Publications
Published
| 22. |
Linking of Lagrangian Tori and Embedding Obstructions in Symplectic 4-Manifolds (with L. Côté)
⊞ Abstract
⊟ Abstract
| We classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a first corollary, we strengthen a result due independently to Eliashberg-Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$ which are homologous to the zero section. As a second corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1 \cup K_2$ is not even smoothly isotopic to a Lagrangian. In the second part of the paper, we study linking of Lagrangian tori in $(\mathbb{R}^4, \omega)$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions. |
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| 21. |
Bulky Hamiltonian isotopies of Lagrangian tori with applications,
Proceedings of the Gökova Geometry-Topology Conference 2018/2019. Gökova Geometry/Topology Conference (GGT), Gökova, 2021, 138–163.
⊞ Abstract
⊟ Abstract
| We exhibit monotone Lagrangian tori inside the standard symplectic four-dimensional unit ball that become Hamiltonian isotopic to the Clifford torus, i.e. the standard product torus, only when considered inside a strictly larger ball (they are not even symplectomorphic to a standard torus inside the unit ball). These tori are then used to construct new examples of symplectic embeddings of toric domains into the unit ball which are symplectically knotted in the sense of J. Gutt and M. Usher. We also give a characterisation of the Clifford torus inside the ball as well as the projective plane in terms of quantitative considerations; more specifically, we show that a torus is Hamiltonian isotopic to the Clifford torus whenever one can find a symplectic embedding of a sufficiently large ball in its complement. |
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| 20. |
The persistence of the Chekanov–Eliashberg algebra (with M. Sullivan),
Sel. Math. New Ser. 26 69 (2020).
⊞ Abstract
⊟ Abstract
| We apply the barcodes of persistent homology theory to the Chekanov–Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov–Eliashberg algebra to admit an augmentation as we linearize the algebra only below a certain action level. As an application we show that it is not possible to C0-approximate a stabilized Legendrian by a Legendrian that admits an augmentation. |
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| 19. |
An energy-capacity inequality for Legendrian submanifolds (with M. Sullivan),
J. Topol. Anal. 12 No. 3 (2020), 547–623.
⊞ Abstract
⊟ Abstract
| We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the ℤ2-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the established invariance enables us to use two different contact forms: one for the count of Reeb chords and another for the measure of the smallest length, under the assumption that there is a suitable symplectic cobordism from the latter to the former. The size of the contact isotopy is measured in terms of the oscillation of the contact Hamiltonian, together with the maximal factor by which the contact form is shrunk during the isotopy. The main tool used is a Mayer–Vietoris sequence for Lagrangian Floer homology, obtained by “neck-stretching” and “splashing”. |
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| 18. |
Floer theory for Lagrangian cobordisms (with B. Chantraine, P. Ghiggini, & R. Golovko),
J. Differential Geom. 114 no. 3 (2020), 393–465.
⊞ Abstract
⊟ Abstract
| In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov–Eliashberg algebras of the negative ends of the cobordisms admit augmentations. From this theory we derive several long exact sequences relating the Morse homology of an exact Lagrangian cobordism with the bilinearised contact homologies of its ends. These are then used to investigate the topological properties of exact Lagrangian cobordisms. |
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| 17. |
The classification of Lagrangians nearby the Whitney immersion,
Geom. Topol. 23 (2019), 3367–3458.
⊞ Abstract
⊟ Abstract
| The Whitney immersion is a Lagrangian sphere inside the four-dimensional symplectic vector space which has a single transverse double point of Whitney self-intersection number +1. This Lagrangian also arises as the Weinstein skeleton of the complement of a binodal cubic curve inside the projective plane, and the latter Weinstein manifold is thus the "standard" neighbourhood of Lagrangian immersions of this type. We classify the Lagrangians inside such a neighbourhood which are homologically essential, and which are either embedded or immersed with a single double point; they are shown to be Hamiltonian isotopic to either product tori, Chekanov tori, or rescalings of the Whitney immersion. |
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| 16. |
Positive Legendrian isotopies and Floer theory (with B. Chantraine & V. Colin),
Ann. Inst. Fourier (Grenoble) 69 no. 4 (2019), 1679–1737.
