GAP XII — Sanya

Schedule

Abstracts

Androulidakis, Iakovos (University of Athens, Greece):
Integration of singular subalgebroids

Examples of singular subalgebroids appear in abundance in Geometric Mechanics, for instance as images of Lie algebroid morphisms (every moment map gives rise to such a morphism), also in the study of log symplectic manifolds. In this presentation we explain how the construction of the holonomy groupoid for singular foliations can be adapted to achieve the integration of any singular subalgebroid. The groupoid we obtain is longitudinally smooth, although its overall topology is very bad. Nevertheless, we are able to construct its convolution algebra. This is joint work with Marco Zambon.

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Ciccoli, Nicola (Università di Perugia, Italy):
Geometric quantization of multiplicative integrable systems

The purpose is to discuss some aspects of geometric quantization via symplectic groupoids of Poisson homogeneous spaces. Such Poisson structures does not admit, in general, multiplicative Lagrangian polarizations in the sense of Hawkins. The weaker notion of multiplicative integrable system allows to construct, nevertheless, a discrete quantization groupoid whose convolution C∗ algebra is the quantum homogeneous space. This quantization procedure is in some ways functorial, in that it allows quantization of poisson embeddings of Poisson homogeneous spaces and this aspect will be discussed.

de Leon, Manuel (Consejo Superior de Investigaciones Científicas, Spain):
Reduction of the Hamilton–Jacobi equation

Reduction theory has played a major role in the study of Hamiltonian systems. On the other hand the Hamilton–Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems. The natural question that we try to answer in this paper is how this two topics fit together and how to obtain a reduction and reconstruction procedure for the Hamilton–Jacobi equation.

Gay-Balmaz, François (École Normale Supéreure de Paris, France):
Applications of geometric mechanics: strands dynamics, constraints, and fluid-structure interaction

We will present several applications of geometric mechanics to continuum systems, by focusing on the case of geometrically exact strands. In the first part, we will describe the geometric formulation of strand dynamics, based on the affine Euler–Poincaré reduction. Contrary to the usual Kirchhoff description, the exact geometric approach allows the inclusion of nonlocal interactions and the treatment of tree-like structures encountered in molecular dynamics. The second part will include the study of elastic rods with perfect rolling contact, via the use of the Lagrange–d'Alembert principle for nonholonomic mechanics. We will also present the associated nonholonomic wave equations and discrete models. A formulation of the reduction process in terms of Dirac structures will be given. As an example of a fluid-structure interaction problem, we will finally derive a three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The equations of motion are obtained by coupling the Euler–Poincaré equations of the tube and the fluid, via a holonomic constraint. We discuss the linear stability of the system since this question has been the main focus of previous literature on the subject, namely, the well-known "garden hose instability". We will conclude the course with the study of explicit solutions of the system.

Grabowski, Janusz (Polish Academy of Sciences, Poland):
Statics — Mechanics — Field Theory

We present a geometric approach to dynamical equations of physics, based on the very effective idea of the Tulczyjew triple.

We show the evolution of the corresponding concepts, starting with the roots lying in the variational calculus for statics, through Lagrangian and Hamiltonian mechanics, and concluding with Tulczyjew triples for classical field theories, illustrated with a numer of important examples.

Our approach is covariant and complete, containing not only the Lagrangian side of the picture and Euler–Lagrange equations coming from the variational calculus, but also the description of the phase space, the phase dynamics, and the Hamiltonian formalism of the theory.

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Lin, Runliang (Tsinghua University, China):
Bilinear equations for an extended KP hierarchy

In this talk, we construct the bilinear equations for an extended Kadomtsev–Petviashvili (KP) hierarchy. By introducing an auxiliary parameter, whose flow corresponds to the so-called squared eigenfunction symmetry of KP hierarchy, we find the tau-function for this extended KP hierarchy. It is shown that the bilinear equations generate all the Hirota's bilinear equations for the zero-curvature forms of the extended KP hierarchy, which includes two types of KP equation with self-consistent sources (KPSCS). It seems that the Hirota's bilinear equations obtained in this paper for KPSCS are in a simpler form by comparing with the existing results.

