| 1. |
- Carlsson, Tobias, et al.
(author)
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Algorithm for generating a Brownian motion on a sphere
- 2010
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In: Journal of physics A: Mathematical and theoretical. - : IOP Publishing. - 1751-8113 .- 1751-8121. ; 43:50, s. 505001-
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Journal article (peer-reviewed)abstract
- We present a new algorithm for generation of a random walk on a two-dimensional sphere. The algorithm is obtained by viewing the 2-sphere as the equator in the 3-sphere surrounded by an infinitesimally thin band with boundary which reflects Brownian particles and then applying known effective methods for generating Brownian motion on the 3-sphere. To test the method, the diffusion coefficient was calculated in computer simulations using the new algorithm and, for comparison, also using a commonly used method in which the particle takes a Brownian step in the tangent plane to the 2-sphere and is then projected back to the spherical surface. The two methods are in good agreement for short time steps, while the method presented in this paper continues to give good results also for larger time steps, when the alternative method becomes unstable.
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| 2. |
- Asplund, Johan
(author)
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Chekanov-Eliashberg dg-algebras and partially wrapped Floer cohomology
- 2021
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Doctoral thesis (other academic/artistic)abstract
- This thesis consists of an introduction and two research papers in the fields of symplectic and contact geometry. The focus of the thesis is on Floer theory and symplectic field theory.In Paper I we show that the partially wrapped Floer cohomology of a cotangent fiber stopped by the unit conormal of a submanifold, is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension two knots in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of the knot. This allows us to exhibit codimension two knots with infinite cyclic knot group such that the union of the unit conormal of the knot and the boundary of a cotangent fiber is not Legendrian isotopic to the union of the unit conormal of the unknot union the boundary and the same cotangent fiber.In Paper II we study the Chekanov-Eliashberg dg-algebra which is a holomorphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov-Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non-singular, establishes a proof of a conjecture by Ekholm-Lekili and independently by Sylvan. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomology and the stop removal formulas of Ganatra-Pardon-Shende.
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| 3. |
- Asplund, Johan, et al.
(author)
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Chekanov-Eliashberg dg-algebras for singular Legendrians
- 2022
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In: The Journal of Symplectic Geometry. - : International Press of Boston. - 1527-5256 .- 1540-2347. ; 20:3, s. 509-559
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Journal article (peer-reviewed)abstract
- The Chekanov–Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams [22]. It also leads to a proof of a conjectured isomorphism [17, 25] between partially wrapped Floer cohomology and Chekanov–Eliashberg dg-algebras with coefficients in chains on the based loop space.
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| 4. |
- Bourgeois, Frederic, et al.
(author)
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Effect of Legendrian Surgery
- 2012
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In: Geometry and Topology. - : Geometry and Topology Publications. - 1465-3060 .- 1364-0380. ; 16:1, s. 301-389
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Journal article (peer-reviewed)abstract
- The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in a forthcoming paper. In the Appendix written by S Ganatra and M Maydanskiy it is shown that the results of this paper imply P Seidel’s conjecture from [Proc. Sympos. Pure Math. 80, Amer. Math. Soc. (2009) 415–434].
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| 5. |
- Cieliebak, Kai, et al.
(author)
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Compactness for holomorphic curves with switching Lagrangian boundary conditions
- 2010
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In: The Journal of Symplectic Geometry. - 1527-5256 .- 1540-2347. ; 8:3, s. 267-298
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Journal article (peer-reviewed)abstract
- We prove a compactness result for holomorphic curves with boundary on an immersed Lagrangian submanifold with clean self-intersection. As an important consequence, we show that the number of intersections of such holomorphic curves with the self-intersection locus is uniformly bounded in terms of the Hofer energy.
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| 6. |
- Dimitroglou Rizell, Georgios, 1982-, et al.
(author)
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Refined disk potentials for immersed lagrangian surfaces
- 2022
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In: Journal of differential geometry. - : INTERNATIONAL PRESS BOSTON, INC. - 0022-040X .- 1945-743X. ; 121:3, s. 459-539
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Journal article (peer-reviewed)abstract
- We define a refined Gromov-Witten disk potential of monotone immersed Lagrangian surfaces in a symplectic 4-manifold that are self-transverse as an element in a capped version of the Chekanov- Eliashb erg dg-algebra of the singularity links of the double points (a collection of Legendrian Hopf links). We give a surgery formula that expresses the potential after smoothing a double point. We study refined potentials of monotone immersed Lagrangian spheres in the complex projective plane and find monotone spheres that cannot be displaced from complex lines and conics by symplectomorphisms. We also derive general restrictions on sphere potentials using Legendrian lifts to the contact 5-sphere.
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| 7. |
- Ekholm, Tobias, 1970-, et al.
(author)
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A complete knot invariant from contact homology
- 2018
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In: Inventiones Mathematicae. - : SPRINGER HEIDELBERG. - 0020-9910 .- 1432-1297. ; 211:3, s. 1149-1200
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Journal article (peer-reviewed)abstract
- We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant.
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| 8. |
- Ekholm, Tobias, 1970-, et al.
(author)
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A duality exact sequence for legendrian contact homology
- 2009
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In: Duke mathematical journal. - : Duke University Press. - 0012-7094 .- 1547-7398. ; 150:1, s. 1-75
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Journal article (peer-reviewed)abstract
- We establish a long exact sequence for Legendrian submanifolds L⊂P×R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L to P off of itself. In this sequence, the singular homology H* maps to linearized contact cohomology CH*, which maps to linearized contact homology CH*, which maps to singular homology. In particular, the sequence implies a duality between Ker(CH*→H*) and CH*/Im(H*). Furthermore, this duality is compatible with Poincaré duality in L in the following sense: the Poincaré dual of a singular class which is the image of a∈CH* maps to a class α∈CH* such that α(a)=1. The exact sequence generalizes the duality for Legendrian knots in R3 (see [26]) and leads to a refinement of the Arnold conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [7]
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| 9. |
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| 10. |
- Ekholm, Tobias, 1970-, et al.
(author)
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Branches, quivers, and ideals for knot complements
- 2022
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In: Journal of Geometry and Physics. - : Elsevier. - 0393-0440 .- 1879-1662. ; 177
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Journal article (peer-reviewed)abstract
- We generalize the F-K invariant, i.e. (Z) over cap for the complement of a knot Kin the 3-sphere, the knots-quivers correspondence, and A-polynomials of knots, and find several interconnections between them. We associate an F-K invariant to any branch of the A-polynomial of K and we work out explicit expressions for several simple knots. We show that these F-K invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using R-matrices. We generalize the quantum a-deformed A-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter a, and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide t-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d N = 2 theory T[M-3] and to the data of the associated modular tensor category MTC[M-3].
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