Jo Nelson Research Statement

1 Motivation and Background

Figure 1: Periodic orbits ofthe Reeb vector field on S3

parametrized by S2, aka theHopf fibration (credit: NilesJohnson).

My research is in symplectic and contact geometry with a focus on mod-uli problems in contact topology. I have been working to provide foun-dations and computations of contact homology invariants. These Floer-theoretic invariants are closely related to symplectic homology and haveapplications to singularity theory, dynamics, string theory, mirror sym-metry, knot theory, and symplectic embedding problems. My researchcombines techniques from differential topology and geometry, partialdifferential equations, complex geometry, and algebraic topology andgeometry.

Symplectic and contact structures first arose in the study of classicalmechanical systems such as those describing planetary motion and wavepropagation; see [Ar78, Ne16]. Solutions of these systems are flow linesof either the Hamiltonian vector field on a symplectic manifold or the Reeb

vector field on a contact manifold, see Figure 1. Understanding the evolution and distinguishingtransformations of these systems necessitated the development of global invariants of symplec-tic and contact manifolds which encode structural aspects of the Reeb and/or Hamiltonian flows.These invariants are built out of moduli spaces of pseudoholomorphic curves [Gr85], which are solu-tions to a nonlinear Cauchy-Riemann equation. They originated from Floer’s breakthrough [Fl88]to combine classical variational methods, Gromov’s theory of pseudoholomorphic curves, and Wit-ten’s interpretation of Morse theory [Wi82].

Figure 2: Integrable vs. contact structures on R3.

Formally, a contact structure ξ is a maximally non-integrable hyperplane distribution. This is in con-trast to an integrable hyperplane distribution, whosehyperplanes are given by the tangent spaces of asubmanifold; see Figure 2. Any 1-form λ whosekernel defines a contact structure is called a con-tact form. The Reeb vector field Rλ depends on thechoice of contact form λ and is defined by λ(Rλ) =1, dλ(Rλ, ·) = 0. A closed Reeb orbit of period T > 0 is defined modulo reparametrization by:

γ : R/TZ→ M, γ(t) = Rλ(γ(t)), (1)

and is said to be simple, or equivalently embedded, whenever (1) is injective. The linearized Reebflow for time T defines a symplectic linear map of (ξ, dλ) and if this map does not have 1 as aneigenvalue then γ is said to be nondegenerate. The contact form λ is called nondegenerate if all Reeborbits are nondegenerate; generic contact forms have this property.

2 Overview of resultsConstructions of moduli based invariants looked promising with the advent of a comprehensivesymplectic field theory, announced in 2000 by Eliashberg, Givental, and Hofer [EGH00]. Symplecticfield theory generalizes Floer and Gromov-Witten theories by studying moduli spaces of pseudoholo-morphic curves from punctured Riemann surfaces to certain noncompact symplectic manifolds withasymptotics on periodic orbits of the Hamiltonian or Reeb vector field. There is a Fredholm theorydescribing these moduli spaces as the zero set of an infinite dimensional bundle. To obtain contactand symplectic invariants one must first regularize these moduli spaces so that the (compactified)moduli spaces can be given the structure of a smooth manifold or orbifold, possibly with boundaryand corners.

When multiply covered curves are present, as in symplectic field theory, there is typically afailure of transversality of the zero set which describes the moduli space these curves. The most

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Jo Nelson Research Statement

general approach to regularize these moduli spaces is to abstractly perturb the standard nonlinearCauchy-Riemann equation. However, the use of abstract perturbations is a lengthy technical en-deavor which is often not suitable for applications and in some cases not well understood by thecommunity.

My research, in part joint with Hutchings, makes use of more direct methods to extend thetransversality theory for the standard pseudoholomorphic curve equation and to isolate precisephenomena which can be accounted for with established geometric and analytic methods. In par-ticular, we have provided foundations for a subset of symplectic field theory known as (cylindrical)contact homology, whose chain complex is generated by certain closed Reeb orbits and whose differ-ential counts pseudoholomorphic cylinders interpolating between these Reeb orbits, by extendingthe methods of my thesis [Ne13, Ne15]. Our work [HN1, HN2, HN3] gives a rigorous definitionof cylindrical contact homology for dynamically convex contact forms in three dimensions (cf. Def-inition 3.1), invariance in the dynamically convex case, a definition of local contact homology inany dimension, and a substitute for cylindrical contact homology in higher dimensions in the ab-sence of contractible Reeb orbits. These foundational results were not previously available in theliterature, cf. [Bo02, Bo09, BO09, BCE07, Us99, MLY04]. Section 2.1 gives an overview of my foun-dational results with Hutchings. My work on applications of contact invariants and my supervisionof associated undergraduate research projects can be found in Section 2.2.

