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We discuss some families of integrable and superintegrable systems in -dimensional Euclidean space which are invariant under rotations. The invariant Hamiltonian is integrable with integrals of motion and an additional integral of motion , which are first- and fourth-order polynomials in momenta, respectively. [ABSTRACT FROM AUTHOR]

We consider the task of classifying relative equilibria for mechanical systems with rotational symmetry. We divide relative equilibria into two natural groups: a generic class which we call normal, and a non-generic abnormal class. The eigenvalues of the locked inertia tensor descend to shape-space and endow it with the geometric structure of a three-web with the property that any normal relative equilibrium occurs as a critical point of the potential restricted to a leaf from the web. To demonstrate the utility of this web structure we show how the spherical three-body problem gives rise to a web of Cayley cubics on the three-sphere, and use this to fully classify the relative equilibria for the case of equal masses. [ABSTRACT FROM AUTHOR]

We compute univariate and multigraded Hilbert series of invariants and covariants of representations of the circle and orthogonal group O 2 (R) . The multigradings considered include the maximal grading associated to the decomposition of the representation into irreducibles as well as the bigrading associated to a cotangent-lifted representation, or equivalently, the bigrading associated to the holomorphic and antiholomorphic parts of the real invariants and covariants. This bigrading induces a bigrading on the algebra of on-shell invariants of the symplectic quotient, and the corresponding Hilbert series are computed as well. We also compute the first few Laurent coefficients of the univariate Hilbert series, give sample calculations of the multigraded Laurent coefficients, and give an example to illustrate the extension of these techniques to the semidirect product of the circle by other finite groups. We describe an algorithm to compute each of the associated Hilbert series. Communicated by Ellen Kirkman [ABSTRACT FROM AUTHOR]

We provide an analytical construction of the gluing map for stable affine vortices over the upper half plane with the Lagrangian boundary condition. This result is a necessary ingredient in studies of the relation between gauged sigma model and nonlinear sigma model, such as the closed or open quantum Kirwan map. [ABSTRACT FROM AUTHOR]

We show that contact reductions can be described in terms of symplectic reductions in the traditional Marsden–Weinstein–Meyer as well as the constant rank picture. The point is that we view contact structures as particular (homogeneous) symplectic structures. A group action by contactomorphisms is lifted to a Hamiltonian action on the corresponding symplectic manifold, called the symplectic cover of the contact manifold. In contrast to the majority of the literature in the subject, our approach includes general contact structures (not only co-oriented) and changes the traditional view point: contact Hamiltonians and contact moment maps for contactomorphism groups are no longer defined on the contact manifold itself, but on its symplectic cover. Actually, the developed framework for reductions is slightly more general than purely contact, and includes a precontact and presymplectic setting which is based on the observation that there is a one-to-one correspondence between isomorphism classes of precontact manifolds and certain homogeneous presymplectic manifolds. [ABSTRACT FROM AUTHOR]

Suppose given a Hamiltonian and holomorphic action of G = U (2) on a compact Kähler manifold M, with nowhere vanishing moment map. Given an integral coadjoint orbit O for G, under transversality assumptions we shall consider two naturally associated 'conic' reductions. One, which will be denoted M ¯ O G , is taken with respect to the action of G and the cone over O ; another, which will be denoted M ¯ ν T , is taken with respect to the action of the standard maximal torus T ⩽ G and the ray R + ı ν along which the cone over O intersects the positive Weyl chamber. These two reductions share a common 'divisor', which may be viewed heuristically as bridging between their structures. This point of view motivates studying the (rather different) ways in which the two reductions relate to the the latter divisor. In this paper we provide some indications in this direction. Furthermore, we give explicit transversality criteria for a large class of such actions in the projective setting, as well as a description of corresponding reductions as weighted projective varieties, depending on combinatorial data associated to the action and the orbit. [ABSTRACT FROM AUTHOR]

We compute the Hilbert series of the graded algebra of real regular functions on a linear symplectic quotient by the 2-torus as well as the first four coefficients of the Laurent expansion of this Hilbert series at t = 1 . We describe an algorithm to compute the Hilbert series as well as the Laurent coefficients in explicit examples. [ABSTRACT FROM AUTHOR]

