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• 1.
Columbia Univ, Dept Math, New York, NY 10027 USA.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
On the immersion classes of nearby Lagrangians2016In: Journal of Topology, ISSN 1753-8416, E-ISSN 1753-8424, Vol. 9, no 1, p. 232-244Article in journal (Refereed)

We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.

• 2.
Columbia Univ, Dept Math, Room 509,MC 4406 2990 Broadway, New York, NY 10027 USA.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Simple homotopy equivalence of nearby Lagrangians2018In: Acta Mathematica, ISSN 0001-5962, E-ISSN 1871-2509, Vol. 220, no 2, p. 207-237Article in journal (Refereed)

Given a closed exact Lagrangian in the cotangent bundle of a closed smooth manifold, we prove that the projection to the base is a simple homotopy equivalence.

• 3.
Georg August Univ Gottingen, SUB, Pl Gottinger Sieben 1, D-37073 Gottingen, Germany.
Imperial Coll London, Blackett Lab, Theory Grp, Prince Consort Rd, London SW7 2AZ, England. Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics. Imperial Coll London, Blackett Lab, Theory Grp, Prince Consort Rd, London SW7 2AZ, England. SUNY Stony Brook, CN Yang Inst Theoret Phys, Stony Brook, NY 11794 USA.
T-duality in (2,1) superspace2019In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, no 6, article id 138Article in journal (Refereed)

We find the T-duality transformation rules for 2-dimensional (2,1) supersymmetric sigma-models in (2,1) superspace. Our results clarify certain aspects of the (2,1) sigma model geometry relevant to the discussion of T-duality. The complexified duality transformations we find are equivalent to the usual Buscher duality transformations (including an important refinement) together with diffeomorphisms. We use the gauging of sigma-models in (2,1) superspace, which we review and develop, finding a manifestly real and geometric expression for the gauged action. We discuss the obstructions to gauging (2,1) sigma-models, and find that the obstructions to (2,1) T-duality are considerably weaker.

• 4.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Contact Homology of Legendrian Knots in Five-Dimensional Circle Bundles2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 5.
Humboldt-University Berlin.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Conformal Differential Geometry: Q-curvature and Conformal Holonomy2010 (ed. 1)Book (Refereed)
• 6.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Cohomology of the toric arrangement associated with A(n)2019In: Journal of Fixed Point Theory and Applications, ISSN 1661-7738, E-ISSN 1661-7746, Vol. 21, no 1, article id 15Article in journal (Refereed)

We compute the total cohomology of the complement of the toric arrangement associated with the root system An as a representation of the corresponding Weyl group via fixed point theory of a twisted action of the group. We also provide several proofs of an explicit formula for the Poincare polynomial of the complement of the toric arrangement associated with An.

• 7.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Equivariant cohomology of moduli spaces of genus three curves with level two structure2019In: Geometriae Dedicata, ISSN 0046-5755, E-ISSN 1572-9168, Vol. 202, no 1, p. 165-191Article in journal (Refereed)

We study the cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked point and strata of the moduli space of Abelian differentials of genus three.

• 8.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Equivariant Cohomology of the Moduli Space of Genus Three Curves with Symplectic Level Two Structure via Point CountsIn: European Journal of Mathematics, ISSN 2199-675X, E-ISSN 2199-6768Article in journal (Refereed)
• 9.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Relations in the tautological ring of the universal curveIn: Communications in analysis and geometry, ISSN 1019-8385, E-ISSN 1944-9992Article in journal (Refereed)
• 10.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
The equivariant Euler characteristic of A_3[2]In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145Article in journal (Refereed)
• 11.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Ändliga kroppar2016Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 12.
INFN, Sez Firenze, I-50019 Sesto Fiorentino, Italy.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. INFN, Sez Firenze, I-50019 Sesto Fiorentino, Italy.
Complete integrability from Poisson-Nijenhuis structures on compact hermitian symmetric spaces2018In: The Journal of Symplectic Geometry, ISSN 1527-5256, E-ISSN 1540-2347, Vol. 16, no 5, p. 1167-1208Article in journal (Refereed)