2.1 Foundational resultsCylindrical contact homology is in principle an invariant of contact manifolds (Y, ξ) that admit anondegenerate contact form λ without Reeb orbits of certain gradings. The cylindrical contact ho-mology of (Y, ξ) is defined by choosing a nondegenerate contact form λ, taking the homology ofa chain complex over Q which is generated by “good” Reeb orbits and denoted by CCQ

∗ (Y, λ, J),and whose differential ∂Q counts J-holomorphic cylinders in R× Y for a suitable almost complexstructure J. The grading on the complex is given by the Conley-Zehnder index, a winding number as-sociated to the path of symplectic matrices obtained from linearizing the flow along γ, restricted toξ. The Conley-Zehnder index is denoted by CZτ(γ) and typically depends on a choice of trivializa-tion τ. Unfortunately, in many cases there is no way to choose J so as to obtain the transversality forholomorphic cylinders needed to define ∂Q, to show that (∂Q)2 = 0, and to prove that the homologyis invariant of the choice of J and λ.

In [HN1] we extended the methods from my thesis [Ne13, Ne15] and showed that the cylin-drical contact homology differential ∂Q can be defined by directly counting pseudoholomorphiccylinders in dimension 3 for the class of dynamically convex contact forms, cf. Definition 3.1 with-out any abstract perturbation of the Cauchy-Riemann equation. Our work [HN2, HN3] establishesthe invariance of cylindrical contact homology under choices of J and λ.

Theorem 2.1. [HN1, HN2, HN3] Let λ be a nondegenerate, dynamically convex contact form on a closedthree-manifold Y. Suppose further that:

(*) A contractible Reeb orbit γ has CZ(γ) = 3 only if γ is embedded.

Then for generic λ-compatible almost complex structures J on R× Y, (CCQ(Y, λ, J), ∂Q), is a well-definedchain complex and its homology is invariant under choices of dynamically convex λ defining the contactstructure ξ and generic J,

CHQ∗ (Y, ξ) := H∗(CCQ(Y, λ, J), ∂Q).

Under the assumptions of Theorem 2.1 we also define the following related contact homologytheories. Further details are provided in Section 3.1.

Theorem 2.2. [HN2, HN3] Under the assumptions of Theorem 2.1,(i) There is a well-defined nonequivariant contact homology defined with coefficients in Z which is a contact

invariant and denoted by NCH∗.

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Jo Nelson Research Statement

(ii) There is a well-defined equivariant contact homology defined with coefficients in Z which is a contactinvariant and denoted by CHZ

∗ .(iii) There is a canonical isomorphism, CHQ

∗ ' CHZ∗ ⊗Q.

As a result of Theorem 2.2(iii) we call CHZ∗ an integral lift of cylindrical contact homology. Our

constructions also produce a definition of local contact homology in any dimension, a theory whichhas important applications in dynamics [GG09, HM15]. Long term dynamical applications includeexistence results for homoclinic orbits in Hamiltonian systems [HW90], the study of periodic orbitsof iterated disk maps [BrHof12], and the relationship between the topology of a contact 3-manifoldand the minimal number of simple Reeb orbits.

Figure 3: Two views of the right handed Legen-drian trefoil in (R3, ξstd), courtesey of Lenny Ng.

Our work is connected to several other contact andsymplectic invariants. In particular, the nonequivari-ant and integral theories are expected to be isomor-phic1 to positive symplectic homology and positive S1-equivariant symplectic homology, respectively. Sym-plectic homology is a Floer type invariant of symplec-tic manifolds with contact type boundary which detectsthe symplectic structure of the interior and the Reeb dy-namics of the boundary [CFHW, Se08, Vi99]. The pos-itive symplectic homology complex is a quotient com-plex of total complex by the complex generated by critical points in the interior of the symplecticmanifold.

