Summary RESEARCH STATEMENT

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1 RESEARCH STATEMENT WEIWEI WU Summary My research lies in symplectic geometry and topology. Specically, I have focused on three dierent themes: existence and classication of Lagrangian submanifolds, topology of symplectomorphism groups, and nite group actions on symplectic manifolds. (1) Lagrangian submanifolds. Research on Lagrangian submanifolds is a central topic in symplectic geometry; a fundamental problem is to classify them in nice situations. For example, one of the most famous problems in the eld, the Arnold's nearby conjecture, asserts that exact Lagrangian submanifolds in a cotangent bundle are all Hamiltonian isotopic to the zero section. In [47], Wu13 I provided a complete classication of exact Lagrangians up to Hamiltonian isotopy in A n -surface singularities [47] Wu13 (see Section 1.2), s:uniqueness yielding a solution of a generalized version of Arnold's conjecture in this case. In compact cases, I established a series of results showing Lagrangian uniqueness in rational surfaces in dierent cases (up to Hamiltonian isotopy/symplectomorphism/smooth isotopy), as well as giving exotic examples of Lagrangian RP 2 embeddings [7]. BLW12 Finally, I found an exotic Lagrangian torus in CP 2 [46]. Wu12 This torus is realized as a ber of a semi-toric bration and is superheavy in the sense of symplectic quasi-morphism theory, exhibiting a new phenomenon that resolves a question regarding stem (see Section 1.4) s:lagin due to Entov and Polterovich [11]. EP2 (2) Symplectomorphism groups. Topology and geometry of symplectomorphism groups has been an important problem since Gromov's original work [21], Gromov which shows the striking result that the symplectomorphism group of monotone S 2 S 2 deformation retracts to SO(3) SO(3). I have shown a series of results in this direction (see [24, 32, 47, 30, 35]). HPW12,LW12, Wu13, LLW, PW13 Among others, the upcoming joint work with M. Pinsonnault resolve the long-awaiting problem of contractibility of the group of Hamiltonians on Del Pezzo surfaces of degree 6. Along the way I developed a new construction of symplectomorphisms which we call ballswappings. This construction provides a bridge from Lagrangian Dehn twists to the whole symplectomorphism group of certain symplectic manifolds. Aside from these applications, such symplectomorphisms seem to be of independent interests. (3) Symplectic Cremona maps. In algebraic geometry, a Cremona map is a birational automorphism on P n. A classical result in birational geometry, due to Noether, asserts that a birational transformation on CP 2 can be decomposed into a series of quadratic transformations. In [32], LW12 this result was recovered on the homological level in the symplectic category. A further fundamental result in complex Cremona theory says, for a nite group G, a minimal rational G-surface either admits a conic bundle structure or is a Del Pezzo surface. Recently in the joint with with W. Chen and T.-J. Li [8], CLW13 we showed the same result holds true for symplectic G-actions on Kahler surfaces, by coupling symplectic techniques 1

2 2 WEIWEI WU with the celebrated Mori theory. This serves as a cornerstone for further study in nite Cremona subgroups in the symplectic category. As we proceed, I will also explain various connections from my current work to the future plans of research on Hofer geometry. s:lag 1. Topology of Lagrangian submanifolds 1.1. Non-emptiness of L. Recall that a symplectic rational surface is a symplectic blow-up of CP 2 or S 2 S 2. It is known to Eliashberg-Givental-Hofer that no Lagrangian submanifold with genus> 1 could occur. Lagrangian tori appear in abundance, where it even becomes an industry to produce new examples as well as to distinguish them (see Section 1.4). s:lagin In contrast, general results on Lagrangian spheres is missing in the current literature. Jointly with T.-J. Li, in LW12 [32] I proved the following existence result for Lagrangian S 2 when M is a symplectic rational surface. Theorem 1. Given homology class 2 H 2 (M; Z), let K be the canonical class of M. has an embedded Lagrangian spherical representative if and only if (i)!