Poisson-Nijenhuis (PN) structures have been proven to be relevant for the quantization of Poisson manifolds, through the notion of multiplicative integrable model on the symplectic groupoid. We study in this paper a class of PN structures defined by the compatible Bruhat-Poisson structure and KKS symplectic form on compact hermitian symmetric spaces. We determine the spectrum of the Nijenhuis tensor and prove complete integrability. In the case of Grassmannians, this leads to a bihamiltonian approach to Gelfand-Tsetlin variables. Our results provide a tool for the quantization of the Bruhat-Poisson structure on compact hermitian symmetric spaces.

• 13.
ULB.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. Stanford University. MIT. 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)

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].

• 14.
Univ Oxford, Math Inst, Woodstock Rd, Oxford OX2 6GG, England.
SUNY Stony Brook, Simons Ctr Geometry & Phys, Stony Brook, NY 11794 USA. Northeastern Univ, Dept Phys, Boston, MA 02115 USA. Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics. Univ Calif Santa Barbara, Dept Math, Santa Barbara, CA 93106 USA. Univ Oxford, Math Inst, Woodstock Rd, Oxford OX2 6GG, England.
Infinitely many M2-instanton corrections to M-theory on G(2)-manifolds2018In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, no 9, article id 077Article in journal (Refereed)

We consider the non-perturbative superpotential for a class of four-dimensional N = 1 vacua obtained from M-theory on seven-manifolds with holonomy G(2). The class of G(2)-holonomy manifolds we consider are so-called twisted connected sum (TCS) constructions, which have the topology of a K3-fibration over S-3. We show that the non-perturbative superpotential of M-theory on a class of TCS geometries receives infinitely many inequivalent M2-instanton contributions from infinitely many three-spheres, which we conjecture are supersymmetric (and thus associative) cycles. The rationale for our construction is provided by the duality chain of [1], which relates M-theory on TCS G(2) manifolds to E-8 x E-8 heterotic backgrounds on the Schoen Calabi-Yau threefold, as well as to F-theory on a K3-fibered Calabi-Yau fourfold. The latter are known to have an infinite number of instanton corrections to the superpotential and it is these contributions that we trace through the duality chain back to the G(2)-compactification.

• 15.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Konstruktionen av en regelbunden 17-hörning2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 16. Cederbaum, Carla
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Asymptotically flat extensions of CMC Bartnik data2017In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382Article in journal (Refereed)
• 17.
Univ Nantes, F-44035 Nantes 1, France.
Univ Nantes, F-44035 Nantes 1, France. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Positive Legendrian Isotopies And Floer Theory2019In: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 69, no 4, p. 1679-1737Article in journal (Refereed)

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.

• 18. Chantraine, Baptiste
Floer homology and Lagrangian concordance2015In: Proceedings of the Gökova Geometry-Topology Conference 2014, Gökova Geometry/Topology Conference (GGT), Gökova , 2015, p. 76-113Conference paper (Refereed)
• 19. Chantraine, Baptiste
Noncommutative augmentation categories2016In: Proceedings of the Gökova Geometry-Topology Conference 2015, Gökova Geometry/Topology Conference (GGT), Gökova , 2016, p. 116-150Conference paper (Refereed)
• 20.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden.
The vanishing cycles of curves in toric surfaces I2018In: Compositio Mathematica, ISSN 0010-437X, E-ISSN 1570-5846, Vol. 154, no 8, p. 1659-1697Article in journal (Refereed)

This article is the first in a series of two in which we study the vanishing cycles of curves in toric surfaces. We give a list of possible obstructions to contract vanishing cycles within a given complete linear system. Using tropical means, we show that any non-separating simple closed curve is a vanishing cycle whenever none of the listed obstructions appears.