This is a joint work with Xiaojun Liu (China Agricultural University) and Yunbo Zeng (Tsinghua University).

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Mackenzie, Kirill (University of Sheffield, United Kongdom):
Canonical diffeomorphisms for iterated co/tangent bundles and double Lie algebroids

Between the various combinations of the tangent and cotangent functors, there are several well-known diffeomorphisms: the canonical involution J:T2M→T2M, the map T∗(T∗M)→T(T∗M) defined by the canonical symplectic structure, a map R:T∗(T∗M)→T∗(TM) of Legendrian type which was formulated intrinsically by Ping Xu and the speaker (1994), and the map Θ:T(T∗M)→T∗(TM) which arises by dualizing J. We begin by recalling the intrinsic formulations of these maps.

More general formulations of these maps arise naturally in the infinitesimal structure of a double Lie groupoid. Applying the Lie functor either way to a double Lie groupoid yields an LA-groupoid and applying Pradines' duality to this produces a Poisson groupoid with an additional linear structure. These two Poisson groupoids are dual in the sense of Weinstein: that is, they define the same Lie bialgebroid. We show that the maps by which these results are established provide general forms of the above canonical diffeomorphisms.

Peng, Linyu (Waseda University, Japan):
Multisymplectic integrators with the aid of variational bicomplexes

Multisymplectic structure can be understood from either the Lagrangian formalism or the Hamiltonian formalism, each of which leads to numerical methods preserving multisymplecticity. For first order field theories, with the aid of the difference variational bicomplex, we provide a systematic algorithm for conservation of discrete multisymplecticity no matter which formalism we begin with.

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Ratiu, Tudor (École Polytechnique Fédérale de Lausanne, Switzerland):
Continuum mechanics and control

In these lectures I will present several results in continuum mechanics (fluids and solids) from the point of view of geometric mechanics. I will discuss the three fundamental representations of the equations of motion: material, spatial, convective. Then I will specialize to complex fluids with special emphasis on liquid crystals, where I will show how reduction theory helps solve an old problem relating two different models of nematodynamics. Time permitting, I will also show how certain mechanical problems can be cast from the point of view of control theory.

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Schlichenmaier, Martin (Université du Luxembourg, Luxembourg):
Some naturally defined star products for Kähler manifolds

We give for the Kähler manifold case an overview of the constructions of some naturally defined star products. In particular, the Berezin–Toeplitz, Berezin, Geometric Quantization, Bordemann–Waldmann, and Karbegov standard star product are introduced. With the exception of the Geometric Quantization case they are of separation of variables type. The classifying Karabegov forms and the Deligne–Fedosov classes are given. Besides the Bordemann–Waldmann star product they are all equivalent.

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TienZung, Nguyen (Université de Toulouse, France):
Singular foliations

This minicourse is about an approach to the problems of deformations, stability, and normal forms of singular foliations (of any dimension and codimension), using integrable differential forms and the so-called Nambu structures.

I'll talk about:

– Examples of (singular) foliations arising from geometry and physics.

– Integrable differential forms and Nambu structures, deformation cohomology, its computation.

– Some results and open questions about reduction, normal forms, deformations, and stability.

Wang, Hong (Nankai University, China):
Symmetric Reductions and Hamilton–Jacobi Theory of Controlled Hamiltonian Systems

The theory of controlled mechanical systems is a very important subject, following the theoretical development of geometric mechanics, a lot of important problems about this subject are being explored and studied. In this report, we introduce briefly some recent developments of regular symplectic reduction, optimal reduction, Poisson reduction and Hamilton–Jacobi theory of controlled Hamiltonian (CH) systems with symmetry, as well as their applications in the rigid spacecraft with an internal rotor and underwater vehicle with two internal rotors, which show the effect on controls in regular symplectic reduction (by stages) and Hamilton–Jacobi theory. One can show from the following references for more details.

