Dusa McDuff and

Felix Schlenk

Dusa McDuff has led three mathematical lives. In her early in particular groups. This brought twenties she worked on von Neumann algebras and estab- her to symplectic geometry, which was revolutionized lished the existence of uncountably many different alge- around 1985 by Gromov’s introduction of 퐽-holomorphic 1 braic types of II1-factors. After an inspiring six months curves. studying with Gel’fand in and a two-year post- In this short text I will describe some of McDuff’s won- doc in Cambridge studying topology, she wrote a “sec- derful contributions to symplectic geometry. After review- ond thesis” while working with on configu- ing what is meant by “symplectic” I will mostly focus on ration spaces and the group-completion theorem, and in- her work on symplectic embedding problems. Some of vestigated the topology of various diffeomorphism groups, her other results in symplectic geometry are discussed at the end. More personal texts about Dusa can be found Felix Schlenk is a professor of mathematics at the Universit´ede Neuchâtel. His in [11b]. Parts of this text overlap with the “Perspective” email address is [email protected]. in [11b] written jointly with Leonid Polterovich. 1For a version of this text with more than 20 references see arXiv:2011.03317. Communicated by Notices Associate Editor Chikako Mese. 1. Symplectic For permission to reprint this article, please contact: There are many strands to and from symplectic geometry. [email protected]. The most important ones are classical mechanics and al- DOI: https://doi.org/10.1090/noti2238 gebraic geometry. I do not list these strands but refer you

346 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 3 to [17b] and to my 2018 survey in the Bulletin. Here, I a symplectomorphism of 푀 is then a diffeomorphism that simply give the definition. preserves this local structure. Definition 1. Let 푀 be a smooth manifold. A symplectic The standard symplectic structure of ℝ2푛 is thus given form on 푀 is a nondegenerate closed 2-form 휔. A diffeo- by assigning to a closed curve the sum of the signed areas morphism 휑 of 푀 is symplectic (or a symplectomorphism) if of the 푛 curves obtained by projecting to the coordinate ∗ 2 휑 휔 = 휔. planes ℝ (푥푖, 푦푖). And a symplectic structure on a manifold The nondegeneracy condition implies that symplectic is a coherent way of assigning a signed area to sufficiently manifolds are even-dimensional. An example is ℝ2푛 with local closed curves. The equivalence of the two definitions the constant differential 2-form follows from 푛 Darboux’s Theorem. Around every point of a symplectic 휔0 = ∑ 푑푥푖 ∧ 푑푦푖. 푖=1 manifold (푀, 휔) there exists a coordinate chart 휑 such that 휑∗휔 = 휔. Other examples are surfaces endowed with an area form, 0 their products, and Kähler manifolds. The group of of a symplectic manifold is very large. Indeed, for every compactly sup- If you begin your first lecture on symplectic geometry ported smooth function 퐻 ∶ 푀 × [0, 1] → ℝ each time-푡 like this, you may very well find yourself alone the follow- map 휙푡 of its Hamiltonian flow is a symplectomorphism. ing week. You may thus prefer to start in a more elemen- 퐻 Symplectomorphisms of this form are called Hamiltonian tary way. Let 훾 be a closed oriented piecewise smooth curve diffeomorphisms. The Hamiltonian flow is the flow gen- in ℝ2. If 훾 is embedded, assign to 훾 the signed area of the erated by the vector field 푋 implicitly defined by disc 퐷 bounded by 훾, namely area(퐷) or − area(퐷), as in 퐻 Figure 1.1. 휔(푋퐻 , ⋅) = −푑퐻(⋅). 2푛 For (ℝ , 휔0) one has 푋퐻 = 퐽0 ∇퐻, where 퐽0 is the usual 2 complex structure on ⨁푖 ℝ (푥푖, 푦푖). 2. Symplectic Embedding Problems By Darboux’s Theorem, symplectic manifolds have no lo- cal invariants beyond the dimension. But there are several Figure 1.1. The sign of the signed area of an embedded closed ways to associate global numerical invariants to symplectic curve in ℝ2. manifolds. One of them is by looking at embedding prob- 2푛 lems. Take a compact subset 퐾 of (ℝ , 휔0). By a symplec- tic embedding 퐾 → 푀 we mean the restriction to 퐾 of a If 훾 is not embedded, successively decompose 훾 into smooth embedding 휑∶ 푈 → 푀 of an open neighborhood closed embedded pieces as illustrated in Figure 1.2, and ∗ of 퐾 that is symplectic, 휑 휔 = 휔0. In this case we write define 퐴(훾) as the sum of the signed areas of these pieces. 푠 퐾 ,−→(푀, 휔). For every 퐾, the largest number 휆 such that the dilate 휆퐾 symplectically embeds into (푀, 휔) is then a symplectic invariant of (푀, 휔). Seven further reasons to study symplectic embedding problems can be found in my Bulletin article. Figure 1.2. Splitting a closed curve into embedded pieces. 2.1. The Nonsqueezing Theorem. Now take the closed ball B2푛(푎) of radius √푎/휋 centered at the origin of ℝ2푛. Definition 2. The standard symplectic structure of ℝ2푛 is (The notation reflects that symplectic measurements are 2- the map dimensional.) By what we said above, there are very many 2푛 푠 2푛 푛 symplectic embeddings B (푎) ,−→ℝ . However, none of 2푛 퐴(훾) = ∑ 퐴(훾푖), 훾 = (훾1, … , 훾푛) ⊂ ℝ . them can make the ball thinner, as Gromov proved in his 푖=1 pioneering 1985 paper. 2푛 푠 A symplectomorphism 휑 of ℝ is a diffeomorphism that Nonsqueezing Theorem. B2푛(푎) ,−→B2(퐴) × ℝ2푛−2 only if preserves the signed area of closed curves: 푎 ≤ 퐴. 퐴(휑(훾)) = 퐴(훾) 훾 ⊂ ℝ2푛. for all closed curves The identity embedding thus already provides the A symplectic structure on a manifold 푀 is an atlas whose largest ball that symplectically fits into the cylinder B2(퐴)× transition functions are (local) symplectomorphisms, and ℝ2푛−2 of infinite volume. While there are many forms