• 21. Dimitroglou Rizell, Georgios
Exact Lagrangian caps and non-uniruled Lagrangian submanifolds2015In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 53, no 1, p. 37-64Article in journal (Refereed)
• 22.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Legendrian ambient surgery and Legendrian contact homologyManuscript (preprint) (Other academic)

Let $L$ be a Legendrian submanifold of a contact manifold $Y$ and let $S \subset L$ be a framed sphere bounding a subcritical isotropic disk in $Y \setminus L$. We may perform an ambient surgery on $L$ along $S$, obtaining a Legendrian submanifold $L_S \subset Y$ which is Lagrangian cobordant to $L$. We produce an isomorphism of the Legendrian contact homology algebra of $L_S$ with an algebra obtained from the algebra of $L$ after twisting the differential by a count of holomorphic disks with boundary points mapping to $S$. This isomorphism induces a bijection between the sets of augmentations for the algebras of $L$ and $L_S$.

• 23. Dimitroglou Rizell, Georgios
Legendrian ambient surgery and Legendrian contact homology2016In: The Journal of Symplectic Geometry, ISSN 1527-5256, E-ISSN 1540-2347, Vol. 14, no 3, p. 811-901Article in journal (Refereed)
• 24. Dimitroglou Rizell, Georgios
Lifting pseudo-holomorphic polygons to the symplectisation of $P\times\BbbR$ and applications2016In: Quantum Topology, ISSN 1663-487X, E-ISSN 1664-073X, Vol. 7, no 1, p. 29-105Article in journal (Refereed)
• 25. Dimitroglou Rizell, Georgios
Nontriviality results for the characteristic algebra of a DGA2017In: Mathematical proceedings of the Cambridge Philosophical Society (Print), ISSN 0305-0041, E-ISSN 1469-8064, Vol. 162, no 3, p. 419-433Article in journal (Refereed)
• 26.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Surgeries on Legendrian Submanifolds2012Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of a summary of two papers dealing with questions related to Legendrian submanifolds of contact manifolds together with exact Lagrangian cobordisms between Legendrian submanifolds. The focus is on studying Legendrian submanifolds from the perspective of their handle decompositions. The techniques used are mainly from Symplectic Field Theory.

In Paper I, a series of examples of Legendrian surfaces in standard contact 5-space are studied. For every g > 0, we produce g+1 Legendrian surfaces of genus g, all with g+1 transverse Reeb chords, which lie in distinct Legendrian isotopy classes. For each g, exactly one of the constructed surfaces has a Legendrian contact homology algebra admitting an augmentation. Moreover, it is shown that the same surface is the only one admitting a generating family. Legendrian contact homology with Novikov coefficients is used to classify the different Legendrian surfaces. In particular, we study their augmentation varieties.

In Paper II, the effect of a Legendrian ambient surgery on a Legendrian submanifold is studied. Given a Legendrian submanifold together which certain extra data, a Legendrian ambient surgery produces a Legendrian embedding of the manifold obtained by surgery on the original submanifold. The construction also provides an exact Lagrangian handle-attachment cobordism between the two submanifolds. The Legendrian contact homology of the submanifold produced by the Legendrian ambient surgery is then computed in terms of pseudo-holomorphic disks determined by data on the original submanifold. Also, the cobordism map induced by the exact Lagrangian handle attachment is computed. As a consequence, it is shown that a sub-critical standard Lagrangian handle attachment cobordism induces a one-to-one correspondence between the augmentations of the Legendrian contact homology algebras of its two ends.

1. Knotted Legendrian surfaces with few Reeb chords
Open this publication in new window or tab >>Knotted Legendrian surfaces with few Reeb chords
2011 (English)In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 11, no 5, p. 2903-2936Article in journal (Refereed) Published
##### Abstract [en]

For g > 0, we construct g + 1 Legendrian embeddings of a surface of genus g into J(1)(R-2) = R-5 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 Z(2), 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 J(1)(S-2) from a similar perspective.