() = 0, and (ii) is a K-null spherical class. Here a K-null spherical class means pairs trivially with K and is represented by a smooth embedded sphere. The \only if" part follows from Weinstein's neighborhood theorem. Therefore, the theorem essentially guarantees the existence of Lagrangian spherical representatives as long as the homology class under investigation satises minimal necessary conditions. The proof also classies classes which are K-null by providing a simple algorithm, and involves a heavy use of the correspondence between Gromov-Taubes invariants and Seiberg-Witten invariants, as well as the Symplectic Field Theory. :uniqueness (L )-isotopy uniqueness of Lagrangians. Uniqueness up to smooth isotopy of embedded 2-surfaces in 4-manifolds has attracted major interests since the work of Donaldson-Kroheimer- Mrowka. When it comes to uniqueness up to Hamiltonian (symplectomorphism) group actions, the problem is veiled with another layer of mystery due to Seidel's exotic Lagrangian embedding constructions. These problems are the topics for the current section Non-compact symplectic manifolds and exact Lagrangian uniqueness. Recall that a symplectic manifold (M;!) is called exact if! = d for some one form. A Lagrangian submanifold L is called exact if j L is again an exact one form. Arnold's nearby conjecture states that all exact Lagrangians in a cotangent bundle endowed with a canonical exact symplectic structure are all Hamiltonian isotopic to the zero section. Many versions of Arnold's conjecture are dened and veried. In higher dimensions, the most relevant work available among others is [1], Ab12 which proved any exact Lagrangian embedding is a homotopy equivalence to the cotangent bundle. Richard Hind in [23] Hi12 got the rst result in the original form of Arnold's conjecture, proving that all Lagrangian S 2 and RP 2 's are Hamiltonian isotopic to the zero section in their own cotangent bundles. In the joint work with T.-J. Li [32], LW12 I found a new proof of Hind's results.

3 RESEARCH STATEMENT 3 In a more general framework, note that the zero section of a cotangent bundle is the isotropic skeleton of the cotangent bundle. Roughly speaking, an isotropic skeleton is a CW-complex with the restriction! vanishing wherever it is smooth. See for example [6] Bi01 for its precise denition. Slightly generalizing Arnold's question, one wonders what the relation is between embedded exact Lagrangians and the isotropic skeleton in an exact symplectic manifold. This philosophy is originally due to Kontsevich. One simplest example of exact symplectic manifolds with a non-smooth isotropic skeleton is the A n -singularity, which is the ane variety in C m+1 cut out by the equation: x x 2 m + x n+1 m+1 = 1: We will denote this variety by A m n. When m = 2, we call this variety an A n -surface singularity. Symplectically, A m n -singularties can be identied with the plumbing of n copies of T S m, and their zero sections are called standard spheres. Because of the special symmetry of Lagrangian spheres, Se12 A n -singularities acquires rather rich geometry, attracting a lot of eorts (e.g. [42]) but already exhibits much complications. This more general version of Arnold's nearby conjecture was also considered by Richard Hind, who proved in [23] Hi12 that, for A 1 and A 2, Lagrangian spheres are unique up to Hamiltonian isotopy and Dehn twists along spheres. In my recent work [47], Wu13 I showed that the same holds true for all A n -surface singularities. Coupling with a result of Ritter [36] Ri10 which asserts that exact Lagrangians in A n -surface singularities are all spheres, my result yields the following corollary: Theorem 2 ([47]). Wu13 Exact Lagrangians in A n -surface singularites are isotopic to each other up to a composition of Lagrangian Dehn twists along n standard spheres. This seems the rst complete result for the generalized version of Arnold's nearby conjecture. In particular, it implies that any exact Lagrangian in A n -surface singularity is generated by the standard spheres in the sense of Fukaya category. This was originally proved in [25] IU05 [26] IUU10 from the point of