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  • 1. Bourgeois, Frederic
    et al.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Cieliebak, Kai
    A note on Reeb dynamics on the tight 3-sphere2007In: Journal of Modern Dynamics, ISSN 1930-5311 (print) ISSN 1930-532X (electronic), Vol. 1, no 4, p. 597-613Article in journal (Refereed)
    Abstract [en]

    We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space.

  • 2. Bourgeois, Frederic
    et al.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Eliashberg, Yakov
    Symplectic homology product via Legendrian surgery2011In: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 108, no 20, p. 8114-8121Article in journal (Refereed)
    Abstract [en]

    This research announcement continues the study of the symplectic homology of Weinstein manifolds undertaken by the authors [Bourgeois F, Ekholm T, Eliashberg Y (2009) arXiv:0911.0026] where the symplectic homology, as a vector space, was expressed in terms of the Legendrian homology algebra of the attaching spheres of critical handles. Here, we express the product and Batalin-Vilkovisky operator of symplectic homology in that context.

  • 3.
    Bourgeois, Frederic
    et al.
    ULB.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Eliashberg, Yakov
    Stanford University.
    Ganatra, Sheel
    MIT.
    Maydanskiy, Maksim
    Stanford University.
    Effect of Legendrian Surgery2012In: Geometry and Topology, ISSN 1465-3060, E-ISSN 1364-0380, Vol. 16, no 1, p. 301-389Article in journal (Refereed)
    Abstract [en]

    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].

  • 4.
    Carlsson, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Physical and Analytical Chemistry.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Elvingson, Christer
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Physical and Analytical Chemistry.
    Algorithm for generating a Brownian motion on a sphere2010In: Journal of physics A: Mathematical and theoretical, ISSN 1751-8113, Vol. 43, no 50, p. 505001-Article in journal (Refereed)
    Abstract [en]

    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.

  • 5.
    Cieliebak, Kai
    et al.
    LMU, Munich.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Latschev, Janko
    Univ Hamburg.
    Compactness for holomorphic curves with switching Lagrangian boundary conditions2010In: The Journal of Symplectic Geometry, ISSN 1527-5256, E-ISSN 1540-2347, Vol. 8, no 3, p. 267-298Article in journal (Refereed)
    Abstract [en]

    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.

  • 6.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    A version of rational SFT for exact Lagrangian cobordisms in 1-jet spaces2009In: New perspectives and challenges in symplectic field theory: , Providence, RI: Amer. Math. Soc. , 2009, p. 173-199Conference paper (Refereed)
  • 7.
    Ekholm, Tobias
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Differential 3-knots in 5-space with and without self-intersections2001In: TOPOLOGY, ISSN 0040-9383, Vol. 40, no 1, p. 157-196Article in journal (Refereed)
    Abstract [en]

    Regular homotopy classes of immersions $S^3 \to R^5$ constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self-intersection. Geometric properties of self-intersections are used to construct two invariants J and St of generic immersions which are analogous to Arnold's invariants of plane curves [1]. We prove that J and St are independent first-order invariants and that any first-order invariant is a linear combination of these. As by-products, some invariants of immersions are obtained. Using them, we find restrictions on the topology of self-intersections

  • 8.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Double points of exact Lagrangian immersions and Legendrian contact homology2006In: Clay Mathematics Proceedings, ISSN 1534-6455, Vol. 5, p. 181-Article in journal (Refereed)
  • 9.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Immersions in the metastable range and spin structures on surfaces1998In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 83, no 1, p. 5-41Article in journal (Refereed)
  • 10.
    Ekholm, Tobias
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Invariants of generic immersions2001In: Pacific Journal of Mathemtics, Vol. 199, no 2, p. 321-345Article in journal (Refereed)
  • 11.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Knutar och total krökning2007In: Normat, ISSN 0801-3500, Vol. 55, no 4, p. 145-156Article, review/survey (Other (popular science, discussion, etc.))
  • 12.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Morse flow trees and Legendrian contact homology in 1-jet spaces2007In: Geometry and Topology, ISSN 1465-3060, E-ISSN 1364-0380, Vol. 11, no 2, p. 1083-1224Article in journal (Refereed)
    Abstract [en]

    Let L ⊂ J1(M) be a Legendrian submanifold of the 1–jet space of a Riemannian n–manifold M. A correspondence is established between rigid flow trees in M determined by L and boundary punctured rigid pseudo-holomorphic disks in T*M, with boundary on the projection of L and asymptotic to the double points of this projection at punctures, provided n≤2, or provided n>2 and the front of L has only cusp edge singularities. This result, in particular, shows how to compute the Legendrian contact homology of L in terms of Morse theory.