Mathematics
##### Identifiers
urn:nbn:se:uu:diva-172252 (URN)10.2140/agt.2011.11.2903 (DOI)000299576600013 ()
Available from: 2012-04-04 Created: 2012-04-03 Last updated: 2017-12-07Bibliographically approved
2. Legendrian ambient surgery and Legendrian contact homology
Open this publication in new window or tab >>Legendrian ambient surgery and Legendrian contact homology
##### Abstract [en]

Let $L$ be a Legendrian submanifold of a contact manifold $Y$ and let $S \subset L$ be a framed sphere bounding a subcritical isotropic disk in $Y \setminus L$. We may perform an ambient surgery on $L$ along $S$, obtaining a Legendrian submanifold $L_S \subset Y$ which is Lagrangian cobordant to $L$. We produce an isomorphism of the Legendrian contact homology algebra of $L_S$ with an algebra obtained from the algebra of $L$ after twisting the differential by a count of holomorphic disks with boundary points mapping to $S$. This isomorphism induces a bijection between the sets of augmentations for the algebras of $L$ and $L_S$.

Geometry
Mathematics
##### Identifiers
Available from: 2012-07-02 Created: 2012-07-02 Last updated: 2012-10-05
• 27. Dimitroglou Rizell, Georgios
Uniqueness of extremal Lagrangian tori in the four-dimensional disc2016In: Proceedings of the Gökova Geometry-Topology Conference 2015, Gökova Geometry/Topology Conference (GGT), Gökova , 2016, p. 151-167Conference paper (Refereed)
• 28. Dimitroglou Rizell, Georgios
Exotic spheres and the topology of symplectomorphism groups2015In: Journal of Topology, ISSN 1753-8416, E-ISSN 1753-8424, Vol. 8, no 2, p. 586-602Article in journal (Refereed)
• 29. Dimitroglou Rizell, Georgios
Unlinking and unknottedness of monotone Lagrangian submanifolds2014In: Geometry and Topology, ISSN 1465-3060, E-ISSN 1364-0380, Vol. 18, no 2, p. 997-1034Article in journal (Refereed)
• 30. Dimitroglou Rizell, Georgios
Estimating the number of Reeb chords using a linear representation of the characteristic algebra2015In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 15, no 5, p. 2887-2920Article in journal (Refereed)
• 31. Dimitroglou Rizell, Georgios
On homological rigidity and flexibility of exact Lagrangian endocobordisms2014In: International Journal of Mathematics, ISSN 0129-167X, Vol. 25, no 10, p. 1450098-24Article in journal (Refereed)
• 32. Dimitroglou Rizell, Georgios
The number of Hamiltonian fixed points on symplectically aspherical manifolds2017In: Proceedings of the Gökova Geometry-Topology Conference 2016, Gökova Geometry/Topology Conference (GGT), Gökova , 2017, p. 138-150Conference paper (Refereed)
• 33.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Charles Univ Prague, Fac Math & Phys, Sokolovska 83, CR-18000 Prague 8, Czech Republic.
The stable Morse number as a lower bound for the number of Reeb chords2018In: The Journal of Symplectic Geometry, ISSN 1527-5256, E-ISSN 1540-2347, Vol. 16, no 5, p. 1209-1248Article in journal (Refereed)

Assume that we are given a closed chord-generic Legendrian sub-manifold Lambda subset of P x R of the contactisation of a Liouville manifold, where Lambda moreover admits an exact Lagrangian filling L-Lambda subset of R x P x 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.