We also expect that our work should allow for further progress on the classification of Legendrianknots up to Legendrian isotopy in closed dynamically convex contact 3-manifolds. Legendrian knotsare smooth knots whose tangent vectors live in the contact structure, see Figure 3 and Legendrianisotopic are those which are isotopic through a family of Legendrian knots. The classification ofLegendrian knots is an interesting problem because Legendrian knots are surgery loci for construc-tions of new contact manifolds and have close ties with smooth knots via geometric and quantumknot invariants. Connections with our work to symplectic homology [BO] and Legendrian contacthomology [BEE12, EES07, EENS13] are included in Section 3.2. Additionally, there are the followingalternate approaches to defining contact homology invariants.

Remark 2.3. (On related approaches)(Alternatives to CHQ) Bao and Honda [BaHon1] give a definition of a contact invariant akin to

cylindrical contact homology for contact manifolds which admit no contractible Reeb orbitsin dimension 3, defined with coefficients in Q. It remains to show this definition agrees withthe “classical” one from [EGH00]. Gutt [Gu] has shown that the positive equivariant portionof symplectic homology is also an invariant for some contact manifolds, see also Remark 3.3.

(Kuranishi) Pardon’s approach [Pa] to defining full contact homology via virtual fundamental cy-cles yields a definition of cylindrical contact homology in the absence of contractible Reeborbits. Bao and Honda [BaHon2] give a definition of the full contact homology differentialgraded algebra for any closed contact manifold in any dimension. These approaches makeuse of Kuranishi structures to construct contact and symplectic invariants and while they holdmore generally, they are also more abstract and hence more difficult to work with in compu-tations and applications; see also [FO99], [FO31]-[FO34], [IP], [Joy], [MW1]-[MW3], [Pa16],[TZ1], [TZ2].

(Polyfolds) Hofer, Wysocki, and Zehnder have developed the abstract analytic framework [HWZI]-[HWZIII], [HWZ-gw], collectively known as polyfolds, to systematically resolve issues of reg-ularizing moduli spaces. Contact homology awaits foundations via polyfolds and the use of

1See Section 3.2 for a more precise explanation.

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Jo Nelson Research Statement

abstract perturbations can make computations difficult. Hofer and I plan to investigate theconnections between the polyfold approach and the one taken by Hutchings and myself.

2.2 Computational methods and undergraduate research supervisionI am currently applying the computational methods I developed in [Ne13, Ne15, Ne] to prove aFloer theoretic interpretation of the McKay correspondence [IM96] of the links of the simple sin-gularities. The simple singularities can be characterized as the absolutely isolated double pointquotient singularity of C2/Γ, where Γ is a finite subgroup of SL(2; C). The link of these singularitiesis (contact) diffeomorphic to S3/Γ.

Theorem in Progress 2.4. The cylindrical contact homology of the link of a simple singularity is a free Q[u]module of rank equal to the number of conjugacy classes of the respective finite subgroup Γ of SL(2; C).

My method of proof demonstrates the cohomological McKay correspondence via the Reeb dy-namics of these links. This is related to work in progress by McLean and Ritter, who are formulatinga McKay correspondence for links of crepant singularities in terms of symplectic homology. Furtherdetails on these computational methods as well as their expected extension to more general Seifertfiber spaces and connections with Chen-Ruan cohomology and string topology may be found inSection 4.

Many aspects of computing Floer theoretic invariants can be made appropriate research projectsfor graduate students and talented undergraduates. I supervised an REU project over two summersresulting in the verification of the case of the An singularity via direct computations [AHNS]. Ourapproach involved the construction of an explicit contact form on L(n + 1, n) which admitted ex-actly two simple Reeb orbits and computed the associated Conley-Zehnder indices of these orbitsand their iterates.

In recent work with Christianson [CN], a former REU mentee, we found new obstructions tosymplectic embeddings of the four-dimensional polydisk,

P(a, 1) = {(z1, z2) ∈ C2 | π|z1|2 ≤ a, π|z2|2 ≤ 1},

into the ball,B(c) =

{(z1, z2) ∈ C2 ∣∣ π|z1|2 + π|z2|2 ≤ c

},

extending work done by Hind-Lisi [HL15] and Hutchings [Hu16]. We made use of Hutchings’refinement of embedded contact homology from [Hu16], which established a necessary conditionfor one convex toric domain, such as an ellipsoid or polydisk, to symplectically embed into another.

Theorem 2.5 ([CN]). Let 2 ≤ a <√

7−1√7−2

= 2.54858.... If the four dimensional polydisk P(a, 1) symplecti-cally embeds into the four dimensional ball B(c) = E(c, c) then c ≥ 2 + a

2 .