  • 13.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Notes on topological strings and knot contact homology.2013In: Proceedings of the Gokova Geometry-Topology Conference 2013, ISSN 978-1-57146-285-5, p. 1-32Article, review/survey (Refereed)
    Abstract [en]

    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].

  • 14.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Rational SFT, Linearized Legendrian Contact Homology, and Lagrangian Floer Cohomology2012In: Perspectives in Analysis, Geometry, and Topology: On the Occasion of the 60th Birthdayof Oleg Viro / [ed] Ilia Itenberg, Burglind Jöricke, Mikael Passare, Springer Science+Business Media B.V., 2012, p. 109-145Conference paper (Refereed)
    Abstract [en]

    We relate the version of rational symplectic field theory for exact Lagrangian cobordisms introduced in [6] to linearized Legendrian contact homology. More precisely, if LXis an exact Lagrangian submanifold of an exact symplectic manifold with convex end ΛY, 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 ΛS 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.

  • 15.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Rational symplectic field theory over Z_2 for exact Lagrangian cobordisms2008In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 10, no 3, p. 641-704Article in journal (Refereed)
    Abstract [en]

    We construct a version of rational symplectic field theory for pairs (X, L), where X is an exact symplectic manifold, where L ⊂ X is an exact Lagrangian submanifold with components subdivided into k subsets, and where both X and L have cylindrical ends. The theory associates to (X, L) a Z-graded chain complex of vector spaces over Z_2 , filtered with k filtration levels. The corresponding k -level spectral sequence is invariant under deformations of (X, L) and has the following property: if (X, L) is obtained by joining a negative end of a pair (X, L) to a positive end of a pair (X, L), then there are natural morphisms from the spectral sequences of (X, L) and of (X ,L) to the spectral sequence of (X, L). As an application, we show that if \Lambda ⊂ Y is a Legendrian submanifold of a contact manifold then the spectral sequences associated to (Y × R, \Lambda_s × R), where Y × R is the symplectization of Y and where \Lambda_s ⊂ Y is the Legendrian submanifold consisting of s parallel copies of \Lambda subdivided into k subsets, give Legendrian isotopy invariants of \Lambda.

  • 16.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Regular homotopy and total curvature. I. Circle immersions into surfaces2006In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 6, p. 459-492Article in journal (Refereed)
  • 17.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Regular homotopy and total curvature. II. Sphere immersions into 3-space2006In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 6, p. 493-512Article in journal (Refereed)
  • 18.
    Ekholm, Tobias
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
    Regular homotopy and Vassiliev Invariants of generic immesrions $S^k \to R^{2k-1}$1998In: Journal of not Theory and Its Ramifcations, Vol. 7, p. 1041-1064Article in journal (Refereed)
  • 19.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Self-intersection surfaces, regular homotopoy, and finite order invariants2000In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 11, p. 909-929Article in journal (Refereed)
  • 20.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    The complex shade of a real space, and its applications2002In: Algebra i Analiz, ISSN 0234-0852, Vol. 14, no 2, p. 56-91Article in journal (Refereed)
  • 21.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Aganagic, Mina
    UC Berkeley.
    Ng, Lenhard
    Duke.
    Vafa, Cumrun
    Harvard.
    Topological Strings, D-Model, and Knot Contact Homology2014In: Advances in Theoretical and Mathematical Physics, ISSN 1095-0761, E-ISSN 1095-0753, Vol. 18, no 4, p. 827-956Article in journal (Other academic)
    Abstract [en]

    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.

  • 22.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Eliashberg, Yakov
    Murphy, Emmy
    Smith, Ivan
    Constructing exact Lagrangian immersions with few double points2013In: Geometric and Functional Analysis, ISSN 1016-443X, E-ISSN 1420-8970, Vol. 23, no 6, p. 1772-1803Article in journal (Refereed)
    Abstract [en]

    We establish, as an application of the results from Eliashberg and Murphy (Lagrangian caps,2013), an h-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space with exactly one transverse double point. Our construction also yields a Lagrangian embedding S1×S2→R6 with vanishing Maslov class.

  • 23.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Etnyre, John
    Invariants of knots, embeddings and immersions via contact geometry2005In: Fields Inst. Commun., Vol. 47, p. 77-Article in journal (Refereed)
  • 24.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Etnyre, John B.
    Sullivan, Michael G.
    Legendrian contact homology in P x R2007In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 359, no 7, p. 3301-3335Article in journal (Refereed)
    Abstract [en]

    A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P × ℝ, where P is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of ℝn and, more generally, invariants of self transverse immersions into ℝn up to restricted regular homotopies. When n = 3, this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.

