5,521 research outputs found
Symplectomorphism groups and isotropic skeletons
The symplectomorphism group of a 2-dimensional surface is homotopy equivalent
to the orbit of a filling system of curves. We give a generalization of this
statement to dimension 4. The filling system of curves is replaced by a
decomposition of the symplectic 4-manifold (M, omega) into a disjoint union of
an isotropic 2-complex L and a disc bundle over a symplectic surface Sigma
which is Poincare dual to a multiple of the form omega. We show that then one
can recover the homotopy type of the symplectomorphism group of M from the
orbit of the pair (L, Sigma). This allows us to compute the homotopy type of
certain spaces of Lagrangian submanifolds, for example the space of Lagrangian
RP^2 in CP^2 isotopic to the standard one.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper21.abs.htm
Complexity and integrability in 4D bi-rational maps with two invariants
In this letter we give fourth-order autonomous recurrence relations with two
invariants, whose degree growth is cubic or exponential. These examples
contradict the common belief that maps with sufficiently many invariants can
have at most quadratic growth. Cubic growth may reflect the existence of
non-elliptic fibrations of invariants, whereas we conjecture that the
exponentially growing cases lack the necessary conditions for the applicability
of the discrete Liouville theorem.Comment: 16 pages, 2 figure
Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds
Motivated by the geometric theory of differential equations and the
variational approach to the equivalence problem for geometric structures on
manifolds, we consider the problem of equivalence for distributions with fixed
submanifolds of flags on each fiber. We call them flag structures. The
construction of the canonical frames for these structures can be given in the
two prolongation steps: the first step, based on our previous works, gives the
canonical bundle of moving frames for the fixed submanifolds of flags on each
fiber and the second step consists of the prolongation of the bundle obtained
in the first step. The bundle obtained in the first step is not as a rule a
principal bundle so that the classical Tanaka prolongation procedure for
filtered structures can not be applied to it. However, under natural
assumptions on submanifolds of flags and on the ambient distribution, this
bundle satisfies a nice weaker property. The main goal of the present paper is
to formalize this property, introducing the so-called quasi-principle frame
bundles, and to generalize the Tanaka prolongation procedure to these bundles.
Applications to the equivalence problems for systems of differential equations
of mixed order, bracket generating distributions, sub-Riemannian and more
general structures on distributions are given.Comment: 49 pages. The Introduction was extended substantially: we demonstrate
how flag structures appear in the geometry of double fibrations and, using
this language, we discuss the motivating examples in more detai
Monodromy invariants in symplectic topology
This text is a set of lecture notes for a series of four talks given at
I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick
overview of symplectic topology and its main tools: symplectic manifolds,
almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants
and Floer homology. The second and third lectures focus on symplectic Lefschetz
pencils: existence (following Donaldson), monodromy, and applications to
symplectic topology, in particular the connection to Gromov-Witten invariants
of symplectic 4-manifolds (following Smith) and to Fukaya categories (following
Seidel). In the last lecture, we offer an alternative description of symplectic
4-manifolds by viewing them as branched covers of the complex projective plane;
the corresponding monodromy invariants and their potential applications are
discussed.Comment: 42 pages, notes of lectures given at IPAM, Los Angele
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