 โ, 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 โsymplecticโ is a calque introduced byHermann Weyl in his textbook on the classical groups.That is, it is a root-by-root translation of the wordโcomplexโ from the Latin roots
com โ plexus,meaning โtogether โ braided,โ into the Greek roots withthe same meaning,
๐๐๐ โ ๐๐๐๐ ๐๐๐ รณ๐.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 โline complexgroupโ or the โAbelian linear groupโ (after Abel, whostudied the group).
A differential 2-form๐ on (real) manifold๐ is a gadgetthat at any point ๐ โ ๐ eats two tangent vectors andspits out a real number in a skew-symmetric, bilinearway. That is, it gives a family of skew-symmetric bilinearfunctions
๐๐ โถ ๐๐๐ร๐๐๐ โ โ
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 [email protected].
For permission to reprint this article, please contact:[email protected]: http://dx.doi.org/10.1090/noti1450
depending smoothly on the point ๐ โ ๐. A 2-form๐ โ ฮฉ2(๐) is symplectic if it is both closed (its exteriorderivative satisfies ๐๐ = 0) and nondegenerate (eachfunction ๐๐ is nondegenerate). Nondegeneracy is equiva-lent to the statement that for each nonzero tangent vector๐ฃ โ ๐๐๐, there is a symplectic buddy: a vector ๐ค โ ๐๐๐so that ๐๐(๐ฃ,๐ค) = 1. A symplectic manifold is a (real)manifold ๐ equipped with a symplectic form ๐.
Nondegeneracy has important consequences. Purely interms of linear algebra, at any point ๐ โ ๐wemay choosea basis of ๐๐๐ that is compatible with ๐๐, using a skew-symmetric analogue of the Gram-Schmidt procedure. Westart by choosing any nonzero vector๐ฃ1 and then finding asymplectic buddy๐ค1. These must be linearly independentby skew-symmetry. We then peel off the two-dimensionalsubspace that ๐ฃ1 and ๐ค1 span and continue recursively,eventually arriving at a basis
๐ฃ1,๐ค1,โฆ ,๐ฃ๐,๐ค๐,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 ๐ฃ๐ and ๐ค๐ span a complexcoordinate subspace. Moreover, the top wedge power,๐๐ โ ฮฉ2๐(๐), is nowhere-vanishing, since at each tangentspace,
๐๐(๐ฃ1,โฆ ,๐ค๐) โ 0.In other words,๐๐ is a volume form, and๐ is necessarilyorientable.
Symplectic geometry enjoys connections to algebraiccombinatorics, algebraic geometry, dynamics, mathemati-cal physics, and representation theory. The key examplesunderlying these connections include:(1) ๐ = ๐2 = โ๐1 with ๐๐(๐ฃ,๐ค) = signed area of the
parallelogram spanned by ๐ฃ and ๐ค;(2) ๐ any Riemann surface as in Figure 1 with the area
for ๐ as in (1);
โฏFigure 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) ๐ = โ2๐ with ๐๐ ๐ก๐ = โ๐๐ฅ๐ โง๐๐ฆ๐;(4) ๐ = ๐โ๐, the cotangent bundle of any manifold
๐, thought of as a phase space, with ๐-coordinatescoming from ๐ being positions, ๐-coordinates inthe cotangent directions representing momentumdirections, and ๐ = โ๐๐๐ โง๐๐๐;
(5) ๐ any smooth complex projective variety with ๐induced from the Fubini-Study form (this includessmooth normal toric varieties, for example);
(6) ๐ = ๐ช๐ a coadjoint orbit of a compact connectedsemisimple Lie group๐บ, equipped with the Kostant-Kirillov-Souriau form ๐. For the group ๐บ = ๐๐(๐),this class of examples includes complex projectivespace โ๐๐โ1, Grassmannians G ๐๐(โ๐), the full flagvariety โฑโ(โ๐), 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, Stokesโ theorem guaranteesthat [๐] โ 0 โ ๐ป2(๐;โ). In other words, compactsymplectic manifolds must exhibit nonzero topology indegree 2 cohomology. The only sphere with this propertyis ๐2.
Example (3) gains particular importance because of thenineteenth century work of Jean Gaston Darboux on thestructure of differential forms. A consequence of his workisDarbouxโs Theorem. Let๐ be a two-dimensional symplec-tic manifold with symplectic form ๐. Then for every point๐ โ ๐, there exists a coordinate chart ๐ about ๐ withcoordinates ๐ฅ1,โฆ , ๐ฅ๐, ๐ฆ1,โฆ ,๐ฆ๐ so that on this chart,
๐ =๐โ๐=1
๐๐ฅ๐ โง๐๐ฆ๐ = ๐๐ ๐ก๐.
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.Darbouxโs 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 ๐1 acting on ๐2 byrotation.
In this case, the conserved quantity is angular momen-tum, and the height function is the simplest example ofa perfect Morse function on ๐2. When the Lie group is aproduct of circles ๐ = ๐1 รโฏร๐1, we say that the mani-fold is a Hamiltonian ๐-space, and the momentum mapis denoted ฮฆ โถ ๐ โ โ๐. In 1982 Atiyah and independentlyGuillemin and Sternberg proved the Convexity Theorem(see Figure 3):Convexity Theorem. If ๐ is a compact Hamiltonian ๐-space, then ฮฆ(๐) is a convex polytope. It is the convex hullof the images ฮฆ(๐๐) of the ๐-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. Atiyahโs proofdemonstrates that the momentum map provides Morsefunctions on ๐ (in the sense of Bott), bringing the powerof differential topology to bear on global topologicalquestions about ๐. 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. Inโ2๐ we let ๐ต2๐(๐) denotethe ball of radius ๐. In1985 Gromov proved thenonsqueezing theorem (seeFigure 4):
Nonsqueezing Theorem.There is an embedding
๐ต2๐(๐ ) โช ๐ต2(๐) ร โ2๐โ2
preserving ๐๐ ๐ก๐ if and only if ๐ โค ๐.One direction is straightforward: if ๐ โค ๐, then
๐ต2๐(๐ ) โ ๐ต2(๐) ร โ2๐โ2. To find an obstruction tothe existence of such a map, Gromov used a pseudo-holomorphic curve in ๐ต2(๐) ร โ2๐โ2 and the symplecticembedding to produce a minimal surface in ๐ต2๐(๐ ),forcing ๐ โค ๐.
On the other hand, a volume-preserving map existsfor any ๐ and ๐ . Colloquially, you cannot squeeze asymplectic camel through the eye of a needle.
Gromovโs 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 โTheSymplectic Camelโ by Ian Stewartโpublished in theSeptember 1987 issue of Natureโwe 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 AMSโMAA 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 AMSโMAA Invited Ad-dress at MathFest 2016 held in August. Tara S. Holm
Ph
oto
by
Mel
issa
Totm
an.
A Decade Ago in the Notices: PseudoholomorphicCurves
โWHAT ISโฆSymplectic Geometry?โ discusses Gromovโsโnonsqueezing theorem,โ 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 โWHAT ISโฆa Pseudoholomorphic Curve?โ bySimon Donaldson, which appeared in the October 2005issue of the Notices. โThe 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,โ 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: โFirst, as geometricprobes to explore symplectic manifolds: for examplein Gromovโs result, later extended by Taubes, on theuniqueness of the symplectic structure on the complexprojective planeโฆSecond, as the source of numericalinvariants: GromovโWitten invariants.โ
Another related piece โWHAT ISโฆa Toric Variety?โby Ezra Miller appeared in the May 2008 issue ofthe Notices.
1254 Notices of the AMS Volume 63, Number 11