view of mirror symmetry and computations on derived category of sheaves, and already found interesting applications [31]. LM12 Our method should deal with more general plumbings with clever choices of compactications, which will be a direction of future study. This seems to involve understandings of contact llings. See more discussions in Section 4.2. s:more exact Lagrangian uniqueness in compact manifolds. For compact symplectic manifolds, we also obtained the following uniqueness result: Theorem 3 ( LW12,LLW [32, 30]). In symplectic rational surfaces with 7, homologous Lagrangian spheres and RP 2 's are all Hamiltonian isotopic to each other. This result is known to Evans [13] Ev10 in the monotone cases (i.e. when [c 1 ] = [!]). Again one wonders what can be said about rational surfaces with large Euler numbers. It seems unlikely that uniqueness up to Hamiltonian isotopies can be obtained any more, due to a counterexample of Seidel [41] Se08 when = 8. However, we still have the following uniqueness result, which also holds for Lagrangian RP 2 for 11. Theorem 4 ([7]). BLW12 Let M = CP 2 #kcp 2 be a rational symplectic manifold. Then,

4 4 WEIWEI WU (1) the symplectic Torelli group Symp h (M;!) acts transitively on homologous Lagrangian spheres, and (2) homologous Lagrangian spheres are smoothly isotopic to each other. The same holds true for Lagrangian RP 2 when k 8 for both items. The question remains open when it comes to Lagrangian RP 2 for k 9. following example in [7] BLW12 which exhibits the diculty of this problem. But we have the p:krp2 Proposition 5. Let M = CP 2 #kcp 2. (1) If k 9, then there exists L a Lagrangian RP 2 and S a symplectic ( 1) sphere for some symplectic form! on M, such that L and S have trivial Z2 intersection, but L and S cannot be made disjoint with a smooth isotopy. (2) If k 10, then there exists L 0 and L 1 that are Z2-homologous smoothly embedded RP 2 's, which are not smoothly isotopic. Furthermore there are deformation equivalent symplectic forms! 0 and! 1 on M so that L i (M;! i ) are Lagrangians. This example (and its proof) shows that, to answer whether Lagrangian RP 2 's are unique up to smooth isotopies or symplectomorphisms, one should consider a ball-packing size problem. This explicit application of ball-packing problem seems new and worth further investigation Weak homotopy type of L. It is in general believed that L is \as dicult as" Symp(M) according to known computations. In particular, the only available result before in non-compact symplectic manifolds is due to R. Hind [23], Hi12 where he proved the space of Lagrangian S 2 in T S 2 is contractible, and conjectured the same holds true for Lagrangian RP 2 in T RP 2. I was able to resolved this problem in the joint work [24] HPW12 with R. Hind and M. Pinsonnault by showing: Theorem 6. The space of Lagrangian RP 2 in T RP 2 is weakly contractible. s:lagin The method we used applies equally well for T S 2, which gives a shorter proof than that in the existing literature Non-displaceability of Lagrangian submanifolds Background. A prevalent approach to the non-displaceability of Lagrangian submanifolds is to compute the Lagrangian Floer cohomology (see for example [16] FOOO for a framework set up for general symplectic manifolds). In recent years, a novel approach due to Entov and Polterovich shed new lights on the problem[11, EP2,EP1 10]. Using spectral invariants dened by Hamiltonian Floer homology, they introduced the notions of symplectic quasi-morphisms and symplectic quasi-states, which are certain Hamiltonian-invariant functionals on the space of Hamiltonian functions satisfying a list of axioms. Given a symplectic quasi-state and a subset S M, S is called heavy if: (F ) inf x2s F (x); 8F 2 C1 (M); and superheavy if (F ) sup F (x); x2s 8F 2 C 1 (M):