⊞ Abstract
⊟ Abstract
| Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds filled by a Liouville domain with non-zero symplectic homology are strongly orderable in the sense of Liu. |
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| 15. |
The stable Morse number as a lower bound for the number of Reeb chords (with R. Golovko),
J. Symplectic Geom. 14 no. 3 (2016) 811–901.
⊞ Abstract
⊟ Abstract
| Assume that we are given a closed chord-generic Legendrian submanifold $\Lambda \subset P \times \mathbb{R}$ of the contactisation of a Liouville manifold, where $\Lambda$ moreover admits an exact Lagrangian filling $L_{\Lambda} \subset \mathbb{R} \times P \times \mathbb{R}$ inside the symplectisation. Under the further assumptions that this filling is spin and has vanishing Maslov class, we prove that the number of Reeb chords on $\Lambda$ is bounded from below by the stable Morse number of $L_{\Lambda}$. Given a general exact Lagrangian filling $L_{\Lambda}$, we show that the number of Reeb chords is bounded from below by a quantity depending on the homotopy type of $L_{\Lambda}$, following Ono–Pajitnov’s implementation in Floer homology of invariants due to Sharko. This improves previously known bounds in terms of the Betti numbers of either $\Lambda$ or $L_{\Lambda}$. |
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| 14. |
The number of Hamiltonian fixed points on symplectically aspherical manifolds (with R. Golovko),
Proceedings of the Gökova Geometry-Topology Conference 2016. Gökova Geometry/Topology Conference (GGT), Gökova, 2017, 138–150.
⊞ Abstract
⊟ Abstract
| We show that a generic Hamiltonian diffeomorphism on a closed symplectic manifold which is symplectically aspherical has at least the stable Morse number of fixed points - this is in line with a conjecture by Arnold. |
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| 13. |
Lagrangian isotopy of tori in S2×S2 and ℂP2 (with E. Goodman & A. Ivrii),
Geom. Funct. Anal. 26 no. 5 (2016) 1297–1358.
⊞ Abstract
⊟ Abstract
| We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space ℝ4, the projective plane ℂP2, and the monotone S2×S2. The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for T*𝕋2, i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section. |
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| 12. |
Nontriviality results for the characteristic algebra of a DGA,
Math. Proc. Cambridge 162 no. 3 (2017) 419–433.
⊞ Abstract
⊟ Abstract
| Assume that we are given a semifree noncommutative differential graded algebra (DGA for short) whose differential respects an action filtration. We show that the canonical unital algebra map from the homology of the DGA to its characteristic algebra, i.e. the quotient of the underlying algebra by the two-sided ideal generated by the boundaries, is a monomorphism. The main tool that we use is the weak division algorithm in free noncommutative algebras due to P. Cohn. |
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| 11. |
Uniqueness of extremal Lagrangian tori in the four-dimensional disc,
Proceedings of the Gökova Geometry-Topology Conference 2015. Gökova Geometry/Topology Conference (GGT), Gökova, 2016, 151–167.
⊞ Abstract ⊞ Remarks
⊟ Abstract ⊞ Remarks
| The following interesting quantity was introduced by K. Cieliebak and K. Mohnke for a Lagrangian submanifold L of a symplectic manifold: the minimal positive symplectic area of a disc with boundary on L. They also showed that this quantity is bounded from above by π/n for a Lagrangian torus inside the 2n-dimensional unit disc equipped with the standard symplectic form. A Lagrangian torus for which this upper bound is attained is called extremal. We show that an extremal Lagrangian torus inside the four-dimensional unit disc is contained in the boundary ∂D4=S3, and is hence Hamiltonian isotopic to the product torus S11/√2 × S11/√2 ⊂ S3. This provides an answer to a question by L. Lazzarini in the four-dimensional case. |
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| 10. |
Noncommutative augmentation categories (with B. Chantraine, P. Ghiggini, & R. Golovko),
Proceedings of the Gökova Geometry-Topology Conference 2015. Gökova Geometry/Topology Conference (GGT), Gökova, 2016, 116–150.
