| 1. |
- Carlsson, Tobias, et al.
(författare)
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Algorithm for generating a Brownian motion on a sphere
- 2010
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Ingår i: Journal of physics A: Mathematical and theoretical. - : IOP Publishing. - 1751-8113 .- 1751-8121. ; 43:50, s. 505001-
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Tidskriftsartikel (refereegranskat)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. |
- Bourgeois, Frederic, et al.
(författare)
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Effect of Legendrian Surgery
- 2012
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Ingår i: Geometry and Topology. - : Geometry and Topology Publications. - 1465-3060 .- 1364-0380. ; 16:1, s. 301-389
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Tidskriftsartikel (refereegranskat)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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| 3. |
- Cieliebak, Kai, et al.
(författare)
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Compactness for holomorphic curves with switching Lagrangian boundary conditions
- 2010
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Ingår i: The Journal of Symplectic Geometry. - 1527-5256 .- 1540-2347. ; 8:3, s. 267-298
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Tidskriftsartikel (refereegranskat)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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| 4. |
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| 5. |
- Ekholm, Tobias, 1970-, et al.
(författare)
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Filtrations on the knot contact homology of transverse knots
- 2013
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Ingår i: Mathematische Annalen. - : Springer. - 0025-5831 .- 1432-1807. ; 355:4, s. 1561-1591
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Tidskriftsartikel (refereegranskat)abstract
- We construct a new invariant of transverse links in the standard contactstructure on R^3. This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link, see (Ekholm et al., Knot contacthomology, Arxiv:1109.1542, 2011; Ng, Duke Math J 141(2):365–406, 2008). Herethe knot contact homology of a link in R3is the Legendrian contact homology DGAof its conormal lift into the unit cotangent bundle SR^3of R^3, and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links andapply it to show that the new invariant is independent of previously known invariantsof transverse links.
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| 6. |
- Ekholm, Tobias, 1970-, et al.
(författare)
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Knot contact homology
- 2013
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Ingår i: Geometry and Topology. - : Mathematical Sciences Publishers. - 1465-3060 .- 1364-0380. ; 17:2, s. 975-1112
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Tidskriftsartikel (refereegranskat)abstract
- The conormal lift of a link K in ℝ3 is a Legendrian submanifold ΛK in the unit cotangent bundle U∗ℝ3 of ℝ3 with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of K, is defined as the Legendrian homology of ΛK, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization ℝ × U∗ℝ3 with Lagrangian boundary condition ℝ × ΛK.We perform an explicit and complete computation of the Legendrian homology of ΛK for arbitrary links K in terms of a braid presentation of K, confirming a conjecture that this invariant agrees with a previously defined combinatorial version of knot contact homology. The computation uses a double degeneration: the braid degenerates toward a multiple cover of the unknot, which in turn degenerates to a point. Under the first degeneration, holomorphic disks converge to gradient flow trees with quantum corrections. The combined degenerations give rise to a new generalization of flow trees called multiscale flow trees. The theory of multiscale flow trees is the key tool in our computation and is already proving to be useful for other computations as well.
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| 7. |
- Ekholm, Tobias, 1970-
(författare)
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Notes on topological strings and knot contact homology.
- 2013
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Ingår i: Proceedings of the Gokova Geometry-Topology Conference 2013. ; , s. 1-32
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Forskningsöversikt (refereegranskat)abstract
- We give an introduction to the physics and mathematics involved in the recently observed relation between topological string theory and knot contact homology and then discuss this relation. This note is based on two lectures given at the 2013 Gökova Geometry and Topology Conference, and reports on joint work by Aganagic, Ng, Vafa, and the author [1].
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| 8. |
- Ekholm, Tobias, 1970-
(författare)
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Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology
- 2012
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Ingår i: Perspectives in Analysis, Geometry, and Topology. - Boston, MA : Springer Science+Business Media B.V.. - 9780817682767 ; , s. 109-145
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Konferensbidrag (refereegranskat)abstract
- We relate the version of rational symplectic field theory for exact Lagrangian cobordisms introduced in [6] to linearized Legendrian contact homology. More precisely, if L ⊂ Xis an exact Lagrangian submanifold of an exact symplectic manifold with convex end Λ ⊂ Y, where Yis a contact manifold and Λis a Legendrian submanifold, and if Lhas empty concave end, then the linearized Legendrian contact cohomology of Λ, linearized with respect to the augmentation induced by L, equals the rational SFT of (X,L). Following ideas of Seidel [15], this equality in combination with a version of Lagrangian Floer cohomology of Lleads us to a conjectural exact sequence that in particular implies that if X=Cn , then the linearized Legendrian contact cohomology of Λ ⊂ S 2n − 1is isomorphic to the singular homology of L. We outline a proof of the conjecture and show how to interpret the duality exact sequence for linearized contact homology of [7] in terms of the resulting isomorphism.
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| 9. |
- Ekholm, Tobias, 1970-, et al.
(författare)
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Topological Strings, D-Model, and Knot Contact Homology
- 2014
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Ingår i: Advances in Theoretical and Mathematical Physics. - Boston : International Press of Boston. - 1095-0761 .- 1095-0753. ; 18:4, s. 827-956
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed A-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the "D-model." In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Q-deformation of the extension of A-polynomials associated with the link complement.
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