Schlenk’s folding construction permits us to conclude our bound on c is optimal, [Sc15, Prop.4.3.9]. Additionally, we proved that if certain symplectic embeddings of four dimensional convextoric domains exist then a modified version of the combinatorial criterion from [Hu16] must hold,thereby reducing the computational complexity of the original criterion from O(2n) to O(n2).

3 Technical aspects of contact homology and related invariantsThis section provides further details on Theorems 2.1 and 2.2, in regards to the aforementioned classof dynamically convex class of contact forms.

Definition 3.1. (cf. [HWZ99]) Let λ be a nondegenerate contact form on a closed three-manifold Y.We say that λ is dynamically convex if either:• λ has no contractible Reeb orbits, or

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Jo Nelson Research Statement

• c1(ξ)|π2(Y) = 0, and every contractible Reeb orbit γ has CZ(γ) ≥ 3.

Generic convex, compact, star-shaped hypersurfaces Y in R4 admit a dynamically convex con-tact form via the restriction of λ = 1

2 ∑2k=1(xkdyk − ykdxk) to Y.

Theorem 2.1 required the following additional assumption, which we briefly explain.

Remark 3.2 (On the hypotheses of Theorem 2.1 and related future work).(a) The hypothesis (*) automatically holds when π1(Y) contains no torsion:

(*) A contractible Reeb orbit γ has CZ(γ) = 3 only if γ is embedded,(b) In general, the hypothesis (*) can be removed from Theorem 2.1 assuming a certain technical

conjecture on the asymptotics of holomorphic curves.(c) We expect that the hypothesis of dynamical convexity can be further weakened.

3.1 Discussion of Theorem 2.2Our work [Ne13, Ne15, HN1] demonstrates that generic S1-independent almost complex structuresyield sufficient transversality to ensure that ∂Q is well-defined and that

(∂Q)2

= 0 for large classesof contact manifolds in dimension 3. However, S1-independent almost complex structures do notgive sufficient transversality to count index zero cylinders in cobordisms, which is necessary todefine the chain maps. This is because the covers of index zero cylinders can live in a modulispace of negative virtual dimension. In order to prove topological invariance of CHQ

∗ (Y, λ, J) in[HN2, HN3] we make use of S1-dependent almost complex structures, similar to the set up in [BO].

Breaking the S1 symmetry invalidates certain properties needed to prove(∂Q)2

= 0 and thechain map and chain homotopy equations. As a result, the use of S1-dependent almost complexstructures leads to a Morse-Bott version of the chain complex. The homology of this chain com-plex is not the desired cylindrical contact homology but rather a nonequivariant version of it. Thisnonequivariant version is denoted by NCH∗ and the content of Theorem 2.2(i). Moreover, the pres-ence of contractible Reeb orbits necessitates the use of obstruction bundle gluing [HT07, HT09],producing a correction term in the expression of the nonequivariant differential [HN3]. The de-sired cylindrical contact homology CHQ

∗ can be regarded as an “S1- equivariant” version of non-equivariant contact homology, and recovering this [HN2, HN3] requires additional family Floertheoretic constructions in the spirit of [BO, Hu08, SeSm10].

An outcome of these constructions is our so called integral lift of contact homology, denotedby CHZ

∗ , and the content of Theorem 2.2(ii)-(iii). We have examples which show that this integrallift of contact homology also contains interesting torsion information pertaining to the qualitativebehavior of the Reeb dynamics. The entirety of Theorem 2.2 allows us to conclude that CHQ

∗ is acontact invariant.

3.2 Connections with symplectic homology and Legendrian contact homologyThe preprint of Bourgeois and Oancea [BO] suggests that an isomorphism between the positive partof S1-equivariant symplectic homology and linearized contact homology should exist in general.Their proof requires the assumption that an almost complex structure J can be chosen so that allrelevant moduli spaces of J-holomorphic curves are cut out transversely.2

Our work indicates that the construction of a geometric isomorphism must involve an obstruc-tion bundle contribution term in the presence of contractible orbits. We plan to pursue this avenueof study in the coming year, and expect an affirmative answer to the following conjecture.

2This assumption is only true for contact manifolds arising as unit cotangent bundles DT ∗ L with dim L ≥ 5 or thoseRiemannian manifolds L which admit no contractible closed geodesics.