Consider the dynamics of two point masses on a surface of constant curvature subject to an attractive force analogue of Newton's inverse square law, that is under a 'cotangent' potential. When the distance between the bodies is sufficiently small, the reduced equations of motion may be seen as a perturbation of an integrable system. We take suitable action-angle coordinates to average these perturbing terms and describe dynamical effects of the curvature on the motion of the two-bodies. [ABSTRACT FROM AUTHOR]

We investigate the complex geometry of total spaces of reductive principal bundles over compact base spaces and establish a close relation between the Kähler property of the base, momentum maps for the action of a maximal compact subgroup on the total space, and the Kähler property of special equivariant compactifications. We provide many examples illustrating that the main result is optimal. [ABSTRACT FROM AUTHOR]

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$ -space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$ , where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$ -spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new. [ABSTRACT FROM AUTHOR]

We study multisymplectic structures taking values in vector bundles with connections from the viewpoint of the Hamiltonian symmetry. We introduce the notion of bundle-valued n -plectic structures and exhibit some properties of them. In addition, we define bundle-valued homotopy momentum sections for bundle-valued n -plectic manifolds with Lie algebroids to discuss momentum map theories in both cases of quaternionic Kähler manifolds and hyper-Kähler manifolds. Furthermore, we generalize the Marsden-Weinstein-Meyer reduction theorem for symplectic manifolds and construct two kinds of reductions of vector-valued 1-plectic manifolds. [ABSTRACT FROM AUTHOR]

Insect flight research is propelled by their unmatched flight capabilities. However, complex underlying aerodynamic phenomena make computational modeling of insect-type flapping flight a challenging task, limiting our ability in understanding insect flight and producing aerial vehicles exploiting same aerodynamic phenomena. To this end, novel mid-fidelity approach to modeling insect-type flapping vehicles is proposed. The approach is computationally efficient enough to be used within optimal design and optimal control loops, while not requiring experimental data for fitting model parameters, as opposed to widely used quasi-steady aerodynamic models. The proposed algorithm is based on Helmholtz–Hodge decomposition of fluid velocity into curl-free and divergence-free parts. Curl-free flow is used to accurately model added inertia effects (in almost exact manner), while expressing system dynamics by using wing variables only, after employing symplectic reduction of the coupled wing-fluid system at zero level of vorticity (thus reducing out fluid variables in the process). To this end, all terms in the coupled body-fluid system equations of motion are taken into account, including often neglected terms related to the changing nature of the added inertia matrix (opposed to the constant nature of rigid body mass and inertia matrix). On the other hand—in order to model flapping wing system vorticity effects—divergence-free part of the flow is modeled by a wake of point vortices shed from both leading (characteristic for insect flight) and trailing wing edges. The approach is evaluated for a numerical case involving fruit fly hovering, while quasi-steady aerodynamic model is used as benchmark tool with experimentally validated parameters for the selected test case. The results indicate that the proposed approach is capable of mid-fidelity accurate calculation of aerodynamic loads on the insect-type flapping wings. [ABSTRACT FROM AUTHOR]

Let K be a compact group. For a symplectic quotient Mλ of a compact Hamiltonian Kähler K-manifold, we show that the induced complex structure on Mλ is locally invariant when the parameter λ varies in Lie(K)*. To prove such a result, we take two different approaches: (i) use the complex geometry properties of the symplectic implosion construction; (ii) investigate the variation of geometric invariant theory (GIT) quotients. [ABSTRACT FROM AUTHOR]

We show that spheroidal wave functions viewed as the essential part of the joint eigenfunctions of two commuting operators on L2(S2) have a defect in the joint spectrum that makes a global labeling of the joint eigenfunctions by quantum numbers impossible. To our knowledge, this is the first explicit demonstration that quantum monodromy exists in a class of classically known special functions. Using an analog of the Laplace–Runge–Lenz vector we show that the corresponding classical Liouville integrable system is symplectically equivalent to the C. Neumann system. To prove the existence of this defect, we construct a classical integrable system that is the semiclassical limit of the quantum integrable system of commuting operators. We show that this is a generalized semitoric system with a nondegenerate focus–focus point, such that there is monodromy in the classical and the quantum systems. [ABSTRACT FROM AUTHOR]