  • 25.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Etnyre, John
    Ng, Lenhard
    Sullivan, Michael
    Filtrations on the knot contact homology of transverse knots2013In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 355, no 4, p. 1561-1591Article in journal (Refereed)
    Abstract [en]

    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.

  • 26.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Etnyre, John
    Geogia Tech.
    Ng, Lenhard
    Duke univ.
    Sullivan, Michael
    U Mass.
    Knot contact homology2013In: Geometry and Topology, ISSN 1465-3060, E-ISSN 1364-0380, Vol. 17, no 2, p. 975-1112Article in journal (Refereed)
    Abstract [en]

    The conormal lift of a link K in ℝ3 is a Legendrian submanifold ΛK in the unit cotangent bundle U3 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 × U3 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.

  • 27.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Etnyre, John
    Department of mathematics, Georgia Tech.
    Sabloff, Josh
    Department of mathematics, Haverford College.
    A duality exact sequence for legendrian contact homology2009In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 150, no 1, p. 1-75Article in journal (Refereed)
    Abstract [en]

    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]

  • 28.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Etnyre, John
    Sullivan, Michael
    Non-isotopic Legendrian submanifolds in R^{2n+1}2005In: J. Differential Geom., Vol. 71, no 1, p. 85-128Article in journal (Refereed)
  • 29.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Etnyre, John
    Sullivan, Michael
    Orientations in Legendrian contact homology and exact Lagrangian immersions.2005In: Internat. J. Math., Vol. 16, no 5, p. 453-Article in journal (Other (popular science, discussion, etc.))
  • 30.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Etnyre, John
    Sullivan, Michael
    The contact homology of Legendrian submanifolds in R^{2n+1}2005In: J. Differential Geom., Vol. 71, no 2, p. 177-Article in journal (Refereed)
  • 31.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Kragh, Thomas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Smith, Ivan
    Univ Cambridge, Ctr Math Sci, Wilberforce Rd, Cambridge CB3 0WB, England.
    Lagrangian exotic spheres2016In: Journal of Topology and Analysis, Vol. 8, no 3, p. 375-397Article in journal (Refereed)
    Abstract [en]

    Let k > 2. We prove that the cotangent bundles T*Sigma and T*Sigma' of oriented homotopy (2k -1)-spheres Sigma and Sigma' are symplectomorphic only if [Sigma] = [+/-Sigma'] is an element of Theta(2k-1)/bP(2k), where Theta(2k-1) denotes the group of oriented homotopy (2k -1)-spheres under connected sum, bP(2k) denotes the subgroup of those that bound a parallelizable 2k-manifold, and where -Sigma denotes Sigma with orientation reversed. We further show that if n = 4k -1 and RPn#Sigma admits a Lagrangian embedding in CPn, then [Sigma#Sigma] is an element of bP(4k). The proofs build on [1] and [18] in combination with a new cut-and-paste argument; that also yields some interesting explicit exact Lagrangian embeddings, for instance of the sphere S-n into the plumbing T*Sigma(n)#T-pl*Sigma(n) of cotangent bundles of certain exotic spheres. As another application, we show that there are re-parametrizations of the zero-section in the cotangent bundle of a sphere that are not Hamiltonian isotopic (as maps rather than as submanifolds) to the original zero-section.

  • 32.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. Institut Mittag-Leffler, Aurav 17, 182 60 Djursholm, Sweden.
    Kucharski, Piotr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Longhi, Pietro
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics. Institute for Theoretical Physics, ETH Zurich, CH - 8093, Zurich, Switzerland.
    Physics and geometry of knots-quivers correspondence2018Manuscript (preprint) (Other academic)
    Abstract [en]

    The recently conjectured knots-quivers correspondence relates gauge theoretic invariants of a knot K in the 3-sphere to representation theory of a quiver QK associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of the large N duality of Ooguri and Vafa, that relates knot invariants to counts of holomorphic curves with boundary on LK, the conormal Lagrangian of the knot in the resolved conifold, and corresponding M-theory considerations. From the physics side, we show that the quiver encodes a 3d N=2 theory T[QK] whose low energy dynamics arises on the worldvolume of an M5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of QK. From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on LK is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver QK and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to QK and (doubly) refined version of LMOV invariants.

