Publications
Published
| 14. |
The number of Hamiltonian fixed points on symplectically aspherical manifolds,
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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