In this paper we will extend to non-abelian groups inverse spectral results, proved by us in an earlier paper (Guillemin and Wang, 2016), for compact abelian groups, i.e. tori. More precisely, Let G be a compact Lie group acting isometrically on a compact Riemannian manifold X. We will show that for the Schrödinger operator − ħ 2 Δ + V with V ∈ C ∞ (X) G , the potential function V is, in some interesting examples, determined by the G -equivariant spectrum. The key ingredient in this proof is a generalized Legendrian relation between the Lagrangian manifolds Graph (d V) and Graph (d F) , where F is a spectral invariant defined on an open subset of the positive Weyl chamber. [ABSTRACT FROM AUTHOR]

We identify a family of torus representations such that the corresponding singular symplectic quotients at the 0-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may have graded isomorphic algebras of real regular functions and graded Poisson isomorphic complex symplectic quotients yet not be graded regularly diffeomorphic nor graded regularly symplectomorphic. [ABSTRACT FROM AUTHOR]

We develop a quantization method, that we name decomposable Weyl quantization, which ensures that the constants of motion of a prescribed finite set of Hamiltonians are preserved by the quantization. Our method is based on a structural analogy between the notions of reduction of the classical phase space and diagonalization of selfadjoint operators. We obtain the spectral decomposition of the emerging quantum constants of motion directly from the quantization process. If a specific quantization is given, we expect that it preserves constants of motion exactly when it coincides with decomposable Weyl quantization on the algebra of constants of motion. We obtain a characterization of when such property holds in terms of the Wigner transforms involved. We also explain how our construction can be applied to spectral theory. Moreover, we discuss how our method opens up new perspectives in formal deformation quantization and geometric quantization. [ABSTRACT FROM AUTHOR]

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an SU 2 -module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at t = 1. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most 1 0. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary SL 2 - or SU 2 -module as well as its first three Laurent coefficients. [ABSTRACT FROM AUTHOR]

We provide a short and self-contained proof of an equality of Maslov indices in a linear symplectic reduction and apply it to obtain an equality of Maslov focal indices after reducing by symmetries an electromagnetic Lagrangian system. [ABSTRACT FROM AUTHOR]

Let $Q$ be a closed manifold admitting a locally free action of a compact Lie group $G$. In this paper, we study the properties of geodesic flows on $Q$ given by suitable G-invariant Riemannian metrics. In particular, we will be interested in the existence of geodesics that are closed up to the action of some element in the group $G$ , since they project to closed magnetic geodesics on the quotient orbifold $Q/G$. [ABSTRACT FROM AUTHOR]

This is the second of two companion papers. We describe a generalization of the point vortex system on surfaces to a Hamiltonian dynamical system consisting of two or three points on complex projective space CP2 interacting via a Hamiltonian function depending only on the distance between the points. The system has symmetry group SU(3). The first paper describes all possible momentum values for such systems, and here we apply methods of symplectic reduction and geometric mechanics to analyze the possible relative equilibria of such interacting generalized vortices.The different types of polytope depend on the values of the 'vortex strengths', which are manifested as coefficients of the symplectic forms on the copies of CP2. We show that the reduced space for this Hamiltonian action for 3 vortices is generically a 2-sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction. The other non-trivial reduced spaces are topological spheres with isolated singular points. For 2 generalized vortices, the reduced spaces are just points, and the motion is governed by a collective Hamiltonian, whereas for 3 the reduced spaces are of dimension at most 2. In both cases the system will be completely integrable in the non-abelian sense. [ABSTRACT FROM AUTHOR]

Recently Ohsawa (Lett Math Phys 105:1301–1320, 2015) has studied the Marsden–Weinstein–Meyer quotient of the manifold T ∗ R n × T ∗ R 2 n 2 under a O (2 n) -symmetry and has used this quotient to describe the relationship between two different parametrisations of Gaussian wave packet dynamics commonly used in semiclassical mechanics. In this paper, we suggest a new interpretation of (a subset of) the unreduced space as being the frame bundle F (T ∗ R n) of T ∗ R n . We outline some advantages of this interpretation and explain how it can be extended to more general symplectic manifolds using the notion of the diagonal lift of a symplectic form due to Cordero and de León (Rend Circ Mat Palermo 32:236–271, 1983). [ABSTRACT FROM AUTHOR]