MARCH 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 347 of symplectic rigidity, this theorem is its most fundamen- Lemma. There exists a 퐽-holomorphic sphere 푢(푆2) through tal manifestation. The theorem shows that some volume- Φ(0) that represents the homology class of 푆2(퐴′). preserving mappings cannot be approximated by symplec- tic mappings in the 퐶0-topology. This existence result follows from Gromov’s compact- In [95a], Lalonde and McDuff generalized the Non- ness theorem for 퐽-holomorphic curves in symplectic squeezing Theorem to all symplectic manifolds. manifolds. The key point for the proof of the compactness theorem is that 퐽 is 휔-tame, implying that 퐽-holomorphic General Nonsqueezing Theorem. For any symplectic mani- curves cannot behave too wildly. The compactness theo- fold (푀, 휔) of dimension 2푛 − 2, rem implies the lemma because the class of 푆2(퐴′) is prim- 푠 2푛 2 itive in 퐻2(푃; ℤ). B (푎) ,−→ (B (퐴) × 푀, 휔0 ⊕ 휔) only if 푎 ≤ 퐴. Every good tool in symplectic geometry can be used to prove the Nonsqueezing Theorem. However, the tech- nique of 퐽-holomorphic curves used by Gromov is the most influential one, and also the most important toolin McDuff’s work. An almost complex structure 퐽 on a manifold 푃 is a smooth collection {퐽푝}푝∈푃, where 퐽푝 is a linear endomor- 2 phism of 푇푝푃 such that 퐽푝 = −id. The “almost” indicates that such a structure does not need to be a complex struc- ture, i.e., does not need to come from a holomorphic at- las. Not all symplectic manifolds admit complex struc- tures, but they all admit almost complex structures. A 퐽- holomorphic curve in an almost complex manifold (푃, 퐽) is a map 푢 from a Riemann surface (Σ, 푗) to (푃, 퐽) such that 푑푢 ∘ 푗 = 퐽 ∘ 푑푢. This equation generalizes the Cauchy–Riemann equation defining holomorphic maps ℂ → ℂ푛. In this text, the do- Figure 2.1. The geometric idea of the proof. main of a 퐽-holomorphic curve will always be the usual Riemann sphere, namely the round sphere 푆2 ⊂ ℝ3 whose The Nonsqueezing Theorem readily follows from the 2 휋 complex structure 푗 rotates a vector 푣 ∈ 푇푝푆 by . Even lemma: the set 2 in this case, it is usually impossible to write down a 퐽- 푆 = Φ−1 (Φ(B2푛(푎)) ∩ 푢(푆2)) holomorphic curve for a given 퐽. But this is not a problem, since one usually just wants to know that such a curve ex- is a proper 2-dimensional complex surface in B2푛(푎) ists. through 0. By the Lelong inequality from complex analysis, 2푛 푠 2 Now assume that 휑∶ B (푎) ,−→B (퐴) × ℝ2푛−2. Choose the area of 푆 with respect to the Euclidean inner product 푘 so large that after a translation the image of 휑 is con- is at least 푎. Using also that Φ is symplectic we can now tained in B2(퐴) × (0, 푘)2푛−2. Compactifying the disc to the estimate sphere 푆2(퐴′) with its usual area form of area 퐴′ > 퐴 and ∗ ′ taking the quotient to the torus 푇2푛−2 = ℝ2푛−2/푘ℤ2푛−2, we 푎 ≤ area(푆) = ∫휔0 = ∫Φ 휔 = ∫ 휔 ≤ ∫ 휔 = 퐴 . 2 then obtain a symplectic embedding 푆 푆 Φ(푆) ᵆ(푆 ) 푠 This proof of the Nonsqueezing Theorem also works, Φ∶ B2푛(푎) ,−→푆2(퐴′) × 푇2푛−2 =∶ (푃, 휔), for instance, for closed symplectic manifolds (푀, 휔) for where 휔 is the split symplectic structure on the product 푃. which the integral of 휔 over spheres vanishes. In the gen- We will see that 푎 ≤ 퐴′. Since 퐴′ > 퐴 was arbitrary, Gro- eral case, however, there was trouble with holomorphic mov’s theorem then follows. spheres of negative Chern number. By now, this trouble 2푛 Denote by 퐽0 the usual complex structure on B (푎) ⊂ has been overcome thanks to the definition of Gromov– ℂ푛, and let 퐽 be an almost complex structure on 푃 that re- Witten invariants for general closed symplectic manifolds 2푛 stricts to Φ∗퐽0 on Φ(B (푎)) and that is 휔-tame, meaning by the work of several (teams of) authors. McDuff helped that 휔(푣, 퐽푣) > 0 for all nonzero 푣 ∈ 푇푃. Such an exten- to clarify the approach of Fukaya–Oh–Ono–Ohta in her sion exists, since 휔-tame almost complex structures can be joint work with Wehrheim; see [17c, 19]. In [95a], how- viewed as sections of a bundle over 푃 whose fibers are con- ever, Lalonde and McDuff circumvented all technical is- tractible. sues by using a chain of beautiful geometric constructions