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Jo Nelson Research Statement

Conjecture 1. Under the assumptions of Theorem 2.1 there exist natural isomorphisms,NCH∗(Y, ξ) ' SH+

∗ (R×Y, ξ; Z);CHZ∗ (Y, ξ) ' SH+,S1

∗ (R×Y, ξ; Z).(2)

Remark 3.3. More precisely, NCH∗(Y, ξ) should be isomorphic to the positive symplectic homol-ogy SH+

∗ of a filling of Y. In fact, for a contact manifold with no contractible Reeb orbits, or moregenerally for a contact manifold of dimension 2n− 1 in which every contractible Reeb orbit satisfiesCZ(γ) > 4− n, one can define positive symplectic homology directly in terms of the symplectiza-tion, without reference to a filling [BO, Section 4.1.2(2)]; this is satisfied by the dynamically convexassumption in dimension 3. This positive symplectic homology (and its equivariant version) isshown to be an invariant of the closed contact manifold in [Gu, Theorems 1.2, 1.3].

Additionally we expect that our work can be incorporated into the framework of Legendriancontact homology [BEE12, CELN, EES07, EENS13]. This relative version of contact homology iscurrently defined for Legendrians in 1-jet spaces [EENS13] and for Legendrian links in #k(S1 × S2)[EN15]. We anticipate that our results will permit this theory to be geometrically defined for Leg-endrian knots in closed dynamically convex contact 3-manifolds and that the proof of the followingconjecture is within reach of our methods.

Conjecture 2. The contact homology of Legendrian knots of dynamically convex contact 3-manifolds is well-defined: The stable tame isomorphism class of the DGA associated to a Legendrian knot K is independent ofthe choice of compatible almost complex structure J and is invariant under Legendrian isotopies of K.

There is also the following hope that one can relate the Legendrian DGA to other combinatorialknot invariants as in [ENS, EENS13]. Such a knot invariant in combination with a combinatorialexpression would lead to progress on the following question.

Question 3. Do there exist infinite families of Legendrian knots in dynamically convex contact 3-manifolds,which have the same classical invariants but are pairwise non Legendrian isotopic?

4 Futher details on McKay correspondence and computations

Figure 4: −∇H for H = zwith a fiber over S2, respec-tively S2/Z3.

There is a canonical contact form α0 associated to prequantization spaces[BW58] and certain S1-bundles over symplectic orbifolds Y; this form isis degenerate, meaning that the closed Reeb orbits are non-isolated. Wecan directly perturb the critical manifolds realized as Reeb orbits witha Morse-Smale function H on the base B that is invariant under the S1-action of the bundle S1 ↪→ Y h→ B. The perturbation,

αε = (1 + εh∗H)α0,

yields the Reeb dynamics,

Rε = (1 + εh∗H)−1R + ε(1 + εh∗H)−2XH,

where XH is a Hamiltonian vector field on S2 and XH its horizontal lift.The only fibers that persist as Reeb orbits of Rε are those over the

critical points of H. Additional Reeb orbits of Rε must come from horizontal lifts of closed orbits ofXH. Since εH and εdH are small, these Reeb orbits all have action (e.g. length) greater than 1/ε.Thus the only closed Reeb orbits left intact by this perturbation up must occur as a multiple coverof a simple Reeb orbit lying over a critical point of H. There is a proportionality between the actionof these persisting orbits and their length. This filtration on both the action and index leads to a

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Jo Nelson Research Statement

formal version of filtered contact homology, allowing one to recover cylindrical contact homologyin a well-defined way [Ne13, Ne15]. In particular, the only orbits which generate the filtered chaincomplex must project to critical points on the base.

The height function on S2 yields the desired computation of CHQ∗ (S3, ξstd), see Figure 4. The

simple Reeb orbit projecting to the north pole has index 2 and the one projecting to the south polehas index 0. The index increases by 4 after each iteration of the Reeb orbit. 3. We thus recover thewell known computation of cylindrical contact homology for the sphere (S3, ξstd).Example 4.1 ([Ne13]). The cylindrical contact homology for the sphere (S3, ξstd) is given by

CHQ∗ (S

3, ξstd) =

{Q ∗ ≥ 2, even;0 ∗ else.

If we consider the lens space (L(n + 1, n), ξstd) with its contact structure induced by the oneon S3, then the free homotopy classes of the orbits of Rε over critical points of H are in 1-1 corre-spondence with the conjugacy classes of L(n + 1, n). While (*) from Theorem 2.1 is not satisfied,the methods of [Ne15] still allow us to conclude that we get a well-defined chain complex which isinvariant under the choice of contact form and almost complex structure.

