8,403 research outputs found
Open problems, questions, and challenges in finite-dimensional integrable systems
The paper surveys open problems and questions related to different aspects
of integrable systems with finitely many degrees of freedom. Many of the open
problems were suggested by the participants of the conference “Finite-dimensional
Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version
Examples of integrable and non-integrable systems on singular symplectic manifolds
We present a collection of examples borrowed from celestial mechanics and
projective dynamics. In these examples symplectic structures with singularities
arise naturally from regularization transformations, Appell's transformation or
classical changes like McGehee coordinates, which end up blowing up the
symplectic structure or lowering its rank at certain points. The resulting
geometrical structures that model these examples are no longer symplectic but
symplectic with singularities which are mainly of two types: -symplectic
and -folded symplectic structures. These examples comprise the three body
problem as non-integrable exponent and some integrable reincarnations such as
the two fixed-center problem. Given that the geometrical and dynamical
properties of -symplectic manifolds and folded symplectic manifolds are
well-understood [GMP, GMP2, GMPS, KMS, Ma, CGP, GL,GLPR, MO, S, GMW], we
envisage that this new point of view in this collection of examples can shed
some light on classical long-standing problems concerning the study of
dynamical properties of these systems seen from the Poisson viewpoint.Comment: 14 page
Causal structures and causal boundaries
We give an up-to-date perspective with a general overview of the theory of
causal properties, the derived causal structures, their classification and
applications, and the definition and construction of causal boundaries and of
causal symmetries, mostly for Lorentzian manifolds but also in more abstract
settings.Comment: Final version. To appear in Classical and Quantum Gravit
The Construction of Mirror Symmetry
The construction of mirror symmetry in the heterotic string is reviewed in
the context of Calabi-Yau and Landau-Ginzburg compactifications. This framework
has the virtue of providing a large subspace of the configuration space of the
heterotic string, probing its structure far beyond the present reaches of
solvable models. The construction proceeds in two stages: First all
singularities/catastrophes which lead to ground states of the heterotic string
are found. It is then shown that not all ground states described in this way
are independent but that certain classes of these LG/CY string vacua can be
related to other, simpler, theories via a process involving fractional
transformations of the order parameters as well as orbifolding. This
construction has far reaching consequences. Firstly it allows for a systematic
identification of mirror pairs that appear abundantly in this class of string
vacua, thereby showing that the emerging mirror symmetry is not accidental.
This is important because models with mirror flipped spectra are a priori
independent theories, described by distinct CY/LG models. It also shows that
mirror symmetry is not restricted to the space of string vacua described by
theories based on Fermat potentials (corresponding to minimal tensor models).
Furthermore it shows the need for a better set of coordinates of the
configuration space or else the structure of this space will remain obscure.
While the space of LG vacua is {\it not} completely mirror symmetric, results
described in the last part suggest that the space of Landau--Ginburg {\it
orbifolds} possesses this symmetry.Comment: 58 pages, Latex file, HD-THEP-92-1
Collisions of particles in locally AdS spacetimes I. Local description and global examples
We investigate 3-dimensional globally hyperbolic AdS manifolds containing
"particles", i.e., cone singularities along a graph . We impose
physically relevant conditions on the cone singularities, e.g. positivity of
mass (angle less than on time-like singular segments). We construct
examples of such manifolds, describe the cone singularities that can arise and
the way they can interact (the local geometry near the vertices of ).
We then adapt to this setting some notions like global hyperbolicity which are
natural for Lorentz manifolds, and construct some examples of globally
hyperbolic AdS manifolds with interacting particles.Comment: This is a rewritten version of the first part of arxiv:0905.1823.
That preprint was too long and contained two types of results, so we sliced
it in two. This is the first part. Some sections have been completely
rewritten so as to be more readable, at the cost of slightly less general
statements. Others parts have been notably improved to increase readabilit
An Invitation to Singular Symplectic Geometry
In this paper we analyze in detail a collection of motivating examples to
consider -symplectic forms and folded-type symplectic structures. In
particular, we provide models in Celestial Mechanics for every -symplectic
structure. At the end of the paper, we introduce the odd-dimensional analogue
to -symplectic manifolds: -contact manifolds.Comment: 14 pages, 1 figur
Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci
In this paper we continue our study of bifurcations of solutions of
boundary-value problems for symplectic maps arising as Hamiltonian
diffeomorphisms. These have been shown to be connected to catastrophe theory
via generating functions and ordinary and reversal phase space symmetries have
been considered. Here we present a convenient, coordinate free framework to
analyse separated Lagrangian boundary value problems which include classical
Dirichlet, Neumann and Robin boundary value problems. The framework is then
used to {prove the existence of obstructions arising from} conformal symplectic
symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary
value problems. Under non-degeneracy conditions, a group action by conformal
symplectic symmetries has the effect that the flow map cannot degenerate in a
direction which is tangential to the action. This imposes restrictions on which
singularities can occur in boundary value problems. Our results generalise
classical results about conjugate loci on Riemannian manifolds to a large class
of Hamiltonian boundary value problems with, for example, scaling symmetries
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