Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

The symplectic Thom conjecture

Abstract: In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic Thom conjecture: a symplectic surface in a symplectic four-manifold is genus-minimizing in its homology class. Another corollary of the relations is a general adjunction inequality for embedded surfaces of negative self-intersection in four-manifolds.

On the structure of symplectic operators and hereditary symmetries

Abstract: In the last fifteen years, there has been a remarlmble development in the exact analysis of certain nonlinear evolution equations, like tbe Korteweg-de Vries equation. I t is weH known that among the surprising features of these so-called exactly solvable equations is the possession of infinitely many symmetries and conservation laws, of N-soliton solutions and Bäcklund transformations. It has turned out that considering an operator which maps symmetries (1) onto symmetries of a given equation yields a useful approach to all these features. 'l'his operator is called a strong synunetry (2 ) ( or recursion operator (3 ) ). It is particularly useful because its transpose generates conserved covariants from given ones and because its eigenfunctions are also symmetries (which actually characterize the N-soliton solutions). One way of finding strong symmetries is to use the fact that any translation-invariant operator fP(u), possessing the property defined by v below is a strong symmetry for the hierarchy of equations u, = ( fP(u) )\" Uz, n = 0, 1, 2, ... . These operators are called hereditary symmetries. The strong symmetries of all well-known exactly solvable equations are hereditary (2 ). Recently there has also been progress in understanding the Hamiltonian structure of these evolution equations (4). An evolution equations is said to be a Hamiltonian system if it can be written in the form u 1 = O(u)f(u), where O(u) is impletic (which is, roughly speaking, the same as saying that 0-1(u) is sympletic) and where f(u) is the gradient of a suitable potential. For these systems tbe operator-valued function O(u) is of particular interest because it is a N oether operator, i.e. it maps conserved covariants onto symmetries. Our paper is related to .1\\Iagri's work wbo considered bi-Hamiltonion systems u, = = 81(u)j1(u) = 82(u)/2(u) and who sbowed that these equations have fP(u) =81(u) 8;(u) as strong symmetries.

The symplectic methods for the computation of hamiltonian equations

Abstract: The present paper gives a brief survey of results from a systematic study, undertaken by the authors and their colleagues, on the symplectic approach to the numerical computation of Hamiltonian dynamical systems in finite and infinite dimensions. Both theoretical and practical aspects of the symplectic methods are considered. Almost all the real conservative physical processes can be cast in suitable Hamiltonian formulation in phase spaces with symplectic structure, which has the advantages to make the intrinsic properties and symmetries of the underlying processes more explicit than in other mathematically equivalent formulations, so we choose the Hamiltonian formalism as the basis, together with the mathematical and physical motivations of our symplectic approach for the purpose of numerical simulation of dynamical evolutions. We give some symplectic difference schemes and related general concepts for linear and nonlinear canonical systems in finite dimensions. The analysis confirms the expectation for them to behave more satisfactorily, especially in the desirable conservation properties, than the conventional schemes. We outline a general and constructive theory of generating functions and a general method of construction of symplectic difference schemes based on all possible generating functions. This is crucial for the developments of the symplectic methods. A generalization of the above theory and method to the canonical Hamiltonian eqs. in infinite dimensions is also given. The multi-level schemes, including the leapfrog one, are studied from the symplectic point of view. We given an application of symplectic schemes, with some indications of their potential usefulness, to the computation of chaos.

Four dimensions from two in symplectic topology

Abstract: This paper introduces two-dimensional diagrams that are slight general- izations of moment map images for toric four-manifolds and catalogs techniques for reading topological and symplectic properties of a symplectic four-manifold from these diagrams. The paper offers a purely topological approach to toric manifolds as well as methods for visualizing other manifolds, including the K3 surface, and for visualizing certain surgeries. Most of the techniques extend to higher dimensions.

Classical solutions of the quantum Yang-Baxter equation

Abstract: The classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups. Whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a Poisson Lie group, we work directly with a Poisson Lie groupG and its associated symplectic groupoid. The classical analog of the quantumR-matrix is a lagrangian submanifold in the cartesian square of the symplectic groupoid. For any symplectic leafS inG, induces a symplectic automorphism σ ofS×S which satisfies the set-theoretic Yang-Baxter equation. When combined with the “flip” map exchanging components and suitably implanted in each cartesian powerS n , σ generates a symplectic action of the braid groupB n onS n . Application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants, but work on this last step is still in progress.