Theorem 4.2 ([Ne]). The cylindrical contact homology for the sphere (L(n + 1, n), ξstd) is given by

CHQ∗ ((L(n + 1, n), ξstd) =

Qn+1 ∗ ≥ 2, even;Qn ∗ = 0;0 ∗ else.

This computation agrees with what is obtained in [AHNS] for the link of the An singularity,which is to be expected given that these two manifolds are contactomorphic.

The differential ∂Q should behave as follows for perturbations of α0 by H different from theheight function.

Theorem in Progress 4.3. The pseudoholomorphic cylinders counted by the cylindrical contact homologydifferential lie over Morse-Smale trajectories of H in the base.

Figure 5: ∇H and XH = J0∇H

The simple singularities appear as the origin of the va-riety C2/Γ, where Γ ⊂ SU2(C) is a finite subgroup.This perturbation lends itself to a means of comput-ing the cylindrical contact homology of the links of thesimple singularities [Ne], viewed as the quotient spaces(S3/Γ, ξcan). In this set up, we must pick H so that XH isinvariant under the image of Γ in SO(3). In the case ofthe E6 singularity Γ = T∗ is the binary tetrahedral group and taking H = xyz yields the desired T-invariant vector field XH on S3, as in Figure 5. After quotienting out the entire bundle by the actionof Γ, the free homotopy classes of the orbits of Rε over critical points of H are in 1-1 correspondencewith the conjugacy classes of Γ. As a result, the rank of CHQ

∗ is governed by the number of theseconjugacy classes, producing the McKay correspondence theorem mentioned in Section 2.2.

Moreover, this perturbed Reeb flow realizes the presentation of the spherical manifolds S3/Γas Seifert fiber spaces. We obtain S1-bundles over the 2-sphere with the expected number of excep-tional fibers in 1-1 correspondence with the generators of the chain complex. These methods shouldextend to other Seifert spaces, resulting in a connection with Chen-Ruan cohomology [CR04], andsuggesting a generalization of Ritter’s recent work [Ri14] on Floer theory for negative line bundlesvia Gromov-Witten theory of orbifolds.

3A computation for the grading yields |γkp| = CZ(γk

p) − 1 = 4k − 2 + indexp(H). Here p ∈ Crit(H), and γkp is the

k-fold iterate of a simple Reeb orbit γ over p.

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Jo Nelson Research Statement

References[AHNS] L. Abbrescia, I. Huq-Kuruvilla, J. Nelson, and N. Sultani, Reeb Dynamics of the Link of the An Singularity, to

appear in Involve, arXiv:1509.02939

[Ar78] V.I. Arnold, Mathematical Methods in Classical Mechanics. Springer-Verlag, Berlin (1978).

[BaHon1] E. Bao and K. Honda, Definition of Cylindrical Contact Homology in dimension three, arXiv:1412.0276

[BaHon2] E. Bao and K. Honda, Semi-global Kuranishi charts and the definition of contact homology, arxiv:1512.00580

[BW58] W.M. Boothby, H.C. Wang, On contact manifolds. Ann. of Math. (2) 68 (1958), 721-734.

[Bo02] F. Bourgeois, A Morse-Bott approach to contact homology. Ph.D. thesis, Stanford University, 2002.

[Bo09] F. Bourgeois, A survey of contact homology. New perspectives and challenges in symplectic field theor, AMS,2009.

[BEE12] F. Bourgeois, T. Ekholm, Y. Eliashberg, Effect of Legendrian Surgery. Geom. & Topol. 16 (2012) 301-389.

[BCE07] F. Bourgeois, K. Cieliebak, T. Ekholm. A note on Reeb dynamics on the tight 3-sphere. J. Mod. Dyn. 1:597-613, 2007.

[BO09] F. Bourgeois, A. Oancea, An exact sequence for contact and symplectic homology. Invent. Math., 175:611-680, 2009.

[BO] F. Bourgeois, A. Oancea, S1-equivariant symplectic homology and linearized contact homology. arXiv:1212.3731.

[BrHof12] B. Bramham, H. Hofer, First steps towards a symplectic dynamics. Surv. Differ. Geom. 17:127–177, 2012.