• 34. Dimitroglou Rizell, Georgios
Lagrangian isotopy of tori in $S^2\times S^2$ and $\BbbCP^2$2016In: Geometric and Functional Analysis, ISSN 1016-443X, E-ISSN 1420-8970, Vol. 26, no 5, p. 1297-1358Article in journal (Refereed)
• 35.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. Univ Fed Fluminense, Inst Matemat & Estat, Rua Prof Marcos Waldemar de Freitas Reis S-N, BR-24210201 Niteroi, RJ, Brazil.
Univ Mississippi, Dept Math, POB 1848, University, MS 38677 USA.
Morse-Bott split symplectic homology2019In: Journal of Fixed Point Theory and Applications, ISSN 1661-7738, E-ISSN 1661-7746, Vol. 21, no 3, article id 77Article in journal (Refereed)

We associate a chain complex to a Liouville domain ((W) over bar ,d lambda) whose boundary Y admits a Boothby-Wang contact form (i.e.is a prequantization space). The differential counts Floer cylinders with cascades in the completion W of (W) over bar, in the spirit of Morse-Bott homology (Bourgeois in A Morse-Bott approach to contact homology, Ph.D. Thesis. ProQuest LLC, Stanford University, Ann Arbor 2002; Frauenfelder in Int Math Res Notices 42:2179-2269, 2004; Bourgeois and Oancea in Duke Math J 146(1), 71-174, 2009). The homology of this complex is the symplectic homology of W (Diogo and Lisi in J Topol 12:966-1029, 2019). Let X be obtained from (W) over bar by collapsing the boundary Y along Reeb orbits, giving a codimension two symplectic submanifold Sigma. Under monotonicity assumptions on X and Sigma, we show that for generic data, the differential in our chain complex counts elements of moduli spaces of cascades that are transverse. Furthermore, by some index estimates, we show that very few combinatorial types of cascades can appear in the differential.

• 36.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Univ Mississippi, Dept Math, University, MS USA.
Symplectic homology of complements of smooth divisors2019In: Journal of Topology, ISSN 1753-8416, E-ISSN 1753-8424, Vol. 12, no 3, p. 967-1030Article in journal (Refereed)

If (X,omega) is a closed symplectic manifold, and sigma is a smooth symplectic submanifold Poincare dual to a positive multiple of omega, then X set minus sigma can be completed to a Liouville manifold (W,d lambda). Under monotonicity assumptions on X and on sigma, we construct a chain complex whose homology computes the symplectic homology of W. We show that the differential is given in terms of Morse contributions, Gromov-Witten invariants of X relative to sigma and Gromov-Witten invariants of sigma. We use a Morse-Bott model for symplectic homology. Our proof involves comparing Floer cylinders with punctures to pseudoholomorphic curves in the symplectization of the unit normal bundle to sigma.

• 37.
Natl Res Univ, Higher Sch Econ, Fac Math, Usacheva 6, Moscow 119048, Russia.
Univ Melbourne, Sch Math & Stat, Melbourne, Vic 3010, Australia. Ecole Polytech Fed Lausanne, Dept Math, CH-1015 Lausanne, Switzerland. Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics. Inst Informat Transmiss Problems, Moscow 127994, Russia;ITEP, Moscow 117218, Russia. Univ Amsterdam, Korteweg De Vries Inst Math, Postbus 94248, NL-1090 GE Amsterdam, Netherlands.
Primary invariants of Hurwitz Frobenius manifolds2018In: TOPOLOGICAL RECURSION AND ITS INFLUENCE IN ANALYSIS, GEOMETRY, AND TOPOLOGY / [ed] Liu, CCM Mulase, M, AMER MATHEMATICAL SOC , 2018, p. 297-331Conference paper (Refereed)

Hurwitz spaces parameterizing covers of the Riemann sphere can be equipped with a Frobenius structure. In this review, we recall the construction of such Hurwitz Frobenius manifolds as well as the correspondence between semisimple Frobenius manifolds and the topological recursion formalism. We then apply this correspondence to Hurwitz Frobenius manifolds by explaining that the corresponding primary invariants can be obtained as periods of multidifferentials globally defined on a compact Riemann surface by topological recursion. Finally, we use this construction to reply to the following question in a large class of cases: given a compact Riemann surface, what does the topological recursion compute?