348 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 3 that reduced the General Nonsqueezing Theorem essen- Before discussing the proof, we use the theorem to ob- tially to the case proved by Gromov. The most important tain table (2.1). If (푑; 푚1, … , 푚푘) is a solution of (DE), then and influential of these is a multiple folding construction. 2 푘 푘 Its simple version was used in [95a] to show how the Gen- 2 2 2 (3푑 − 1) = (∑ 푚푖) ≤ 푘 ∑ 푚푖 = 푘(푑 + 1), eral Nonsqueezing Theorem implies that for all symplec- 푖=1 푖=1 tic manifolds Hofer’s metric on the group of compactly that is, supported Hamiltonian diffeomorphisms is nondegener- (9 − 푘)푑2 − 6푑 + (1 − 푘) ≤ 0. ate and hence indeed a metric; see §3.3. These two the- orems were the first deep results in symplectic geometry For 푘 ≤ 8, this equation has finitely many solutions 푑, and proven for all symplectic manifolds. so (DE) has finitely many solutions for 푘 ≤ 8. They are readily computed: 2.2. Ball packings. We next try to pack a symplectic mani- fold by balls as densely as possible. Taking the open ball (1; 1,1), (2; 1×5), (3; 2,1×6), (4; 2×3,1×5), (5; 2×6,1,1), (6; 3,2×7), ̊4 B (1) as target, let and all these vectors reduce to (0; −1) by Cremona moves. 푠 4 4 ̊4 푘 Vol(B (푎)) | 4 푠 For instance, for the problem ∐ B (1) ,−→ B (퐴) the 푝 = sup { | ∐ B (푎) ,−→ B̊4(1)} 8 푘 4 | strongest constraint comes from the solution (6; 3, 2×7), Vol(B (1)) 푘 17 that gives 퐴 > . be the percentage of the volume of B̊4(1) that can be filled 6 On the other hand, if (푑; 푚 , … , 푚 ) is a solution by 푘 symplectically embedded equal balls. Then: 1 푘 of (DE) with 푘 ≥ 9, then 푘 1 2 3 4 5 6 7 8 ⩾ 9 푘 푎 푎 푎 푎 ∑ 푚 = (3푑 − 1) < 3푑 ≤ √푘 푑 = 푎 √푘. 1 3 20 24 63 288 푑 푖 푑 푑 푑 푝푘 1 1 1 (2.1) 푖=1 2 4 25 25 64 289 5 5 8 17 푎 푘 푐푘 1 2 2 2 √푘 Hence the constraint 퐴 > ∑푖=1 푚푖 is weaker than the 2 2 3 6 푑 2 2 volume constraint 퐴 > 푘푎 , and so 푝푘 = 1. The lower line gives the capacities The proof of the Ball Packing Theorem is a beautiful | 4 푠 푐 = inf {퐴 | ∐ B (1) ,−→ B̊4(퐴)} story in three chapters, each of which contains an impor- 푘 | 푘 tant idea of McDuff. The original symplectic embedding problem is converted to an increasingly algebraic problem 2 푘 that are related to the packing numbers 푝푘 by 푐푘 = . This in three steps. 푝푘 table was obtained for 푘 ⩽ 5 by Gromov, for 푘 = 6, 7, 8 and The starting point is the symplectic blow-up construc- 푘 a square by McDuff and Polterovich [94b], and for all 푘 tion, that goes back to Gromov and Guillemin–Sternberg, by Biran. This result is a special case of the following alge- and was first used by McDuff [91a] to study symplectic em- braic reformulation of the general ball packing problem beddings of balls. Recall that the complex blow-up Bl(ℂ2) 2 푘 of ℂ at the origin 0 is obtained by replacing 0 by all com- 4 푠 4 plex lines in ℂ2 through 0. At the topological level, this ∐ B (푎푖) ,−→ B̊ (퐴). (2.2) 2 푖=1 operation can be done as follows. First remove from ℂ an open ball B̊4. The boundary 푆3 of ℂ2 ⧵ B̊4 is foliated by the Ball Packing Theorem. An embedding (2.2) exists if and 1 Hopf circles {(훼푧1, 훼푧2) ∣ 훼 ∈ 푆 }, namely the intersections only if 3 2 푘 of 푆 with complex lines. Now Bl(ℂ ) is obtained by replac- 2 2 3 (i) (Volume constraint) 퐴 > ∑푖=1 푎푖 ing each such circle by a point. The boundary sphere 푆 (ii) (Constraint from exceptional spheres) becomes a 2-sphere ℂP1 in Bl(ℂ2) of self-intersection num- 1 푘 2 퐴 > ∑푖=1 푎푖푚푖 for every vector of nonnegative inte- ber −1, called the exceptional divisor. The manifold Bl(ℂ ) 푑 2 gers (푑; 푚1, … , 푚푘) that solves the Diophantine system is diffeomorphic to the connected sum ℂ2#ℂP . 2 2 This construction can be done in the symplectic setting. ∑ 푚푖 = 3푑 − 1, ∑ 푚푖 = 푑 + 1 (DE) 4 4 푖 푖 If one removes B̊ (푎) from ℝ , then there exists a symplec- 4 and can be reduced to (0; −1, 0, … , 0) by repeated Cre- tic form 휔푎 on Bl(ℝ ) such that 휔푎 = 휔0 outside a tubu- 1 mona moves. lar neighborhood of the exceptional divisor ℂP and such 1 that 휔푎 is symplectic on ℂP with ∫ℂP1 휔푎 = 푎. Given Here, a Cremona move takes a vector (푑; 푚1, … , 푚푘) with a symplectic embedding 휑∶ B4(푎) → (푀, 휔) into a sym- 푚1 ⩾ ⋯ ⩾ 푚푘 to the vector plectic 4-manifold, we can apply the same construction to ′ ′ (푑 ; 퐦 ) = (푑 + 훿; 푚1 + 훿, 푚2 + 훿, 푚3 + 훿, 푚4, … , 푚푘), 휑(B4(푎)) in 푀 to obtain the symplectic blow-up of (푀, 휔) ′ where 훿 = 푑 − (푚1 + 푚2 + 푚3), and then reorders 퐦 . by weight 푎.