[CR04] W. Chen, Y. Ruan, A new cohomology theory for orbifold. Comm. Math. Phys. 248, no. 1, 1-31 (2004).

[CN] K. Christianson, J. Nelson, Symplectic embeddings of four-dimensional polydisks into balls. arXiv:1610.00566.

[CFHW] K. Cieliebak, A. Floer, H. Hofer, K. Wysocki, Applications of symplectic homology II: stability of the actionspectrum. Math. Zeit. 223, 27-45 (1996).

[CELN] K. Cieliebak, T. Ekholm, J. Latschev, L. Ng, Knot contact homology, string topology, and the cord algebra.arXiv:1601.02167

[EES07] T. Ekholm, J. Etnyre, M. Sullivan, Legendrian contact homology in P×R. Trans. Amer. Math. Soc. 359 (2007), no.7

[EENS13] T. Ekholm, J. Etnyre, L. Ng, M. Sullivan, Knot contact homology. Geom. Topol. 17 (2013), no. 2, 975–1112.

[EN15] T. Ekholm, L. Ng, Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold. J. DifferentialGeom. 101 (2015), no. 1, 67-157.

[ENS] T. Ekholm, L. Ng, V. Shende, A complete knot invariant from contact homology. arXiv:1606.07050

[EGH00] Y. Eliashberg, A. Givental, H. Hofer, Introduction to symplectic field theory. GAFA (2000), 560-673.

[Fl88] A. Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), no. 6, 775–813

[FO99] K. Fukaya, K. Ono, Arnold conjecture and Gromov-Witten invariant. Topology 38 (1999), no. 5, 933-1048.

[FO31] K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Technical detail on Kuranishi structure and Virtual Fundamental Chain. 257pages. arXiv:1209.4410.

[FO32] K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Shrinking good coordinate systems associated to Kuranishi structures. 19pages. arXiv:1405.1755

[FO33] K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Kuranishi structure, Pseudo-holomorphic curve, and Virtual fundamentalchain: Part 1. 203 pages. arXiv:1503.07631

[FO34] K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono, Exponential decay estimates and smoothness of the moduli space of pseudo-holomorphic curves. 111 pages. arXiv:1603.07026

[GG09] G. Ginzburg, G. Gurel, On the generic existence of periodic orbits in Hamiltonian dynamics. J. Mod. Dyn. 595-610,2009.

[Gr85] M. Gromov, Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), 307-347.

[Gu] J. Gutt, The positive equivariant symplectic homology as an invariant for some contact manifolds. arXiv:1503.01443

[HL15] R. Hind, S. Lisi. Symplectic embeddings of polydisks. Selecta Math. 21(3):1099–1120, 2015.

[HW90] H. Hofer. K. Wysocki, First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math.Ann. 288 (1990), no. 3, 483–503.

[HWZ99] H. Hofer, K. Wysocki, E. Zehnder, A characterization of the tight 3-sphere. II. Comm. Pure Appl. Math. 52 (1999),no. 9, 1139-1177.

[HWZI] H. Hofer, K. Wysocki, E. Zehnder, A General Fredholm Theory I: A Splicing-Based Differential Geometry. JEMS9:841-876, 2007.

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[HWZII] H. Hofer, K. Wysocki, E. Zehnder, A general Fredholm theory II: Implicit function theorems. GAFA 19:206-293, 2009.[HWZIII] H. Hofer, K. Wysocki, E. Zehnder, A general Fredholm theory III: Fredholm Functors and Polyfolds. Geom. & Topol.

13:2279-2387, 2009.[HWZ-gw] H. Hofer, K. Wysocki, E. Zehnder, Applications of Polyfold Theory I: Gromov-Witten Theory. arXiv:1107.2097[HM15] U. Hryniewicz, L. Macarini, Local contact homology and applications, J. Topol. Anal. 7 (2015), no. 2, 167–238.[Hu08] M. Hutchings, Floer homology of families. I. Algebr. Geom. Topol. 8 (2008), no. 1, 435–492.[Hu14] M. Hutchings, Lecture notes on embedded contact homology. Contact and Symplectic Topology, Bolya Society

Mathematical Studies 26 (2014), 389–484, Springer.[Hu16] M. Hutchings. Beyond ECH capacities. Geom. & Topol. 20(2):1085–1126, 2016.[HN1] M. Hutchings, J. Nelson, Cylindrical contact homology for dynamically convex contact forms in three dimensions. to

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