5 RESEARCH STATEMENT 5 Further, when is xed, a superheavy subset is always heavy, and any -heavy subset is not displaceable. It is not hard to show that the symplectic quasi-states can be integrated over a moment polytope, which implies that, if all but one special ber of the moment polytope are displaceable, then the special ber must be superheavy with respect to any symplectic quasi-states. Such a special ber is called a stem. This idea is extended to moment polytopes with singularities. Question (Entov-Polterovich). Is the standard Lagrangian RP 2 CP 2 a stem? The corresponding question of the anti-diagonal in S 2 S 2 was resolved negatively by several independent groups, see [12] ElP and its introduction part. The spirit of this problem is to understand how many symmetries a Lagrangian needs to acquire to be a stem Results. Towards Entov-Polterovich's question, I considered a semi-toric system on CP 2, which roughly means it has two commuting Hamiltonian functions where one of them is integrable. Such a semi-toric system still has a moment polytope in the complement of the standard RP 2. From this point of view, I showed the following result, which answers Entov-Polterovich's question negatively: Theorem 7 ([46]). Wu12 One of the toric ber is superheavy with respect to certain symplectic quasi-state and is disjoint from the standard RP 2. In particular, this implies such a ber is non-displaceable, and the standard Lagrangian RP 2 is not a stem. The proof relies on computations of Lagrangian Floer cohomology using the machinery developed FOOO,FOOOtoric I,FOOOtoric II,FOOOdeg,FOOOspectral by Fukaya, Oh, Ohta and Ono [16, 17, 18, 19, 20]. However, using symplectic quasi-states to detect non-displaceable Lagrangian submanifolds also applies to other symplectic manifolds admitting toric action on an open part. In [7], BLW12 and an unpublished work joint with M.S.Borman, I was able to detect non-displaceable bers in certain toric picture of M = CP 2 #3CP 2 without any computation on Lagrangian Floer cohomology of the toric bers. Recently, the construction of this semi-toric ber is extended by J. Oakley and M. Usher to higher dimensions, which led to displaceable monotone Lagrangians, which is a new phenomenon. They were also able to show that the exotic ber is Hamiltonian isotopic to the Chekanov torus Discussions. The torus considered above has a nice topological interpretation. To explain this we start with T RP 2. In a cotangent ber of T RP 2 we take a unit circle by choosing an appropriate induced Riemannian metric. Then the trace under the geodesic ow of this unit circle is a Lagrangian torus. This construction is rst due to Polterovich and constructed in T S 2 [3]. AF Now choose a disk bundle of T RP 2 with radius bigger than 2. By quotienting the circle action generated by geodesic ow on its boundary, we retrieve CP 2. Then the torus under consideration is the proper transform of Polterovich's torus. Having explained this, one immediately sees that an analog of our result in higher dimensions, including both CP n corresponding to T RP n, and the n dimensional quadric Q n corresponding to T S n, should be available. Corresponding Lagrangians in higher dimensions are no longer a torus but S 1 S n 1 except for dimension 3. Similar results should be proved for these Lagrangians, and some of them are already available due to Oakley-Usher. Let's now focus on T S 3, which is particularly interesting. It obtains both S 1 S 2 and a 3-torus using similar constructions. The 3-torus appears as follows: since SO(4) has rank 2, its maximal

6 6 WEIWEI WU torus acts acts as cotangent map on T S 3. This would not work for n > 3 since SO(n + 1) has rank < n 1. Adding the above torus action to the geodesic ow yields a toric action on the complement of the Lagrangian S 3. The preimages of the ray with the best symmetric property in the moment polytope are appropriate analogues of Polterovich's torus. Such tori should have interesting behavior under the so-called conifold transition, which replaces the Lagrangian S 3 by a symplectic S 2. This transition is not local in the symplectic category, which makes the problem much harder to approach. So the rst step of understanding such a torus and the conifold transition eects seems to be nding a convenient interpretation for the transition in some favorable situations. s:symp 2. Topology of Symplectomorphism groups 2.1. Background. The homotopy type of symplectomorphism groups has been extensively studied by various authors since the seminal work of Gromov in [21]. Gromov However, the problem exhibits considerable complications even in symplectic manifolds with very simple structure, which is related to the bubbling pheonomenon in Gromov-Witten theory, see [4]. AM Also, a natural extension of the question is to nd the compactly supported counterpart for non-compact symplectic manifolds. This is an area of little research before to the best of my knowledge Results. On the compact side, a breakthrough was made recently by J.D.Evans (independently, by M. Pinsonnault), who found the weak homotopy type for CP 2 #kcp 2 endowed with the monotone symplectic forms, k 5. However, since Evans' proof heavily depend on choosing a simply-connected curve conguration, it does not generalize to k 6. I overcame this problem in the joint work with M. Pinsonnault, completing the long-awaited picture for symplectic Del Pezzo surfaces since the preprint version [41] Se08 was available in 2003: Theorem 8 ([35]). PW13 Ham(CP 2 #kcp 2 ;!) is weakly contractible for k 6 when! is monotone. On the non-compact side, a natural example is to consider is A n -surface singularity. (See the denition in Section 1). s:lag The symplectomorphism group of A m n was pursued rst in [27], KS02 where Khovanov and Seidel proved that the braid group Br n+1 embeds in 0 (Symp c (A m n )) by exploiting deep relations between Lagrangian Dehn twists [41] Se08 and representations of braid groups on the Fukaya category. Only recently, Jonny Evans [14] Ev11 showed that all i (Symp c (A 2 n)) = 0 for i > 0, and obtained another direction of monomorphism from 0 (Symp c (A m n )) to Br n+1. These results imply that 0 (Symp c (A 2 n)) is a co-hopan subgroup of Br n+1. We strengthen this result to the following theorem and complete the computation: Theorem 9 ([47]). Wu13 0 (Symp c (A 2 n)) = Br n+1. Moreover, both maps from Khovanov-Seidel and Evans are isomorphisms in A 2 n. Another direction remained not being addressed in the literature is the symplectomorphism groups of manifolds with concave ends (i.e. there exists a vector eld X satisfying L X! =! which points inward). In general such objects are quite complicated, so we restricted our scope to the symplectization of lens spaces with spherically symmetric contact forms, which we denoted as sl(n; 1). Jointly with Richard Hind and Martin Pinsonnault, I showed:

7 RESEARCH STATEMENT 7 Theorem 10 ([24]). HPW12 The homotopy type of symplectomorphism groups of sl(n; 1) are exactly the loop groups of contact isometry of L(n; 1). They are in turn a union of countably many free loop groups of SO(3). This result seems the rst computation available on the homotopy type of Symp(M) when M has a concave end. s:crem 3. Symplectic Cremona Maps 3.1. Background. In complex algebraic geometry, a Cremona transformation is a birational automorphism of CP n. We will restrict our attention to n = 2 in the rest of our discussions. Back in the 19th century, Max Noether and Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation. Back then, a problem Castelnuovo and Kantor tried to further address was the nite subgroups in plane Cremona group. The research for this question has extended for a whole century. Despite the interests of this problem, the classication of nite subgroups of the plane Cremona group was only completed very recently by I. Dolgachev and V. Iskovskikh [9]. DI09 For a modern introduction to this subject, see for example [2]. AC02 In symplectic geometry, counterparts of ingredients in plane Cremona theory are evident, that Cremona maps corresponds to symplectomorphisms on symplectic rational surfaces CP 2 #kcp 2, where k can take arbitrary value, and the standard quadratic transformation corresponds to Lagrangian Dehn twists. In this context, we wonder the role of these classical results in the symplectic category Results. The following result recovers Noether and Castelnuovo's theorem on a homological level: Theorem 11 ([32]). LW12 For any f 2 Symp(M) when M is a symplectic rational surface, there is a sequence of Lagrangian Dehn twists i, i = 1; : : : ; r, such that the action of 1 r on H 2 (M; Z) coincides with that of f. The proof relies on classication of homology classes of Lagrangian spheres in rational surfaces in the previous sections. Such a result also extended one by Y. Ruan [37] Ru93 for rational surfaces with Euler number at most 11. One step further, in the upcoming joint work with W. Chen and T.