MARCH 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 349 There is also an inverse construction. Given a sym- symplectic form on 푀푘 in class 퐴ℓ−휀 ∑푖 푎푖푒푖 in such a way plectically embedded −1-sphere Σ in a symplectic 4- that these balls look large. This can be done with the help manifold (푀, 휔) of area ∫Σ 휔 = 푎, one can cut out a tubu- of the inflation method of Lalonde–McDuff [94a, 96a]. lar neighborhood of Σ and glue back B4(푎), to obtain the Inflation Lemma. Let (푀, 휔) be a closed symplectic 4- “symplectic blow-down” of 푀. manifold, and assume that the class 퐶 ∈ 퐻2(푀; ℤ) with Now let 푀푘 be the smooth manifold obtained by blow- 2 2 퐶 ⩾ 0 can be represented by a closed connected embedded ing up the complex projective plane ℂP in 푘 points. Its 퐽-holomorphic curve Σ for some 휔-tame 퐽. Then the class homology 퐻2(푀푘; ℤ) is generated by the class 퐿 of a com- [휔] + 푠 PD(퐶) has a symplectic representative for all 푠 ⩾ 0. plex line and by the classes 퐸1, … , 퐸푘 of the exceptional di- 2 visors. Let ℓ, 푒푖 ∈ 퐻 (푀푘; ℤ) be their Poincar´eduals. Every symplectic form 휔 on 푀푘 defines a first Chern class 푐1(휔), namely the first Chern class of any 휔-tame almost com- plex structure. If we take a symplectic form 휔 on 푀푘 con- structed as above via 푘 symplectic ball embeddings (that always exist if the balls are small enough), then −푐1(휔) is Poincar´edual to the class 퐾 ∶= −3퐿 + ∑푖 퐸푖. Define 2 풞퐾 (푀푘) ⊂ 퐻 (푀푘; ℝ) to be the set of classes represented by symplectic forms with −푐1(휔) = PD(퐾). Compactifying B̊4(퐴) to ℂP2(퐴) with its usual Kähler form integrating to 퐴 over a complex line, and using that the classes 퐸 can be represented by symplectic −1-spheres, 푖 Figure 2.2. The ray in the symplectic cone provided by the McDuff and Polterovich [94b] obtained Inflation Lemma. 푘 4 푠 ̊4 Step 1. There exists an embedding ∐푖=1 B (푎푖) ,−→ B (퐴) if 푘 2 I explain the proof for the case 퐶2 = 0. In this case, and only if the class 훼 ∶= 퐴ℓ − ∑푖=1 푎푖푒푖 ∈ 퐻 (푀푘; ℝ) lies the normal bundle of Σ is trivial, and we can identify a in 풞퐾 (푀푘). tubular neighborhood of Σ with Σ × 퐷, where 퐷 ⊂ ℝ2 is a Our symplectic embedding problem is thus translated disc. Pick a radial function 푓(푟) with support in 퐷 that is into a problem on the symplectic cone 풞퐾 (푀푘). In Käh- non-negative and has ∫퐷 푓 = 1. Let 훽 be the closed 2-form ler geometry, the problem of deciding which on 푀 that equals 훽(푧, 푥, 푦) = 푓(푟) 푑푥 ∧ 푑푦 on Σ × 퐷 and classes can be represented by a Kähler form has a long his- vanishes outside of Σ × 퐷. Then [훽] = PD([Σ]) and the tory and is quite well understood. The solution of our sym- forms 휔 + 푠훽 are symplectic for all 푠 ≥ 0. Indeed, plectic analogue, however, needs different ideas and tools: (휔 + 푠훽)2 = 휔⏟2 + 2푠 휔 ∧ 훽 + 푠2 훽2 > 0, call a class 퐸 ∈ 퐻2(푀푘; ℤ) exceptional if 퐾 ⋅ 퐸 = 1 and ⏟ ⏟ >0 퐸2 = −1, and if 퐸 can be represented by a smoothly em- ≥0 =0 bedded −1-sphere. where for the middle term we used that 휔|Σ is symplectic. 푘 4 푠 If 휔 is a symplectic form on 푀푘 with −푐1(휔) = PD(퐾), ̊4 Now take an embedding ∐푖=1 B (휀푎푖) ,−→ B (퐴) of tiny then any 휔-symplectic embedded −1-sphere represents an balls. By Step 1 we know that the class 훼휀 ∶= 퐴ℓ−휀 ∑푖 푎푖푒푖 exceptional class. Seiberg–Witten–Taubes theory implies has a symplectic representative 휔휀. We wish to inflate this that the converse is also true: every exceptional class 퐸 can form to a symplectic form in class 훼. A first try could be to be represented by an 휔-symplectic embedded −1-sphere. inflate 휔휀 directly in the direction 훼−훼휀 to get up to 훼. But This implies one direction of this does not work, because 2 2 Step 2. 훼 = 퐴ℓ −∑ 푎푖푒푖 ∈ 퐻 (푀푘; ℝ) lies in 풞퐾 (푀푘) if and 2 2 2 푖 (훼 − 훼휀) = (−∑ (1 − 휀)푎푖 푒푖) = −(1 − 휀) ∑ 푎푖 < 0. only if 훼2 > 0 and 훼(퐸) > 0 for all exceptional classes. 푖 푖 However, assuming for simplicity that 퐴 and the 푎푖 are ra- This equivalence is remarkable. Of course, a necessary tional, Seiberg–Witten–Taubes theory implies that there condition for a class 훼 with positive square to have a sym- exists an integer 푛 such that the Poincar´edual of 푛훼 ∈ plectic representative is that 훼 evaluates positively on all 2 퐻 (푀푘; ℤ) can be represented by a connected embedded classes that can be represented by closed symplectically em- 퐽-holomorphic curve for a generic 휔-tame 퐽. We can thus bedded surfaces (and in particular on spheres). But the inflate 휔 in the direction of 푛훼 and obtain that the classes equivalence says that this is also a sufficient condition, and 훼휀 + 푠푛훼 have symplectic representatives for all 푠 ≥ 0. that it is actually enough to check positivity on spheres. 1 Rescaling these forms by , we obtain symplectic forms 푠푛+1 To prove the other direction in Step 2, one starts with an 푠푛+휀 in classes 퐴ℓ − ∑ 푎푖푒푖 that are as close to 훼 as we like. embedding of 푘 tiny balls of size 휀푎푖 and then changes the 푠푛+1 푖

350 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 3 theory used in the last section in such a space. McDuff in [09] simply circumvented the singularities by using a version of the Hirzebruch–Jung resolution of singularities. She removed a bit more than the ellipsoid by successively blowing up finitely many balls, thereby producing a chain of 퐽-spheres. Inflating this chain she reduced the problem 푠 E(1, 푎) ,−→ B̊4(퐴) to the ball packing problem (2.2):

푘 푠 4 4 푠 4 E(1, 푎) ,−→ B̊ (퐴) ⟺ ∐ B (푎푖) ,−→ B̊ (퐴), (2.3) 푖=1

where the 푎푖 are given by

(푎1, … , 푎푘) =∶ 풘(푎) = (⏟⏟⏟1, … , 1, 푤⏟⎵⎵⏟⎵⎵⏟1, … , 푤1, … , 푤⏟⎵⎵⏟⎵⎵⏟푁 , … , 푤푁 )

