Showing papers on "Symplectic vector space published in 2010"

Symplectic Geometric Algorithms for Hamiltonian Systems

Abstract: Preliminaries of Differential Manifolds.- Symplectic Algebra and Geometry Preliminaries.- Hamiltonian Mechanics and Symplectic Geometry.- Symplectic Difference Schemes for Hamiltonian Systems.- The Generating Function Method.- The Calculus of Generating Function and Formal Energy.- Symplectic Runge-Kutta Methods.- Composition Scheme.- Formal Power Series and B-Series.- Volume-Preserving Methods for Source-Free Systems.- Free Systems.- Contact Algorithms for Contact Dynamic Systems.- Poisson Bracket and Lie-Poisson Schemes.- KAM Theorem of Symplectic Algorithms.- Lee-Variational Integrator.- Structure Preserving Schemes for Birkhoff Systems.- Multisymplectic and Variational Integrators.

Almost toric symplectic four-manifolds

Abstract: Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that is a natural generalization of toric manifolds. Notable examples include the K3 surface, the phase space of the spherical pendulum and rational balls useful for symplectic surgeries. The main result of the paper is a complete classification up to diffeomorphism of closed almost toric four-manifolds. A key step in the proof is a geometric classification of the singular affine structures that can occur on the base of a closed almost toric four-manifold.

Notes on monotone Lagrangian twist tori

Abstract: We construct monotone Lagrangian tori in the standard symplectic vector space, in the complex projective space and in products of spheres. We explain how to classify these Lagrangian tori up to symplectomorphism and Hamiltonian isotopy, and how to show that they are not displaceable by Hamiltonian isotopies.

New closed Newton–Cotes type formulae as multilayer symplectic integrators

Abstract: In this paper, we introduce new integrators of Newton–Cotes type and investigate the connection between these new methods, differential methods, and symplectic integrators. From the literature, we can see that several one step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. In this paper, we introduce a new numerical method of closed Newton–Cotes type and we write it as a symplectic multilayer structure. We apply the symplectic schemes in order to solve Hamilton’s equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as integration proceeds.

Altering symplectic manifolds by homologous recombination

Abstract: We use symplectic cohomology to study the non-uniqueness of symplectic structures on the smooth manifolds underlying affine varieties. Starting with a Lefschetz fibration on such a variety and a finite set of primes, the main new tool is a method, which we call homologous recombination, for constructing a Lefschetz fibration whose total space is smoothly equivalent to the original variety, but for which symplectic cohomology with coefficients in the given set of primes vanishes (there is also a simpler version that kills symplectic cohomology completely). Rather than relying on a geometric analysis of periodic orbits of a flow, the computation of symplectic cohomology depends on describing the Fukaya category associated to the new fibration. As a consequence of this and a result of McLean we prove, for example, that an affine variety of real dimension greater than 4 supports infinitely many different (Wein)stein structures of finite type, and, assuming a mild cohomological condition, uncountably many different ones of infinite type. In addition, we introduce a notion of complexity which measures the number of handle attachments required to construct a given Weinstein manifold, and prove that, in dimensions greater than or equal to 12, one may ensure that the infinitely many different Weinstein manifolds smoothly equivalent to a given algebraic variety have bounded complexity.

Semiclassical Limits of Quantized Coordinate Rings

Abstract: This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety V consists of the classical coordinate ring O(V) equipped with an associated Poisson structure. Conjectured relationships between primitive ideals of a generic quantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit Poisson structure on O(V)) are discussed, as are breakdowns in the connections when the symplectic leaves are not algebraic. This prompts replacement of the differential-geometric concept of symplectic leaves with the algebraic concept of symplectic cores, and a reformulated conjecture is proposed: The primitive spectrum of A should be homeomorphic to the space of symplectic cores in V, and to the Poisson-primitive spectrum of O(V). Various examples, including both quantized coordinate rings and enveloping algebras of solvable Lie algebras, are analyzed to support the choice of symplectic cores to replace symplectic leaves.

Low-Dimensional Complex Characters of the Symplectic and Orthogonal Groups

Abstract: We classify the irreducible complex characters of the symplectic groups Sp 2n (q) and the orthogonal groups , Spin 2n+1(q) of degrees up to the bound D, where D = (q n − 1)q 4n−10/2 for symplectic groups, D = q 4n−8 for orthogonal groups in odd dimension, and D = q 4n−10 for orthogonal groups in even dimension.

Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Abstract: Given a Lagrangian sphere in a symplectic 4-manifold $(M, \omega)$ with $b^+=1$, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension $\kappa$ of $(M, \omega)$ is $-\infty$, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is not characteristic and ternary, we show that homologous Lagrangian spheres are smoothly isotopic, and when the Euler number is less than 8, we generalize Hind and Evans' Hamiltonian uniqueness in the monotone case. On the existence side, when $\kappa=-\infty$, we give a characterization of classes represented by Lagrangian spheres, which enables us to describe the non-Torelli part of the symplectic mapping class group.

Symplectic involutions of holomorphic symplectic fourfolds

Abstract: Let X be a holomorphic symplectic fourfold such that b_2=23 and i a symplectic involution of X . The fixed locus F of i is a smooth symplectic submanifold of X; we show that F contains at least 12 isolated points and 1 smooth surface. We conjecture that F is made of 28 isolated fixed points and 1 K3 surface and we provide evidences for the conjecture in some examples, as the Hilbert scheme of a K3 surface, the Fano variety of a cubic in P^5 and the double cover of an EPW sextic.

Some remarks on the size of tubular neighborhoods in contact topology and fillability

Abstract: The well-known tubular neighborhood theorem for contact submanifolds states that a small enough neighborhood of such a submanifold N is uniquely determined by the contact structure on N , and the conformal symplectic structure of the normal bundle. In particular, if the submanifold N has trivial normal bundle then its tubular neighborhood will be contactomorphic to a neighborhood of Nf 0g in the model space N R 2k . In this article we make the observation that if .N; N/ is a 3‐dimensional overtwisted submanifold with trivial normal bundle in .M;/ , and if its model neighborhood is sufficiently large, then .M;/ does not admit a symplectically aspherical filling. 57R17; 53D35 In symplectic geometry, many invariants are known that measure in some way the “size” of a symplectic manifold. The most obvious one is the total volume, but this is usually discarded, because one can change the volume (in case it is finite) by rescaling the symplectic form without changing any other fundamental property of the manifold. The first non-trivial example of an invariant based on size is the symplectic capacity (see Gromov [15]). It relies on the fact that the size of a symplectic ball that can be embedded into a symplectic manifold does not only depend on its total volume but also on the volume of its intersection with the symplectic 2‐planes. Contact geometry does not give a direct generalization of these invariants. The main difficulties stem from the fact that one is only interested in the contact structure, and not in the contact form, so that the total volume is not defined, and to make matters worse the whole Euclidean space R 2nC1 with the standard structure can be compressed by a contactomorphism into an arbitrarily small open ball in R 2nC1 . A more successful approach consists in studying the size of the neighborhood of submanifolds. This can be considered to be a generalization of the initial idea since contact balls are just neighborhoods of points. In the literature this idea has been pursued by looking at the tubular neighborhoods of transverse circles. Let .N; N/ be a closed contact manifold. The product N R 2k carries a contact structure given as

Quantum response of dephasing open systems

Abstract: We develop a theory of adiabatic response for open systems governed by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for residual dissipation even when the ground state is protected by a spectral gap. We give quantum response a geometric interpretation in terms of Hilbert space projections: For a two level system and, more generally, for systems with suitable functional form of the dephasing, the dissipative and non-dissipative parts of the response are linked to a metric and to a symplectic form. The metric is the Fubini-Study metric and the symplectic form is the adiabatic curvature. When the metric and symplectic structures are compatible the non-dissipative part of the inverse matrix of response coefficients turns out to be immune to dephasing. We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: The qubit, coherent states and a model of the integer quantum Hall effect.

Quaternionic Grassmannians and Pontryagin classes in algebraic geometry

Abstract: The quaternionic Grassmannian HGr(r,n) is the affine open subscheme of the ordinary Grassmannian parametrizing those 2r-dimensional subspaces of a 2n-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular there is HP^{n} = HGr(1,n+1). For a symplectically oriented cohomology theory A, including oriented theories but also hermitian K-theory, Witt groups and symplectic and special linear algebraic cobordism, we have A(HP^{n}) = A(pt)[p]/(p^{n+1}). We define Pontryagin classes for symplectic bundles. They satisfy a splitting principle and the Cartan sum formula, and we use them to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 symplectic bundles determine Thom and Pontryagin classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Pontryagin classes.

Symplectic quasi-states on the quadric surface and Lagrangian submanifolds

Abstract: The quantum homology of the monotone complex quadric surface splits into the sum of two fields. We outline a proof of the following statement: The unities of these fields give rise to distinct symplectic quasi-states defined by asymptotic spectral invariants. In fact, these quasi-states turn out to be "supported" on disjoint Lagrangian submanifolds. Our method involves a spectral sequence which starts at homology of the loop space of the 2-sphere and whose higher differentials are computed via symplectic field theory, in particular with the help of the Bourgeois-Oancea exact sequence.