This is the second of two companion papers. We describe a generalization of the point vortex system on surfaces to a Hamiltonian dynamical system consisting of two or three points on complex projective space CP2 interacting via a Hamiltonian function depending only on the distance between the points. The system has symmetry group SU(3). The first paper describes all possible momentum values for such systems, and here we apply methods of symplectic reduction and geometric mechanics to analyze the possible relative equilibria of such interacting generalized vortices.The different types of polytope depend on the values of the 'vortex strengths', which are manifested as coefficients of the symplectic forms on the copies of CP2. We show that the reduced space for this Hamiltonian action for 3 vortices is generically a 2-sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction. The other non-trivial reduced spaces are topological spheres with isolated singular points. For 2 generalized vortices, the reduced spaces are just points, and the motion is governed by a collective Hamiltonian, whereas for 3 the reduced spaces are of dimension at most 2. In both cases the system will be completely integrable in the non-abelian sense. [ABSTRACT FROM AUTHOR]

We discuss, in all generality, the reduction of a Hamilton — Jacobi theory for systems subject to nonholonomic constraints and invariant under the action of a group of symmetries. We consider nonholonomic systems subject to both linear and nonlinear constraints and with different positioning of such constraints with respect to the symmetries. [ABSTRACT FROM AUTHOR]

Abstract We prove a version of the affine Kempf–Ness theorem for non-algebraic symplectic structures and shifted moment maps, and use it to describe hyperkähler quotients of T ∗ G , where G is a complex reductive group. [ABSTRACT FROM AUTHOR]

Let ( G , V ) be a representation with either G a torus or ( G , V ) a locally free stable θ -representation. We study the fiber at 0 of the associated moment map, which is a commuting variety in the latter case. We characterize the cases where this fiber is normal. The quotient ( i.e. the symplectic reduction) turns out to be a specific orbifold when the representation is polar. In the torus case, this confirms a conjecture stated by C. Lehn, M. Lehn, R. Terpereau and the author in a former article. In the θ -case, the conjecture was already known but this approach yield another proof. [ABSTRACT FROM AUTHOR]

We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space we were able to apply powerful methods of differential, symplectic and algebraic geometry to attack the problem of equivalence of states with respect to the strength of correlations, or, in other words, to classify them from this point of view. Such classifications are interpreted as an identification of states with 'the same correlations properties', i.e. ones that can be used for the same information purposes, or, from yet another point of view, states that can be mutually transformed one to another by specific, experimentally accessible operations. It is clear that the latter characterization answers the fundamental question 'what can be transformed into what via available means?'. Exactly such an interpretation, i.e. in terms of mutual transformability, can be clearly formulated in terms of actions of specific groups on the space of states and is the starting point for the proposed methods. [ABSTRACT FROM AUTHOR]

We consider the space M of Euclidean similarity classes of framed loops in R3. Framed loop space is shown to be an infinite-dimensional Kähler manifold by identifying it with a complex Grassmannian. We show that the space of isometrically immersed loops studied by Millson and Zombro is realized as the symplectic reduction of M by the action of the based loop group of the circle, giving a smooth version of a result of Hausmann and Knutson on polygon space. The identification with a Grassmannian allows us to describe the geodesics of M explicitly. Using this description, we show that M and its quotient by the reparameterization group are nonnegatively curved. We also show that the planar loop space studied by Younes, Michor, Shah and Mumford in the context of computer vision embeds in M as a totally geodesic, Lagrangian submanifold. The action of the reparameterization group on M is shown to be Hamiltonian, and this is used to characterize the critical points of the weighted total twist functional. [ABSTRACT FROM AUTHOR]

We calculate the Hörmander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle. [ABSTRACT FROM AUTHOR]