ℓ0 ℓ1 ℓ푁

with the weights 푤푖 > 0 such that 푤1 = 푎 − ℓ0 < 1, 푤2 = Figure 2.3. Inflating 훼휀 = [휔휀] to 훼. 1−ℓ1푤1 < 푤1, and so on. For instance, 풘(3) = (1, 1, 1) and 11 3 1 1 1 퐽 풘( ) = (1, 1, , , , ). The multiplicities ℓ푖 of 풘(푎) give Usually, -holomorphic curves are used to obstruct cer- 4 4 4 4 4 tain symplectic embeddings, like in the Nonsqueezing The- the continued fraction expansion of 푎. For 푎 ∈ ℕ, (2.3) orem. In the Inflation Lemma and Step 2, however, other specializes to 퐽-holomorphic curves are used to construct symplectic em- 푠 ̊4 4 푠 ̊4 beddings. E(1, 푘) ,−→ B (퐴) ⟺ ∐ B (1) ,−→ B (퐴). (2.4) 푘 Step 3. 훼2 > 0 and 훼(퐸) > 0 for all exceptional classes if 4 ̊4 The ball packing problem ∐푘 B (1) → B (퐴) is thus in- and only if (i) and (ii) in the theorem hold. 푠 , ̊4 2 cluded in the 1-parameter problem E(1, 푎) −→ B (퐴). Much of this step is rewriting: the inequality 훼 > 0 The function 푐(푎) was computed in [12b] with the help translates to the volume constraint (i), and the inequality of (2.3). The volume constraint is now 푐(푎) ⩾ √푎. Re- 훼(퐸) > 0 for an exceptional class translates to (ii) if we 푘 call that the Fibonacci numbers are recursively defined write 퐸 in the natural basis, 퐸 = 푑퐿 − ∑ 푚 퐸 . We are 푖=1 푖 푖 by 푓−1 = 1, 푓0 = 0, 푓푛+1 = 푓푛 + 푓푛−1. Denote by thus left with showing that a class 퐸 = (푑; 푚 , … , 푚 ) that 1 푘 푔푛 ∶= 푓2푛−1 the odd-index Fibonacci numbers, hence satisfies (DE) is exceptional exactly if it reduces to (0; −1) (푔0, 푔1, 푔2, 푔3, 푔4, … ) = (1, 1, 2, 5, 13, … ). The sequence under Cremona moves. This follows from a combinatorial 푔푛+1 훾푛 ∶= , argument of Li and Li; see [12b]. 푔푛 2.3. Ellipsoids. We now look at embeddings of 4- 5 13 (훾 , 훾 , 훾 , 훾 ,… ) = (1, 2, , ,… ) , dimensional ellipsoids 0 1 2 3 2 5 2 2 2 휋|푧1| 휋|푧2| 2 1+√5 E(푎, 푏) = {(푧1, 푧2) ∈ ℂ ∣ + ≤ 1}. converges to 휏 , where 휏 ∶= is the Golden Ratio. 푎 푏 2 4 2 Define the Fibonacci stairs as the graph on [1, 휏 ] made This is the ellipsoid in ℂ whose projections to the coordi- from the infinitely many steps shown in Figure 2.5, where nate planes are closed discs of area 푎 and 푏. Again we take 2 푔푛+1 2 푔푛+2 푎푛 = 훾푛 = ( ) and 푏푛 = . The slanted edge starts as target a ball, and after rescaling study the function 푔푛 푔푛 푠 푐(푎) = inf {퐴 ∣ E(1, 푎) ,−→ B̊4(퐴)} , 푎 ⩾ 1. on the volume constraint √푎 and extends to a line through the origin. Since this function is continuous, we can assume that 푎 is rational. Then each leaf of the characteristic foliation Ellipsoid Embedding Theorem. on the boundary of E(1, 푎) is closed. Proceeding as be- (i) On the interval [1, 휏4] the function 푐(푎) is given by 4 2 fore, we compactify B̊ (퐴) to ℂP (퐴) and, given an embed- the Fibonacci stairs. 푠 2 17 ding E(1, 푎) ,−→ B̊4(퐴) ⊂ ℂP (퐴), remove the image and (ii) On the interval [휏4,( )2] we have 푐(푎) = √푎 except 6 collapse the remaining boundary along the characteristic on nine disjoint intervals where 푐 is a step made from foliation. But now this yields a symplectic orbifold with two segments. The first of these steps has the vertex 8 17 one or two cyclic quotient singularities, coming from the at (7, ) and the last at (8, ). 3 6 special leaves in the two coordinate planes. It may be dif- 17 (iii) 푐(푎) = √푎 for all 푎 ⩾ ( )2. ficult to reprove the results from Seiberg–Witten–Taubes 6

MARCH 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 351 probably never know the answer for all 푏. McDuff in [11a] and independently Hutchings proved that 푠 E(푎, 푏) ,−→E(푐, 푑) ⟺ 푁 푘(푎, 푏) ⩽ 푁 푘(푐, 푑) for all 푘, (2.5) where (푁 푘(푎, 푏)) is the nonincreasing sequence obtained by ordering the set {푚푎 + 푛푏 ∣ 푚, 푛 ∈ ℤ≥0}. Using his embedded contact homology (ECH), Hutchings had asso- ciated with every starshaped subset 퐾 ⊂ ℝ4 a sequence of numbers 푐푘(퐾) that are monotone with respect to sym- plectic embeddings, and for an ellipsoid these ECH ca- pacities equal the above sequence, 푐푘(E(푎, 푏)) = 푁 푘(푎, 푏). The McDuff–Hutchings theorem (2.5) therefore implied that ECH-capacities are a complete set of invariants for the problem of embedding one 4-dimensional ellipsoid into another. These results are all in dimension four, and until re- Figure 2.4. The Fibonacci stairs: the graph of 푐(푎) on [1, 휏4]. cently, not much was known about higher-dimensional symplectic embedding problems beyond the Nonsqueez- ing Theorem. The reason is that in dimension four, 퐽- holomorphic curves are a much more powerful tool, be- cause of positivity of intersections of such curves, and because one usually finds them with the help of the4- dimensional Seiberg–Witten–Taubes theory. However, 4- dimensional ellipsoid embeddings also opened the door for understanding certain symplectic embedding prob- lems in higher dimensions. I describe how they led to packing stability in all dimensions. Given a connected symplectic manifold (푀, 휔) of finite volume, let 푝(푀, 휔) be the smallest number (or infinity) such that for every 푘 ≥ 푝(푀, 휔) an arbitrarily large percentage of the volume of (푀, 휔) can be covered by 푘 equal symplectically embed- ded balls. Table (2.1) shows that 푝(B4) = 9. The finiteness 푛 Figure 2.5. The th step of the Fibonacci stairs. of 푝 is now known for many symplectic manifolds, and in particular for balls in all dimensions. In this case, the Thus the graph of 푐(푎) starts with an infinite completely ingredients of the proof are: regular staircase, then has a few more steps, but for 푎 ⩾ 17 1 (1) McDuff’s observation from [09] that the ellipsoid ( )2 = 8 is given by the volume constraint. 6 36 E(1, … , 1, 푘) can be cut into 푘 equal balls. 푠 (2) The Ellipsoid Embedding Theorem for E(1, 푎) ,−→ B̊4(퐴) and the McDuff–Hutchings theorem (2.5). (3) Ellipsoids admit a suspension construction: for The theorem better explains the packing numbers 푐푘 in any vectors 풂, 풃, 풄, table (2.1) in view of the equivalence (2.4); cf. the blue 푠 푠 E(풂) ,−→E(풃) ⟹ E(풂, 풄) ,−→E(풃, 풄). dots in Figure 2.4. The embedding constraint at 푏푛 comes from a really exceptional exceptional sphere, namely one For instance, (2) yields that in an exceptional class 퐸 = (푑; 퐦) such that 퐦 is parallel 푠 푠 E(1, 푘),−→E(푘1/3, 푘2/3) and E(1, 푘2/3),−→B4(푘1/3) to the weight expansion 풘(푏푛). When McDuff first looked at ellipsoid embeddings for all 푘⩾21. in [09], it was not clear at all that this would unveil an in- Together with (1) and (3) we thus obtain that for these 푘 teresting fine structure of symplectic rigidity. Many more 6 푠 푠 1/3 2/3 푠 6 1/3 infinite staircases have been found by now, and results of ∐ B (1) ,−→E(1, 1, 푘) ,−→ E(1, 푘 , 푘 ) ,−→ B (푘 ) Usher show that a simple-looking problem, such as for 푘 which 퐴 does the ellipsoid E(1, 푎) embed into the poly- and hence 푝(B6) ≤ 21. Already Gromov had proved that disc B2(퐴) × B2(푏퐴), is already so intricate that we shall 푝(B6) ≥ 8. Is it true that 푝(B6) = 8?