Quaternionic Grassmannians and Borel classes in algebraic geometry

Abstract: The quaternionic Grassmannian HGr(r,n) is the affine open subscheme of the ordinary Grassmannian parametrizing those 2r-dimensional subspaces of a 2n-dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular there is HP^{n} = HGr(1,n+1). For a symplectically oriented cohomology theory A, including oriented theories but also hermitian K-theory, Witt groups and symplectic and special linear algebraic cobordism, we have A(HP^{n}) = A(pt)[p]/(p^{n+1}). We define Borel classes for symplectic bundles. They satisfy a splitting principle and the Cartan sum formula, and we use them to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes.

Orthogonal and symplectic matrix models: universality and other properties

Abstract: We study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measure. For these models we find the bounds (similar to the case of hermitian matrix models) for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order O(1). We prove also universality of local eigenvalue statistics in the bulk.

Fold-forms for four-folds

Abstract: This paper explains an application of Gromov’s h-principle to prove the existence, on any orientable four-manifold, of a folded symplectic form. That is a closed two-form which is symplectic except on a separating hypersurface where the form singularities are like the pullback of a symplectic form by a folding map. We use the h-principle for folding maps (a theorem of Eliashberg) and the h-principle for symplectic forms on open manifolds (a theorem of Gromov) to show that, for orientable even-dimensional manifolds, the existence of a stable almost complex structure is necessary and sufficient to warrant the existence of a folded symplectic form.

Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres

Abstract: Using Lefschetz fibrations, we construct nonstandard symplectic structures on cotangent bundles of spheres. These structures are of Liouville type, which means exact and convex at infinity.

Zero product determined classical Lie algebras

Abstract: We show that the Lie algebra ℒ of skew-symmetric matrices with respect to either transpose or symplectic involution is zero product determined. This means that every bilinear map {·,·} from ℒ × ℒ into a vector space X is of the form {x, y} = T ([x, y]) for some linear map T provided that the following condition is fulfilled: [x, y] = 0 implies {x, y} = 0.

A Brunn–Minkowski Inequality for Symplectic Capacities of Convex Domains

Abstract: In this work, we prove a Brunn-Minkowski-type inequality in the context of symplectic geometry and discuss some of its applications.

Critical Landscape Topology for Optimization on the Symplectic Group

Abstract: Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over a noncompact symplectic Lie group Sp(2N,ℝ), i.e., minimizing the Frobenius distance from a target symplectic transformation, which can be used to assess the fidelity function over dynamical transformations in classical mechanics and quantum optics. The topology of the set of critical points is proven to have a unique local minimum and a number of saddlepoint submanifolds, exhibiting the absence of local suboptima that may hinder the search for ultimate optimal solutions. Compared with those of previously studied problems on compact Lie groups, such as the orthogonal and unitary groups, the topology is more complicated due to the significant nonlinearity brought by the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group.

The relative symplectic cone and T2-fibrations

Abstract: In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author.

Symplectic Actions of 2-Tori on 4-Manifolds

Abstract: In this paper the author classifies symplectic actions of $2$-tori on compact connected symplectic $4$-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants of the topology of the manifold, of the torus action and of the symplectic form. The author constructs explicit models of such symplectic manifolds with torus actions, defined in terms of these invariants.

The topology of toric symplectic manifolds

Abstract: This is a collection of results on the topology of toric symplectic manifolds. Using an idea of Borisov, we show that a closed symplectic manifold supports at most a finite number of toric structures. Further, the product of two projective spaces of complex dimension at least two (and with a standard product symplectic form) has a unique toric structure. We then discuss various constructions, using wedging to build a monotone toric symplectic manifold whose center is not the unique point displaceable by probes, and bundles and blow ups to form manifolds with more than one toric structure. The bundle construction uses the McDuff--Tolman concept of mass linear function. Using Timorin's description of the cohomology ring via the volume function we develop a cohomological criterion for a function to be mass linear, and explain its relation to Shelukhin's higher codimension barycenters.

On homotopy Poisson actions and reduction of symplectic Q-manifolds

Abstract: We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.

Ruled 4-manifolds and isotopies of symplectic surfaces

Abstract: We study symplectic surfaces in ruled symplectic 4-manifolds which are disjoint from a given symplectic section. As a consequence, in any symplectic 4-manifold, two homologous symplectic surfaces which are C0 close must be Hamiltonian isotopic.

Symplectic reduction of quasi-morphisms and quasi-states

Abstract: We prove that quasi-morphisms and quasi-states on a closed integral symplectic manifold descend under symplectic reduction to symplectic hyperplane sections. Along the way we show that quasi-morphisms that arise from spectral invariants are the Calabi homomorphism when restricted to Hamiltonians supported on stably displaceable sets.