  • 33.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Kutzschebauch, Frank
    Total curvature and area of curves with cusps and of surface maps2005In: Math. Scand., Vol. 96, no 2, p. 224-Article in journal (Other (popular science, discussion, etc.))
  • 34.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Larsson, Ola
    Minimizing singularities of generic plane disks with immersed boundaries2005In: Ark. Mat., Vol. 43, no 2, p. 347-Article in journal (Refereed)
  • 35.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Ng, Lenhard
    Duke University.
    Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold2015In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 101, no 1, p. 67-157Article in journal (Refereed)
    Abstract [en]

    We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in S1×S2 or any connected sum #k(S1×S2), viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplectic homology, this gives a combinatorial description of the symplectic homology of any 4-dimensional Weinstein manifold (and of the linearized contact homology of its boundary). We also study examples and discuss the invariance of the Legendrian homology algebra under deformations, from both the combinatorial and the analytical perspectives

  • 36.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. Inst Mittag Leffler, S-18260 Djursholm, Sweden..
    Ng, Lenhard
    Duke Univ, Dept Math, Durham, NC 27708 USA..
    Shende, Vivek
    Univ Calif Berkeley, Dept Math, 970 Evans Hall, Berkeley, CA 94720 USA..
    A complete knot invariant from contact homology2018In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 211, no 3, p. 1149-1200Article in journal (Refereed)
    Abstract [en]

    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.

  • 37.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Oancea, Alexandru
    UPMC, Sorbonne Univ, Inst Math Jussieu Paris Rive Gauche, France.
    Symplectic and contact differential graded algebras2017In: Geometry and Topology, ISSN 1465-3060, E-ISSN 1364-0380, Vol. 21, no 4, p. 2161-2230Article in journal (Refereed)
    Abstract [en]

    We define Hamiltonian simplex differential graded algebras (DGA) with differentials that deform the high-energy symplectic homology differential and wrapped Floer homology differential in the cases of closed and open strings in a Liouville manifold of finite type, respectively. The order-m term in the differential is induced by varying natural degree-m coproducts over an (m-1)-simplex, where the operations near the boundary of the simplex are trivial. We show that the Hamiltonian simplex DGA is quasi-isomorphic to the (nonequivariant) contact homology algebra and to the Legendrian homology algebra of the ideal boundary in the closed and open string cases, respectively.

  • 38.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. Inst Mittag Leffler, S-18260 Djursholm, Sweden..
    Smith, Ivan
    Univ Cambridge, Ctr Math Sci, Cambridge CB3 0WB, England..
    EXACT LAGRANGIAN IMMERSIONS WITH A SINGLE DOUBLE POINT2016In: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 29, no 1, p. 1-59, article id PII S 0894-0347(2015)00825-6Article in journal (Refereed)
  • 39.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Smith, Ivan
    Exact Lagrangian immersions with one double point revisited2014In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 358, no 1-2, p. 195-240Article in journal (Refereed)
    Abstract [en]

    We study exact Lagrangian immersions with one double point of a closed orientable manifold into . We prove that if the Maslov grading of the double point does not equal then is homotopy equivalent to the sphere, and if, in addition, the Lagrangian Gauss map of the immersion is stably homotopic to that of the Whitney immersion, then bounds a parallelizable -manifold. The hypothesis on the Gauss map always holds when or when . The argument studies a filling of obtained from solutions to perturbed Cauchy-Riemann equations with boundary on the image of the immersion. This leads to a new and simplified proof of some of the main results of Ekholm and Smith (Exact Lagrangian immersions with a single double point 2011)). which treated Lagrangian immersions in the case by applying similar techniques to a Lagrange surgery of the immersion, as well as to an extension of these results to the odd-dimensional case.

  • 40.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. nst Mittag Leffler, Aurav 17, S-18260 Djursholm, Sweden..
    Smith, Ivan
    Univ Cambridge, Ctr Math Sci, Wilberforce Rd, Cambridge CB3 0WB, England..
    Nearby Lagrangian fibers and Whitney sphere links2018In: Compositio Mathematica, ISSN 0010-437X, E-ISSN 1570-5846, Vol. 154, no 4, p. 685-718Article in journal (Refereed)
    Abstract [en]

    Let n > 3, and let L be a Lagrangian embedding of R-n into the cotangent bundle T*R-n of R-n that agrees with the cotangent fiber T*R-x(n) over a point x not equal 0 outside a compact set. Assume that L is disjoint from the cotangent fiber at the origin. The projection of L to the base extends to a map of the n-sphere S-n into R-n\{0}. We show that this map is homotopically trivial, answering a question of Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in T*R-n, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1-59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195-240] with symplectic field theory.