352 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 3 A symplectic manifold (푀, 휔) is rational if a multiple of Now consider the two paths of symplectic embeddings 푡 4 ̊4 [휔] takes rational values on all integral 2-cycles. All closed 휑푗 ∶ B (푡) → B (퐴), 휀 ≤ 푡 ≤ 1, defined by restricting the 푡 rational symplectic manifolds have packing stability. The embeddings 휑푗. Reparametrizing the inverse of the path 휑1 additional ingredient in the proof is that every such mani- 푡 on 푠 ∈ [0, 1] and the path 휑2 on 푠 ∈ [1, 2] we obtain a fold can be completely filled by an ellipsoid. path of symplectic embeddings into B̊4(퐴) that connects 퐾 ⊂ ℝ2푛 2푛 4 2 2.4. Connectivity. For a compact set and a - 휑1 and 휑2. After compactifying B̊ (퐴) to ℂP (퐴) by adding 1 dimensional symplectic manifold (푀, 휔) let Emb휔(퐾, 푀) the line ℂP and by symplectically blowing up the images, be the space of symplectic embeddings 퐾 → (푀, 휔), with this path of embeddings gives rise to a path of symplectic the 퐶∞-topology. The results discussed above tell us for 2 forms 휔푠 on Bl(ℂP ) in class [휔푠] as shown in Figure 2.7. As some 퐾 and (푀, 휔) whether this space is empty or not. In in §2.2 apply symplectic inflation to each form 휔푠, deform- the latter case one may study its topology. The first task is ing 휔푠 to a form 푠̃휔 in class [휔0] = [휔2] = 퐴ℓ − 푒. Finally to see whether Emb휔(퐾, 푀) is connected or not. 2푛 blow down the exceptional divisor for each 푠 to get a path We first take 퐾 to be a ball B (푎). Then 4 푠 2 2푛 of embeddings 휙푠 ∶ B (1) ,−→ℂP (퐴). Since each excep- Emb휔(B (푎), 푀) need not be connected. The first counter- 1 4 tional divisor is disjoint from ℂP , each ball 휙푠(B (1)) lies example was Gromov’s camel theorem: for 2푛 ≥ 4 the 2 1 in ℂP (퐴)⧵ℂP , and by a theorem of McDuff from [90] this camel space in ℝ2푛 with eye of width 1 is the set set with the symplectic form obtained after blow-down is ∘ 4 2푛 2푛 indeed symplectomorphic to B̊ (퐴); cf. §3.2 below. Fur- 풞 = {푥1 < 0} ∪ {푥1 > 0} ∪ B (1). thermore, 휙푠 connects 휑1 with 휑2. Now take 푎 > 1 and define the two embeddings 2푛 2푛 휑± ∶ B (푎) → 풞 by 휑±(푧) = 푧 ± 푣, where 푣 is a multiple 휕 2푛 of such that 휑+ takes B (푎) to the right half space and 휕푥1 2푛 휑− takes B (푎) to the left half space; see Figure 2.6. Then 휑+ and 휑− are not isotopic.

Figure 2.7. Inflating 휔푠 to ̃휔푠.

4 4 McDuff’s theorem that Emb휔(B (1), B̊ (퐴)) is con- nected is sharp in the following sense. There are con- vex subsets 퐾 of ℝ4 with smooth boundary that are ∘ Emb (퐾, 휆퐾) Figure 2.6. Nonequivalent balls in the camel space. arbitrarily close to a ball and such that 휔 has at least two components, for some 휆 > 1. For the cube 4 2 2 4 4 C (1) = B (1) × B (1) the space Emb휔( C (1), B̊ (3)) even On the other hand, McDuff showed in [91a] that has infinitely many connected components. 4 4 Emb휔(B (1), B̊ (퐴)) is connected for all 퐴 > 1 and ex- tended this result in [98, 09] to embeddings of any collec- 3. Other Contributions to Symplectic Geometry tion of 4-ellipsoids into a 4-ellipsoid. 3.1. Construction of (counter)examples. McDuff again The key to these results is, again, symplectic inflation. I and again has given explicit constructions that are both el- 4 give the idea of the proof for embeddings of one ball B (1). ementary and profound. You can grasp them quickly, yet So assume we have two embeddings they can be used to do many other things. We already en- 4 푠 4 countered some of them in the previous section, and I now 휑1, 휑2 ∶ B (1) ,−→ B̊ (퐴). mention three more. Since 휑 and 휑 are close to linear maps near the origin, 1 2 A simply connected symplectic manifold that is not Käh- we can find 휀 > 0 and a compactly supported symplectic 4 4 ler. While Kähler manifolds are symplectic, the converse isotopy 휓 of B̊ (퐴) such that 휑1 = 휓 ∘ 휑2 on B (휀). We may 4 is not true. For instance, the odd-index Betti numbers thus assume from the start that 휑1 = 휑2 on B (휀).

