Journal ArticleDOI
Pseudo holomorphic curves in symplectic manifolds
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In this article, the authors define a parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J).Abstract:
Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the sphere S 2.read more
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Journal ArticleDOI
Quantum field theory and the Jones polynomial
TL;DR: In this paper, it was shown that 2+1 dimensional quantum Yang-Mills theory with an action consisting purely of the Chern-Simons term is exactly soluble and gave a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms.
MonographDOI
Nonlinear dispersive equations : local and global analysis
TL;DR: In this paper, the Korteweg de Vries equation was used for ground state construction in the context of semilinear dispersive equations and wave maps from harmonic analysis.
Book
Topological methods in hydrodynamics
Vladimir I. Arnold,Boris Khesin +1 more
TL;DR: A group theoretical approach to hydrodynamics is proposed in this article, where the authors consider the hydrodynamic geometry of diffeomorphism groups and the principle of least action implies that the motion of a fluid is described by geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy.
Book ChapterDOI
Homological Algebra of Mirror Symmetry
TL;DR: Mirror symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros).
Book ChapterDOI
Geometry of 2D topological field theories
TL;DR: In this paper, the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories is studied, where WDVV equations and Frobenius manifolds are discussed.
References
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Journal ArticleDOI
The existence of minimal immersions of $2$-spheres
Jonathan Sacks,Karen Uhlenbeck +1 more
Book
Lectures on Symplectic Manifolds
TL;DR: In this article, Lagrangian splittings, real and complex polarizations, Kahler manifolds reduction, the calculus of canonical relations, intermediate polarizations Hamiltonian systems and group actions on symplectic manifolds Normal forms Lagrangians and families of functions Intersection theory of lagrangian submanifolds Quantization on cotangent bundles Quantization and polarizations Quantizing lagrangians, and subspaces, construction of the Maslov bundle References