-J. Li, I investigate the nite subgroup of symplectic rational surfaces. The rst result is: Theorem 12 ([8]). CLW13 For (X;!) a symplectic minimal rational G-surface which admits a Kahler structure, whose complex structure is denoted as J, then either X admits a conic bundle structure, or (X; J) is a Del Pezzo surface. The algebro-geometric counterpart of this result is the cornerstone for the study of nite subgroups of Cremona groups in algebraic geometry, which again dates back to 19th century. We have borrowed inspirations from the modern approach of this algebo-geometric result, and applied G- equivariant Mori theory to the symplectic setting. One expects this theorem should open the door for the study of nite subgroups of symplectic Cremona group. On the other hand, this happen to coincide again with Ruan's idea of introducing Mori program into symplectic geometry in [37]. Ru93

8 8 WEIWEI WU s:more 4. More ongoing projects :more exact 4.1. Hofer geometry and the Zimmer's program. Let (M 2n ;!) be a general symplectic manifold. One attractive feature of the Hamiltonian group Ham(M) is that it comes with a natural bi-invariant metric ( ; ), discovered rst by Helmet Hofer. Let f 2 Ham(M), by denition F f := ff : S 1 M! R : the time 1 ow of X F (t; ) is fg is non-empty. Here df t =!(X Ft ; ). Then its Hofer norm is dened as (id; f) = min F t2f f Z 1 0 Z dt (max F M x2m Now the Hofer metric is dened as (f; g) = (id; f 1 g). geometry is that Conjecture. Ham(M;!) has innite Hofer diameter. min x2m F )!n : A fundamental questions in Hofer The rst proved example is due to L. Polterovich [34], PolS where he proved this result for S 2. This problem has attracted major eorts by various authors, and inspired many new ideas such as symplectic quasi-states [10] EP1 and boundary depth [44]. UshBD One of the most popular approach is to use the lower bound provided by the spectral invariants dened by Schwartz [38] Schw and Oh [33]. OhSp Inspired by the philosophy of nding homotopy type of symplectomorphism, I designed a new approach in my upcoming work [48] Wu13-2 by decomposing the problem into symplectic congurations and their complements. Concretely, we have Usher's weightless lemma [45] UshSub which in particular implies that the Hofer norm of symplectic congurations degenerates. Therefore, one only needs to consider the symplectomorphism group of its complement. This is in turn related to symplectic cohomology. It is desirable to extract more information out of symplectic cohomology due to its rich algebraic structures, which potentially helps understanding our original question. Polterovich also applied a relation between Hofer metric and word norm in Ham(M) to show that, when k 3, homomorphisms from SL(k; Z) to Ham(M) is trivial, when M is symplectically hyperbolic. In the case of surfaces, this result covers surfaces with genus 1 [?]. Pol However, this method does not apply to manifolds with non-trivial 2, even in the simplest case of S 2. On the other hand, J. Franks and M. Handel FH [15] recovered Polterovich's result in dimension 2, and extended it to S 2 using a completely dierent method. It is very tempting to have a uniform approach using symplectic geometry and Floer theory to this problem. One interesting feature in common for both approaches is the application of the distortion elements in Heisenberg group, but the catch in symplectic geometry is that it seems hard to establish the property of distortion when 2 6= 0. This in turn should be related to gradings in symplectic geometry, including the Conley-Zehnder index and Seidel's graded symplectic theory More on exact Lagrangians in Liouville manifolds. The shape of exact Lagrangians in a Liouville manifold, as mentioned in Section 3.1.2, is a natural generalization to Arnold's nearby Lagrangian conjecture. However, not much is known so far, except for the case of cotangent bundles and certain small plumbings, e.g. [1]. Ab12 These results usually computes the whole Fukaya category, which is in general very hard. However, the dilation introduced by Seidel and Solomon [43] SS12 seems to imply non-trivial information on this problem. In particular, solely the existence of dilation on A n -singularites already implies that K(; 1) spaces cannot be embedded as exact Lagrangians