MARCH 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 353 of compact Kähler manifolds are even. The first exam- induces the connection 1-form 훽 of the Levi-Civita connec- ple of a compact symplectic manifold that is not Kähler tion on 푇∗Σ and the radial coordinate 푟 on the fibers. Then was found by Thurston; his example (푀, 휔) is a 푇2-bundle one finds smooth functions 푓, 푔 on [0, ∞) such that on the 2 ∗ 1 over 푇 with 푏1(푀) = 3. In [84], McDuff constructed sim- annulus bundle {푥 ∈ 푇 Σ ∣ ≤ 푟(푥) ≤ 1} the 2-form ply connected examples. She symplectically embedded the 2 Thurston manifold 푀 into ℂP5 and showed that the sym- 휔 = 푑(푓(푟)훽 + 푔(푟)휆) plectic blow-up of ℂP5 along 푀 is simply connected and is symplectic and makes the boundary of contact type. has 푏3 = 푏1(푀) = 3. 3.2. The structure of rational and ruled symplectic 4- Cohomologous symplectic forms that are not diffeo- manifolds. The classification of compact complex sur- morphic. Consider two symplectic forms 휔0 and 휔1 on faces is an old and beautiful topic in complex and Käh- a closed symplectic manifold that are cohomologous. If ler geometry. Before 1990, the only result on symplectic 4- these two forms can be connected by a smooth path 휔푡 of manifolds in this direction was a theorem of Gromov for cohomologous symplectic forms, then there exists an iso- the complex projective plane. In [90], McDuff proved the topy of 푀 deforming 휔0 to 휔1, by Moser’s trick. This is the following generalization. If a closed symplectic 4-manifold case, for instance, for any two area forms on a closed sur- contains a symplectically embedded sphere with nonnegative face. In general, however, there may be no such path, as self-intersection number, then it is symplectomorphic to either McDuff showed in [87]. ℂP2 with its standard Kähler structure, to a ruled symplectic Take 푀 = 푆2 ×푆2 ×푇2 with the standard split symplectic manifold, or to a symplectic blow-up of one of these manifolds. form 휔 giving all factors area one. Define Here, a ruled symplectic 4-manifold is the total space of an 푆2-fibration over a closed oriented surface with a sym- Ψ(푧, 푤, (푠, 푡)) = (푧, 푅 (푤), 푠, 푡), 푧,푡 plectic structure that is nondegenerate on the fibers. where 푅푧,푡 is the rotation by angle 2휋푡 of the round sphere In later work with Lalonde [96a], McDuff classified about the axis through 푧, −푧. Then the symplectic forms ruled symplectic surfaces. If (푀, 휔) is a ruled symplectic 4- 푘 ∗ 휔푘 = (Ψ ) 휔, 푘 ∈ ℤ≥0, are all cohomologous, and they can manifold, then 휔 is determined up to symplectomorphism by be joined by a path of symplectic forms, but by no such path its cohomology class and is isotopic to a standard Kähler form in the same cohomology class. The last part of the statement on 푀. Li–Liu complemented this result by showing that was McDuff’s first result obtained by using 퐽-holomorphic if (푀, 휔) is the total space of an 푆2-fibration over a closed curves. She studied the space of 휔푘-tame 퐽-holomorphic surface, then there is a ruling of 푀 by symplectic spheres, spheres in the class of the first 푆2-factor, and showed that i.e., (푀, 휔) is a ruled symplectic manifold. See [96b] for a such spheres wrap 푘-times around the second factor. Since survey on this classification. a diffeomorphism isotopic to the identity acts trivially on An elementary but important point in the proofs is that homology, an isotopy deforming 휔푘 to 휔ℓ can therefore two cohomologous symplectic forms 휔0 and 휔1 are diffeo- only exist if 푘 = ℓ. morphic if they tame the same almost complex structure 퐽. Also in [87] McDuff improved this construction Indeed, all the cohomologous forms 휔푡 = (1 − 푡)휔0 + 푡휔1, to obtain cohomologous symplectic forms on an 8- 푡 ∈ [0, 1], then tame 퐽 and hence are symplectic, and there- dimensional symplectic manifold that are not even diffeo- fore 휔0 and 휔1 are diffeomorphic by Moser’s argument. morphic. These works showed that symplectic geometry has Disconnected contact boundaries. The natural bound- something to say about 4-manifolds, and they helped es- aries of symplectic manifolds are those of contact type, tablish symplectic geometry as one of the core geometries. meaning that there exists a vector field 푋 defined near 휕푀 McDuff has also done much interesting work on the that is transverse to 휕푀, pointing outwards, and is confor- topology of the group of symplectomorphisms for several 2 2 mally symplectic: ℒ푋 휔 = 푑휄푋 휔 = 휔. Symplectic mani- classes of symplectic manifolds. For 푆 -fibrations over 푆 folds with boundary of contact type are analogous in many with any symplectic form she has in particular shown in ways to complex manifolds with pseudoconvex bound- joint work with Abreu [00] that two symplectomorphisms aries, as was shown by Eliashberg and Gromov in 1989. are isotopic through symplectomorphisms whenever they In the latter situation, the boundary is always connected. are isotopic through diffeomorphisms. However, McDuff [91b] explicitly constructed a compact 3.3. Hofer geometry. Recall that any symplectic mani- symplectic 4-manifold whose boundary is of contact type fold (푀, 휔) locally looks like the standard symplectic vec- and disconnected. tor space of the same dimension. Our dear geometric in- She starts with the cotangent bundle 푇∗Σ over a closed tuition from everyday life, so useful in Riemannian geo- orientable surface of genus ≥ 2, endowed with its canon- metry to see distances and curvatures, is thus useless in ical symplectic form 푑휆, where 휆 = ∑푖 푝푖푑푞푖. Also symplectic geometry. However, on the automorphism take a Riemannian metric of constant curvature on Σ. It group Hamc(푀, 휔) of Hamiltonian diffeomorphisms that

354 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 3 are generated by compactly supported functions, there is a and separates the small fibers from the large ones. In the 푡 bi-invariant Finsler metric, that to some extent serves as key step, one then uses the flow 휙퐻 , 푡 ∈ [0, 1], to lift the a substitute for the absence of local geometry in (푀, 휔). small green ball. The projection of the red band in the 푎 Given a time-dependent Hamiltonian function 퐻 ∶ 푀 × image of 휆 to ℝ2 has area ‖퐻‖. Since 휑(B2푛( )) is disjoint 2 [0, 1] → ℝ with compact support, take the integrated oscil- 2푛 from B (푎), one can thus turn the green ball over the blue lation part to obtain an embedding 1 푠 푎 ‖퐻‖ ∶= ∫ (max 퐻(푥, 푡) − min 퐻(푥, 푡)) 푑푡. B2푛+2(푎) ,−→B2( + ‖퐻‖) × 푀. 푥∈푀 0 푥∈푀 2 푎 For 휑 ∈ Hamc(푀, 휔) now define Hence ‖퐻‖ ≥ by the General Nonsqueezing Theorem. 2 푑(휑, id) = inf ‖퐻‖, You can readily grasp the construction from the three fig- 퐻 ures on pages 473, 474, 475 of [17b]. Symplectic folding where 퐻 runs over all compactly supported Hamiltonian found many other applications to symplectic geometry. functions whose time-1 flow map is 휑. Then 푑(휑, 휓) = In [95b], Lalonde and McDuff made a deep study of 푑(휑휓−1, id) defines a bi-invariant metric on Hamc(푀, 휔). geodesics in this Finsler geometry on Hamc(푀, 휔), that The only difficult point to check is nondegeneracy. This lead to completely new geometric intuitions on this group. was done by Hofer for ℝ2푛 by variational methods, by It then became possible to think about Hamiltonian dy- Polterovich for tame rational symplectic manifolds by us- namics in geometric terms such as geodesic, conjugate ing a rigidity property of Lagrangian submanifolds, and point, cut-locus, etc. See Polterovich’s book from 2001 for for all symplectic manifolds by Lalonde–McDuff [95a] more on this. who used their General Nonsqueezing Theorem discussed The books. While in her papers Dusa shows herself to be in §2.1 and the following symplectic folding construction. an impressive and creative problem solver, the two books she wrote with were (and remain) cru- cial for the foundation of symplectic geometry and its dis- semination. Introduction to symplectic topology [17b] ex- plains the methods and results of the field in clear and modern geometric language. A key for the success of this book is that it discusses the main results in the most im- portant and typical cases, without striving for generality. I was very lucky that this book came out (in 1995) just when I wanted to learn the subject. It is one of the main reasons for the transformation of the then small community of symplectic geometers into a large family. In the third edi- tion from 2017, the authors added a chapter with 54 open problems, proving that symplectic geometry is not a closed chapter but rather remains an exploding field. The second book [12a] provides a rigorous foundation of the theory of 퐽-holomorphic curves and explains their applications to symplectic topology. The exposition is so precise and to the point that one can often just cite the re- sult one needs. This book transformed the formerly some- Figure 3.1. The symplectic folding construction, schematically. what romantic theory of 퐽-holomorphic curves into a well- established tool of enormous impact. In addition to her incredible mathematical legacy, Assume that 휑 ≠ id. Then we find a symplectically em- 2푛 2푛 2푛 Dusa’s extraordinary generosity to young (and not so bedded ball B (푎) in 푀 such that 휑(B (푎)) ∩ B (푎) = ∅. young) researchers, her enthusiasm and her (sometimes Now consider the ball overwhelming) energy, and her heartfelt commitment in B2푛+2(푎) ⊂ B2(푎) × B2푛(푎) ⊂ ℝ2 × 푀, scientific, political, and social issues should be recognized. and assume that 휑 is generated in time 1 by 퐻. In three 푎 ACKNOWLEDGMENTS. I am grateful to the two refer- steps, the small green ball B2푛+2( ) ⊂ B2푛+2(푎) is folded 2 ees for the improvements, and to Jesse Litman for her on top of its complement, as illustrated in Figure 3.1. First, patient help with the English. one views B2푛+2(푎) as a B2푛-fibration over the disc B2(푎)

