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194
GLUING TIGHT CONTACT STRUCTURES
, 2002
"... We prove gluing theorems for tight contact structures. As special cases, we rederive gluing theorems due to V. Colin and S. MakarLimanov and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a ..."
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Cited by 231 (26 self)
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We prove gluing theorems for tight contact structures. As special cases, we rederive gluing theorems due to V. Colin and S. MakarLimanov and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus 4 handlebody which becomes overtwisted after Legendrian −1 surgery and study certain Legendrian surgeries on T³.
Compactness results in symplectic field theory
, 2003
"... Research partially supported by THproject This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [EGH]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov’s compactness ..."
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Cited by 161 (8 self)
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Research partially supported by THproject This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [EGH]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov’s compactness theorem in [Gr] as well as compactness theorems in Floer homology theory, [F1, F2], and in contact geometry, [H, HWZ8]. Contents
Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and GromovWitten invariants
, 2001
"... We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their ..."
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Cited by 93 (2 self)
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We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov Witten classes and their descendents.
Algebraic cobordism revisited
"... Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provid ..."
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Cited by 52 (7 self)
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Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative DonaldsonThomas theory. We use double point cobordism to prove all the degree 0 conjectures in DonaldsonThomas theory: absolute, relative, and equivariant. 0.1. Overview. A first idea for defining cobordism in algebraic geometry is to impose the relation
Quasistates and symplectic intersections
, 2008
"... We establish a link between symplectic topology and a recently emerged branch of functional analysis called the theory of quasistates and quasimeasures (also known as topological measures). In the symplectic context quasistates can be viewed as an algebraic way of packaging certain information co ..."
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Cited by 49 (8 self)
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We establish a link between symplectic topology and a recently emerged branch of functional analysis called the theory of quasistates and quasimeasures (also known as topological measures). In the symplectic context quasistates can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by YongGeun Oh. As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds. This work is a part of a joint project with Paul Biran.
Coherent orientations in symplectic field theory
 Math. Z
"... Abstract. We study the coherent orientations of the moduli spaces of ‘trajectories ’ in Symplectic Field Theory, following the lines of [3]. In particular we examine their behavior at multiple closed Reeb orbits under change of the asymptotic direction. Analogous to the orientation of the unstable t ..."
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Cited by 49 (5 self)
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Abstract. We study the coherent orientations of the moduli spaces of ‘trajectories ’ in Symplectic Field Theory, following the lines of [3]. In particular we examine their behavior at multiple closed Reeb orbits under change of the asymptotic direction. Analogous to the orientation of the unstable tangent spaces of critical points in finite–dimensional Morse theory, the orientations are determined by a certain choice of orientation at each closed Reeb orbit.