9 RESEARCH STATEMENT 9 submanifolds into them. In the case of A n -surface singularities, this excludes the existence of exact Lagrangian embeddings of any surfaces with genus> 0. But the case of RP 2 also follows from the simple topological consideration that any homology classes in A n -surface singularities has vanishing Z2 self-intersections. This in particular implies Ritter's result in A n -surface singularities, which says the only exact Lagrangian embedddings therein can only be Lagrangian S 2 's (note that for surfaces with genus> 1, it is also known to Viterbo, Eliashberg-Givental-Hofer). Notice that Ritter's approach heavily depends on dimension 4, for that only in this dimension, symplectic deformation near a Lagrangian sphere turns the sphere into a symplectic one. On the contrary, analysis on dilation elements applies to more general contexts, so if things work out, the proof should depend less on dimension assumptions. The topological ingredients of dilation comes in via Viterbo functoriality and well-known identications of the symplectic cohomology SH (T L) with the free loop space homology H n (L L). It would be interesting to see if a closer look to Viterbo funtoriality and loop space homology would yield similar results in higher dimensions. Especially for dimension 6, it would be interesting to see what can be said from the geometrization in dimension 3. References Ab12 [1] M. Abouzaid. Nearby Lagrangians with vanishing Maslov class are homotopy equivalent. Invent. Math. 189 (2012), no. 2, AC02 [2] M. Alberich-Carramiana Geometry of the plane Cremona maps. Lecture Notes in Mathematics Springer- Verlag, Berlin, xvi+257 pp. ISBN: X AF [3] P. Albers, U. Frauenfelder. A nondisplaceable Lagrangian torus in T S 2. Comm. Pure Appl. Math. 61 (2008), no. 8, AM [4] M. Abreu, D. McDu. Topology of symplectomorphism groups of rational ruled surfaces. J. Amer. Math. Soc. 13 (2000), no. 4, 971{1009 Lya [5] L. Barreira, Y. Pesin. Lyapunov Exponents and Smooth Ergodic Theory. University Lecture Series, v. 23, AMS, Providence, 2001 ISBN: Bi01 [6] P. Biran. Lagrangian barriers and symplectic embeddings. Geom. Funct. Anal., 11(3):407{464, BLW12 [7] M.S. Borman, T.-J. Li, W. Wu. Spherical Lagrangians via ball packings and symplectic cutting To appear in Selecta Math., CLW13 [8] W. Chen, T.-J. Li, W. Wu. Finite subgroups of symplectic Cremona maps in preparation. DI09 [9] I. Dolgachev, V. Iskovskihk. Finite subgroups of the plane Cremona group. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, , Progr. Math., 269, Birkhuser Boston, Inc., Boston, MA, EP1 [10] M. Entov, L. Polterovich. Rigid subsets of symplectic manifolds. Compos. Math. 145 (2009), no. 3, EP2 [11] M. Entov, L. Polterovich. Calabi quasimorphism and quantum homology. Int. Math. Res. Not. 2003, no. 30, ElP [12] Y. Eliashberg, L. Polterovich. Symplectic quasi-states on the quadric surface and Lagrangian submanifolds, arxiv: Ev10 [13] J.D. Evans. Lagrangian spheres in del Pezzo surfaces. J. Topol., 3(1):181{227, Ev11 [14] J.D. Evans. Symplectic mapping class groups of some Stein and rational surfaces. J. Symplectic Geom., 9 (2011), no. 1, FH [15] J. Franks; M. Handel. Area preserving group actions on surfaces. Geom. Topol., 7 (2003), FOOO [16] K. Fukaya, Y.-G. Oh, H. Ohta, K, Ono. Lagrangian intersection Floer theory: anomaly and obstruction. Part I, II. AMS/IP Studies in Advanced Mathematics, 46.1,2 American Mathematical Society, Providence, RI;International Press, Somerville, MA, FOOOtoric I [17] K. Fukaya, Y.-G. Oh, H. Ohta, K, Ono. Lagrangian Floer theory on compact toric manifolds. I. Duke Math. J. 151 (2010), no. 1, OOOtoric II [18] K. Fukaya, Y.-G. Oh, H. Ohta, K, Ono. Lagrangian Floer theory on compact toric manifolds II : Bulk deformations. to appear in Selecta Math.

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