⊞ Abstract
⊟ Abstract
| To a differential graded algebra with coefficients in a noncommutative algebra, by dualisation we associate an A∞–category whose objects are augmentations. This generalises the augmentation category of Bourgeois and Chantraine to the noncommutative world. |
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| 9. |
Legendrian Ambient Surgery and Legendrian Contact Homology,
J. Symplectic Geom. 14 no. 3 (2016) 811–901.
⊞ Abstract
⊟ Abstract
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Let L ⊂ Y be a Legendrian submanifold of a contact manifold, S ⊂ Y a framed embedded sphere bounding an isotropic disc DS ⊂ Y ∖ L, and use LS to denote the manifold obtained from L by a surgery on S. Given some additional conditions on DS we describe how to obtain a Legendrian embedding of LS into an arbitrarily small neighbourhood of L ∪ DS ⊂ Y by a construction that we call Legendrian ambient surgery. In the case when the disc is subcritical, we produce an isomorphism of the Chekanov–Eliashberg algebra of LS with a version of the Chekanov–Eliashberg algebra of L whose differential is twisted by a count of pseudo-holomorphic discs with boundary-point constraints on S. This isomorphism induces a one-to-one correspondence between the augmentations of the Chekanov-Eliashberg algebras of L and LS. |
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| 8. |
Lifting pseudo-holomorphic polygons to the symplectisation of P × ℝ and applications,
Quantum Topol. 7 no. 1 (2016) 29–105.
⊞ Abstract
⊟ Abstract
| Let ℝ ×(P × ℝ) be the symplectisation of the contactisation of an exact symplectic manifold P, and let ℝ × Λ be a cylinder over a Legendrian submanifold of the contactisation. We show that a pseudo-holomorphic polygon in P having boundary on the projection of Λ can be lifted to a pseudo-holomorphic disc in the symplectisation having boundary on ℝ × Λ. It follows that Legendrian contact homology may be equivalently defined by counting either of these objects. Using this result, we give a proof of Seidel’s isomorphism of the linearised Legendrian contact homology induced by an exact Lagrangian filling and the singular homology of the filling. |
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| 7. |
Floer homology and Lagrangian concordance (with B. Chantraine, P. Ghiggini, & R. Golovko),
Proceedings of the Gökova Geometry-Topology Conference 2014. Gökova Geometry/Topology Conference (GGT), Gökova, 2015, 76–113.
⊞ Abstract
⊟ Abstract
| We derive constraints on Lagrangian concordances from Legendrian sub-manifolds of the standard contact sphere admitting exact Lagrangian fillings. More precisely, we show that such a concordance induces an isomorphism on the level of bilinearised Legendrian contact cohomology. This is used to prove the existence of non-invertible exact Lagrangian concordances in all dimensions. In addition, using a result of Eliashberg–Polterovich, we completely classify exact Lagrangian concordances from the Legendrian unknot to itself in the tight contact-three sphere: every such concordance is the trace of a Legendrian isotopy. We also discuss a high dimen-sional topological result related to this classification. |
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| 6. |
Exotic spheres and the topology of symplectomorphism groups (with J. D. Evans),
J. Topology 8 no. 2 (2015) 586–602.
⊞ Abstract
⊟ Abstract
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We show that, for certain families ϕs of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of T∗Sn with the family of pullbacks ϕs* gives a non-contractible family of compactly supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrized Lagrangian sphere depends up to Hamiltonian isotopy on its parametrization; where the symplectomorphism group is not simply connected, and where the symplectomorphism group does not have the homotopy type of a finite CW complex. We show that these phenomena persist for Dehn twists along the standard matching spheres of the Am-Milnor fibre. The non-triviality is detected by considering the action of symplectomorphisms on the space of parametrized Lagrangian submanifolds. We find related examples of symplectic mapping classes for T∗(Sn×S1) and of an exotic symplectic structure on T∗(Sn×S1) standard at infinity.
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| 5. |
Estimating the number of Reeb chords using a linear representation of the characteristic algebra (with R. Golovko),
Algebr. Geom. Topol. 15 no. 5 (2015) 2887–2920.