MARCH 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 355 References (2) 175 (2012), no. 3, 1191–1282, DOI 10.4007/an- [84] Dusa McDuff, Examples of simply-connected symplectic non- nals.2012.175.3.5. MR2912705 Kählerian manifolds, J. Differential Geom. 20 (1984), no. 1, [17a] Dusa McDuff, My life, Notices Amer. Math. Soc. 267–277. MR772133 64 (2017), no. 8, 892–896, DOI 10.1090/noti1560. [87] Dusa McDuff, Examples of symplectic structures, Invent. MR3676755 Math. 89 (1987), no. 1, 13–36, DOI 10.1007/BF01404672. [17b] Dusa McDuff and Dietmar Salamon, Introduction MR892186 to symplectic topology, 3rd ed., Oxford Graduate Texts [90] Dusa McDuff, The structure of rational and ruled symplectic in Mathematics, Oxford University Press, Oxford, 2017. 4-manifolds, J. Amer. Math. Soc. 3 (1990), no. 3, 679–712, MR3674984 DOI 10.2307/1990934. MR1049697 [17c] Dusa McDuff and Katrin Wehrheim, Smooth Kuranishi [91a] Dusa McDuff, Blow ups and symplectic embeddings in atlases with isotropy, Geom. Topol. 21 (2017), no. 5, 2725– dimension 4, Topology 30 (1991), no. 3, 409–421, DOI 2809, DOI 10.2140/gt.2017.21.2725. MR3687107 10.1016/0040-9383(91)90021-U. MR1113685 [19] Dusa McDuff, Notes on Kuranishi atlases, Virtual fun- [91b] Dusa McDuff, Symplectic manifolds with contact type damental cycles in symplectic topology, Math. Surveys boundaries, Invent. Math. 103 (1991), no. 3, 651–671, DOI Monogr., vol. 237, Amer. Math. Soc., Providence, RI, 2019, 10.1007/BF01239530. MR1091622 pp. 1–109, DOI 10.1070/rm1970v025n06abeh001267. [94a] Dusa McDuff, Notes on ruled symplectic 4-manifolds, MR3931094 Trans. Amer. Math. Soc. 345 (1994), no. 2, 623–639, DOI 10.2307/2154990. MR1188638 [94b] Dusa McDuff and Leonid Polterovich, Symplectic pack- ings and algebraic geometry, Invent. Math. 115 (1994), no. 3, 405–434, DOI 10.1007/BF01231766. With an appendix by Yael Karshon. MR1262938 [95a] François Lalonde and Dusa McDuff, The geometry of symplectic energy, Ann. of Math. (2) 141 (1995), no. 2, 349– 371, DOI 10.2307/2118524. MR1324138 [95b] François Lalonde and Dusa McDuff, Hofer’s 퐿∞- geometry: energy and stability of Hamiltonian flows. I, II, Invent. Math. 122 (1995), no. 1, 1–33, 35–69, DOI Felix Schlenk 10.1007/BF01231437. MR1354953 Credits [96a] François Lalonde and Dusa McDuff, The classification of ruled symplectic 4-manifolds, Math. Res. Lett. 3 (1996), no. 6, Opening photo is courtesy of Alejandro Adem. 769–778, DOI 10.4310/MRL.1996.v3.n6.a5. MR1426534 All article figures and author photo are courtesy of Felix [96b] François Lalonde and Dusa McDuff, 퐽-curves and the Schlenk. classification of rational and ruled symplectic 4-manifolds, Con- tact and symplectic geometry (Cambridge, 1994), Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 3–42. MR1432456 [98] Dusa McDuff, From symplectic deformation to isotopy, Top- ics in symplectic 4-manifolds (Irvine, CA, 1996), First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 85– 99. MR1635697 [00] Miguel Abreu and Dusa McDuff, Topology of symplecto- morphism groups of rational ruled surfaces, J. Amer. Math. Soc. 13 (2000), no. 4, 971–1009, DOI 10.1090/S0894-0347-00- 00344-1. MR1775741 [09] Dusa McDuff, Symplectic embeddings of 4-dimensional ellipsoids, J. Topol. 2 (2009), no. 1, 1–22, DOI 10.1112/jtopol/jtn031. MR2499436 [11a] Dusa McDuff, The Hofer conjecture on embedding symplec- tic ellipsoids, J. Differential Geom. 88 (2011), no. 3, 519– 532. MR2844441 [11b] celebratio.org/McDuff_D [12a] Dusa McDuff and Dietmar Salamon, 퐽-holomorphic curves and symplectic topology, 2nd ed., American Mathemat- ical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI, 2012. MR2954391 [12b] Dusa McDuff and Felix Schlenk, The embedding ca- pacity of 4-dimensional symplectic ellipsoids, Ann. of Math.

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