• 38.
Royal Inst Technol KTH, Dept Math, Stockholm, Sweden..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Asymptotic geometry of discrete interlaced patterns: Part I2015In: International Journal of Mathematics, ISSN 0129-167X, Vol. 26, no 11, article id 1550093Article in journal (Refereed)

A discrete Gelfand-Tsetlin pattern is a configuration of particles in Z(2). The particles are arranged in a finite number of consecutive rows, numbered from the bottom. There is one particle on the first row, two particles on the second row, three particles on the third row, etc., and particles on adjacent rows satisfy an interlacing constraint. We consider the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of a fixed size where the particles on the top row are in deterministic positions. This measure arises naturally as an equivalent description of the uniform probability measure on the set of all tilings of certain polygons with lozenges. We prove a determinantal structure, and calculate the correlation kernel. We consider the asymptotic behavior of the system as the size increases under the assumption that the empirical distribution of the deterministic particles on the top row converges weakly. We consider the asymptotic "shape" of such systems. We provide parameterizations of the asymptotic boundaries and investigate the local geometric properties of the resulting curves. We show that the boundary can be partitioned into natural sections which are determined by the behavior of the roots of a function related to the correlation kernel. This paper should be regarded as a companion piece to the paper [E. Duse and A. Metcalfe, Asymptotic geometry of discrete interlaced patterns: Part II, in preparation], in which we resolve some of the remaining issues. Both of these papers serve as background material for the papers [E. Duse and A. Metcalfe, Universal edge fluctuations of discrete interlaced particle systems, in preparation; E. Duse and K. Johansson and A. Metcalfe, Cusp Airy process of discrete interlaced particle systems, in preparation], in which we examine the edge asymptotic behavior.

• 39. Ekholm, Tobias
Lagrangian exotic spheres2016In: Journal of Topology and Analysis (JTA), ISSN 1793-5253, E-ISSN 1793-7167Article in journal (Refereed)
• 40.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Non-loose Legendrian spheres with trivial contact homology DGA2016In: Journal of Topology, ISSN 1753-8416, E-ISSN 1753-8424, Vol. 9, no 3, p. 826-848Article in journal (Refereed)

Loose Legendrian n-submanifolds, n >= 2, were introduced by Murphy ('Loose Legendrian embeddings in high dimensional contact manifolds', Preprint, 2012, arXiv:1201.2245) and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra (DGA) to a Legendrian submanifold. The DGA of a loose Legendrian submanifold is trivial. We show that the converse is not true by constructing non-loose Legendrian n-spheres in standard contact (2n + 1)-space, n >= 2, with trivial DGA.

• 41.
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)

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].

• 42.
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)

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.

• 43.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
UC Berkeley. Duke. 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)

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.

• 44. Ekholm, Tobias
Constructing exact Lagrangian immersions with few double points.2013In: Geometric and Functional Analysis, ISSN 1016-443X, E-ISSN 1420-8970, Vol. 23, no 6, p. 1772-1803Article in journal (Refereed)
• 45.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
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)

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.

• 46.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
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)

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.

• 47.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Geogia Tech. Duke univ. 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)

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.

• 48.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. Inst Mittag Leffler, Aurav 17, S-18260 Djursholm, Sweden.
Univ So Calif, Los Angeles, CA 90089 USA. Tokyo Inst Technol, Meguro Ku, Tokyo 1528551, Japan.
Legendrian knots and exact Lagrangian cobordisms2016In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, no 11, p. 2627-2689Article in journal (Refereed)

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair (X, L) consisting of an exact symplectic manifold X and an exact Lagrangian cobordism L subset of X which agrees with cylinders over Legendrian links Lambda(+) and Lambda (-) at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of Lambda(+) to that of Lambda (-) .We give a gradient flow tree description of the DGA maps for certain pairs (X, L), which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.

• 49.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry. 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)

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.

• 50.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
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)

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

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