In this note we discuss (weak) dual pairs in Dirac geometry. We show that this notion appears naturally when studying the problem of pushing forward a Dirac structure along a surjective submersion, and we prove a Dirac-theoretic version of Libermann’s theorem from Poisson geometry. Our main result is an explicit construction of self-dual pairs for Dirac structures. This theorem not only recovers the global construction of symplectic realizations from Crainic and Mărcuţ (J Symplectic Geom 9(4):435-444, 2011), but allows for a more conceptual understanding of it, yielding a simpler and more natural proof. As an application of the main theorem, we present a different approach to the recent normal form theorem around Dirac transversals from Bursztyn et al. (J für die reine und angewandte Mathematik (Crelles J), doi:10.1515/crelle-2017-0014, 2017). [ABSTRACT FROM AUTHOR]

The symplectic vortex equations admit a variational description as global minimum of the Yang-Mills-Higgs functional. We study its negative gradient flow on holomorphic pairs where is a connection on a principal -bundle over a closed Riemann surface and is an equivariant map into a Kähler Hamiltonian -manifold. The connection induces a holomorphic structure on the Kähler fibration and we require that descends to a holomorphic section of this fibration. We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the -topology when is equivariantly convex at infinity with proper moment map, is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang-Mills-Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet's Kobayashi-Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment-weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem. [ABSTRACT FROM AUTHOR]

We analyse a monotone lagrangian in CP2 that is hamiltonian isotopic to the standard lagrangian RP2, yet exhibits a distinguishing behaviour under reduction by one of the toric circle actions, namely it intersects transversally the reduction level set and it projects one-to-one onto a great circle in CP1. This lagrangian thus provides an example of embedded composition fitting work of Wehrheim–Woodward and Weinstein. [ABSTRACT FROM AUTHOR]

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the Kähler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting and elucidates the key role played by the following fact: the moment image of et . x increases as t ∊ ℝ increases. [ABSTRACT FROM AUTHOR]

This is a survey article for the mathematical theory of Witten's Gauged Linear Sigma Model, as developed recently by the authors. Instead of developing the theory in the most general setting, in this paper the authors focus on the description of the moduli. [ABSTRACT FROM AUTHOR]

We study the spectral and ergodic properties of Schrödinger operators on a compact connected Riemannian manifold M without boundary in case that the examined system possesses certain symmetries. More precisely, if M carries an isometric and effective action of a compact connected Lie group G , we prove a generalized equivariant version of the semiclassical Weyl law with an estimate for the remainder, using a semiclassical functional calculus for h -dependent functions and relying on recent results on singular equivariant asymptotics. We then deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry-reduced Hamiltonian flow on the principal stratum of the singular symplectic reduction of M is ergodic. In particular, we obtain an equivariant version of the Shnirelman–Zelditch–Colin-de-Verdière theorem, as well as a representation theoretic equidistribution theorem. If M / G is an orbifold, similar results were recently obtained by Kordyukov. When G is trivial, one recovers the classical results. [ABSTRACT FROM AUTHOR]

The multisymplectic formalism of field theories developed over the last fifty years is extended to deal with manifolds that have boundaries. In particular, a multisymplectic framework for first-order covariant Hamiltonian field theories on manifolds with boundaries is developed. This work is a geometric fulfillment of Fock's formulation of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin. This framework leads to a geometric understanding of conventional choices for boundary conditions and relates them to the moment map of the gauge group of the theory. It is also shown that the natural interpretation of the Euler-Lagrange equations as an evolution system near the boundary leads to a presymplectic Hamiltonian system in an extended phase space containing the natural configuration and momenta fields at the boundary together with extra degrees of freedom corresponding to the transversal components at the boundary of the momenta fields of the theory. The consistency conditions for evolution at the boundary are analyzed and the reduced phase space of the system is shown to be a symplectic manifold with a distinguished isotropic submanifold corresponding to the boundary data of the solutions of Euler-Lagrange equations. This setting makes it possible to define well-posed boundary conditions, and provides the adequate setting for the canonical quantization of the system. The notions of the theory are tested against three significant examples: scalar fields, Poisson σ-model and Yang-Mills theories. [ABSTRACT FROM AUTHOR]