⊞ Abstract
⊟ Abstract
| Given a chord-generic, horizontally displaceable Legendrian submanifold Λ ⊂ P × ℝ with the property that its characteristic algebra admits a finite-dimensional matrix representation, we prove an Arnold-type lower bound for the number of Reeb chords on Λ. This result is a generalization of the results of Ekholm, Etnyre, Sabloff and Sullivan, which hold for Legendrian submanifolds whose Chekanov–Eliashberg algebras admit augmentations. We also provide examples of Legendrian submanifolds Λ of ℂn× ℝ, n ≥ 1, whose characteristic algebras admit finite-dimensional matrix representations but whose Chekanov–Eliashberg algebras do not admit augmentations. In addition, to show the limits of the method of proof for the bound, we construct a Legendrian submanifold Λ ⊂ ℂn× ℝ with the property that the characteristic algebra of Λ does not satisfy the rank property. Finally, in the case when a Legendrian submanifold Λ has a non-acyclic Chekanov–Eliashberg algebra, using rather elementary algebraic techniques we obtain lower bounds for the number of Reeb chords of Λ. These bounds are slightly better than the number of Reeb chords it is possible to achieve with a Legendrian submanifold whose Chekanov–Eliashberg algebra is acyclic. |
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| 4. |
Exact Lagrangian caps and non-uniruled Lagrangian submanifolds,
Ark. Mat. 53 no. 1 (2015) 37–64.
⊞ Abstract ⊞ Remarks
⊟ Abstract ⊞ Remarks
| We make the elementary observation that the Lagrangian submanifolds of ℂn, n ≥ 3, constructed by Ekholm, Eliashberg, Murphy and Smith are non-uniruled and, moreover, have infinite relative Gromov width. The construction of these submanifolds involve exact Lagrangian caps, which obviously are non-uniruled in themselves. This property is also used to show that if a Legendrian submanifold inside a contactisation admits an exact Lagrangian cap, then its Chekanov–Eliashberg algebra is acyclic. |
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| 3. |
On homological rigidity and flexibility of exact Lagrangian endocobordisms
(with R. Golovko),
Int. J. Math 25 no. 10 (2014).
⊞ Abstract
⊟ Abstract
| We show that an exact Lagrangian cobordism L ⊂ ℝ ×P × ℝ from a Legendrian submanifold Λ ⊂ P × ℝ to itself satisfies Hi(L; 𝔽) = Hi(Λ; 𝔽) for any field 𝔽, given that Λ admits a spin exact Lagrangian filling and that the concatenation of any spin exact Lagrangian filling of Λ and L is also spin. The main tool used is Seidel's isomorphism in wrapped Floer homology. In contrast to that, for loose Legendrian submanifolds of ℂn × ℝ, we construct examples of such cobordisms whose homology groups have arbitrarily high ranks. In addition, we prove that the front Sm-spinning construction preserves looseness, which implies certain forgetfulness properties of it. |
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| 2. |
Unlinking and unknottedness of monotone Lagrangian submanifolds (with J. D. Evans),
Geom. Topol. 18 no. 2 (2014) 997–1034.
⊞ Abstract
⊟ Abstract
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Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangian immersion. The topological assumptions are satisfied by a large class of manifolds which are realised as monotone Lagrangians, including tori. After some additional homotopy theoretic calculations, we deduce that all monotone Lagrangian tori in the symplectic vector space of odd complex dimension at least five are smoothly isotopic. |
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| 1. |
Knotted Legendrian surfaces with few Reeb chords,
Algebr. Geom. Topol. 11 no. 5 (2011) 2903–2936.
⊞ Abstract ⊞ Remarks
⊟ Abstract ⊞ Remarks
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For g > 0, we construct g + 1 Legendrian embeddings of a surface of genus g into J1(ℝ2)=ℝ5 which lie in pairwise distinct Legendrian isotopy classes and which all have g + 1 transverse Reeb chords (g+1 is the conjecturally minimal number of chords). Furthermore, for g of the g + 1 embeddings the Legendrian contact homology DGA does not admit any augmentation over ℤ2, and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in J1(S2) from a similar perspective.
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⊞ Abstract ⊟ Remarks
| The conjectural bound on the minimal number of Reeb chords for a Legendrian surface is false. D. Sauvaget has constructed a Legendrian embedding of a genus-two surface with a single transverse Reeb chord in standard contact five-space [Curiosités Lagrangiennes en dimension 4, Ann. I. Fourier 54 no. 6 (2004) 1997–2020]. |
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Preprints
Thesis
Here is the introductory summary of my Ph.D. thesis (2012).
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