  • 41.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Szucs, Andras
    The group of immersions of homotopy (4k-1)-spheres2005In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 38, no 1, p. 163-176Article in journal (Refereed)
  • 42.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Szucs, Andras
    Terpai, Tamas
    Cobordisms of fold maps and maps with a prescribed number of cusps2007In: Kyushu Journal of Mathematics, ISSN 1340-6116(Print);1883-2032(Online), Vol. 61, no 2, p. 395-414Article in journal (Refereed)
    Abstract [en]

    A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($\Sigma^{1,1}$-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ($\Sigma^{1,0}$-singularities). Two fold maps are fold bordant if there are cobordisms between their source- and target manifolds with a fold map extending the two maps between the boundaries, if the two targets agree and the target cobordism can be taken as a product with a unit interval then the maps are fold cobordant. We compute the cobordism groups of fold maps of $(2k-1)$-manifolds into $(3k-2)$-space. Analogous cobordism semi-groups for arbitrary closed $(3k-2)$-dimensional target manifolds are endowed with Abelian group structures and described. Fold bordism groups in the same dimensions are described as well.

  • 43.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Szucs, András
    On the triple points of singular maps2002In: Comment. Math. Helv., Vol. 77, no 2, p. 408-414Article in journal (Refereed)
  • 44.
    Ekholm, Tobias
    et al.
    MATHEMATICS I-V. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Szücs, András
    Geometric formulas for Smale invariants of codimension two immersions2003In: Topology, ISSN 0040-9383, E-ISSN 1879-3215, Vol. 42, no 1, p. 171-196Article in journal (Refereed)
  • 45.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
    Takase, Masamichi
    Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions2011In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 43, no 2, p. 251-266Article in journal (Refereed)
    Abstract [en]

    A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self-intersection points equal to-n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. Precomposing these circle bundle immersions with their universal covering maps, we get for n > 0 immersions g(n) of the 3-sphere into 4-space. In this note, we compute the Smale invariants of g(n). The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends g(n). As an application, we determine the classes represented by g(n) in the cobordism group of immersions which is naturally identified with the stable 3-stem. It follows in particular that g(n) represents a generator of the stable 3-stem if and only if n is divisible by 3.

  • 46.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Weistrand, Ola
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematics I -V.
    Total curvatures of holonomic links2000In: JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS, ISSN 0218-2165, Vol. 9, no 7, p. 893-906Article in journal (Refereed)
    Abstract [en]

    A differential geometric characterization of the braid-index of a link is found. After multiplication by 2 pi, it equals the infimum of the sum of total curvature and total absolute torsion over holonomic representatives of the link.

  • 47.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    White, Brian
    Wienholtz, Daniel
    Embeddedness of minimal surfaces with total boundary curvature at most 4 pi2002In: Ann. of Math. (2), Vol. 155, no 1, p. 209-234Article in journal (Refereed)
  • 48.
    Ekholm, Tobias
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Wolff, Maxime
    Framed holonomic knots2002In: Algebr. Geom. Topol. (electronic), Vol. 2, p. 449-463Article in journal (Refereed)
  • 49.
    Kamerlin, Natasha
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Chemistry - Ångström, Physical Chemistry. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Carlsson, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Chemistry - Ångström, Physical Chemistry.
    Elvingson, Christer
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Chemistry - Ångström, Physical Chemistry.
    Construction of a closed polymer network for computer simulations2014In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 141, no 15, p. 154113-Article in journal (Refereed)
    Abstract [en]

    Computer simulations are an important tool for linking the behaviour of polymer materials to the properties of the constituent polymer chains. In simulations, one normally uses periodic boundary conditions to mimic a macroscopic system. For a cross-linked polymer network, this will impose restrictions on the motion of the polymer chains at the borders of the simulation cell. We present a new method for constructing a three-dimensional closed network without periodic boundaries by embedding the system onto the surface of a sphere in four dimensions. This method can also be used to construct finite-sized gel particles for simulating the swelling of particles in a surrounding solvent. The method is described in algorithmic detail to allow the incorporation of the method into different types of simulation programs. We also present the results of Brownian dynamics simulations, analyzing the end-to-end distribution, radial distribution function, and the pore size distribution for different volume fractions and for chains with varying stiffness.

  • 50. Kamerlin, Natasha
    et al.
    Ekholm, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Carlsson, Tobias
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Chemistry - Ångström, Physical Chemistry.
    Elvingson, Christer
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Chemistry - Ångström, Physical Chemistry.
    Construction of a non-periodic closed network for computer simulationsManuscript (preprint) (Other academic)
12 1 - 50 of 52
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