[1] J.E. Marsden, H. Wang, and Z.X. Zhang, Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry, Diff. Geom. Appl., 33(3)(2014), 13–45, (arXiv: 1202.3564).

[2] H. Wang and Z.X. Zhang, Optimal reduction of controlled Hamiltonian system with Poisson structure and symmetry, Jour. Geom. Phys., 62(5)(2012), 953–975.

[3] T.S. Ratiu and H. Wang, Poisson reduction of controlled Hamiltonian system by controllability distribution, (arXiv: 1312.7047).

[4] H. Wang, Hamilton–Jacobi theorem for regular reducible Hamiltonian system on a cotangent bundle, (arXiv: 1303.5840).

[5] H. Wang, Hamilton–Jacobi theorems for regular controlled Hamiltonian system and its reductions, (arXiv: 1305.3457).

[6] H. Wang, Symmetric reduction and Hamilton–Jacobi equation of rigid spacecraft with a rotor, J. Geom. Symm. Phys., 32 (2013), 87–111, (arXiv: 1307.1606).

[7] H. Wang, Symmetric reduction and Hamilton–Jacobi equation of underwater vehicle with internal rotors, (arXiv: 1310.3014).

Wang, Zuoqin (University of Science and Technology of China, China):
Spectral theory of perturbed harmonic oscillators

Wu, Siye (University of Hong Kong, Hong Kong):
Hitchin's equations over a non-orientable manifold

We study Hitchin's equations and Higgs bundles over a non-orientable manifold whose oriented cover is compact Kähler. Using the involution induced by the deck transformation, we show that Hitchin's moduli space is Langrangian/complex with respect to the hyper-Kähler structure on Hitchin's moduli space associated to the oriented cover. We then establish a Donaldson–Corlette type correspondence and relate Hitchin's moduli space to representation varieties. This is a joint work with N.-K. Ho and G. Wilkin.

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Yakimov, Milen (Louisiana State University, USA):
Normal Gelfand–Tsetlin subalgebras, Poisson UFDs and cluster algebras

We will first review how Poisson manifolds arise in the theory of cluster algebras, a construction due to Gekhtman, Shapiro and Vainshtein. The main part of the talk will address the opposite question, how to construct cluster algebra structures on coordinate rings of Poisson manifolds. Many problems about cluster algebras are of this type. We will introduce a generalization of the Gelfand–Tsetlin construction of integrable systems. Instead of taking subalgebras generated by the Casimirs of chains of Poisson subalgebras, we consider the larger subalgebras generated by the Poisson normal elements of the subalgebras in such chains. The latter is a Poisson analog of the notion of normal elements in noncommutative algebras. We will introduce notions of Poisson prime elements and Poisson Unique Factorization Domains and use those to characterize the generalized Gelfand–Tsetlin subalgebras. These techniques will be then applied to construct cluster algebra structures on various interesting examples of coordinate rings of Poisson manifolds.

Yoshimura, Hiroaki (Waseda University, Japan):
Discrete Lagrangian Systems and Variational Integrators for Interconnected Systems

It is known that interconnections in physical systems can be represented by a Dirac structure. Electric circuits and nonholonomic systems are those which can be typically represented by Lagrange–Dirac dynamical systems. In this talk, we explore variational integrators for such interconnected systems. In particular, we show an discrete variational principle for interconnected systems with constraints and also to develop two different variational integrators. We illustrate the numerical validity of our theory by an example of LC-transmission line.

Zenkov, Dmitry (North Carolina State University, USA):
Hamel's Formalism, Configuration Redundancy, and Variational Integrators

Hamel's formalism is a Lagrangian analogue of Hamiltonian mechanics represented in non-canonical coordinates. This talk will elucidate the variational nature of Hamel's formalism and its application to the dynamics of constrained systems and structure-preserving integrators.

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PDF file of the abstracts