4,428 research outputs found
Symplectic maps to projective spaces and symplectic invariants
After reviewing recent results on symplectic Lefschetz pencils and symplectic
branched covers of CP^2, we describe a new construction of maps from symplectic
manifolds of any dimension to CP^2 and the associated monodromy invariants. We
also show that a dimensional induction process makes it possible to describe
any compact symplectic manifold by a series of words in braid groups and a word
in a symmetric group.Comment: 39 pages; to appear in Proc. 7th Gokova Geometry-Topology Conferenc
3-manifolds efficiently bound 4-manifolds
It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for
instance, every 3-manifold has a surgery diagram. There are several proofs of
this fact, including constructive proofs, but there has been little attention
to the complexity of the 4-manifold produced. Given a 3-manifold M of
complexity n, we show how to construct a 4-manifold bounded by M of complexity
O(n^2). Here we measure ``complexity'' of a piecewise-linear manifold by the
minimum number of n-simplices in a triangulation. It is an open question
whether this quadratic bound can be replaced by a linear bound.
The proof goes through the notion of "shadow complexity" of a 3-manifold M. A
shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M.
We prove that, for a manifold M satisfying the Geometrization Conjecture with
Gromov norm G and shadow complexity S, c_1 G <= S <= c_2 G^2 for suitable
constants c_1, c_2. In particular, the manifolds with shadow complexity 0 are
the graph manifolds.Comment: 39 pages, 21 figures; added proof for spin case as wel
Self-boundedness and self-hiddenness for implicit two-dimensional systems
In this paper we introduce and develop the concepts of self-boundedness and self-hiddenness for implicit two-dimensional systems. The aim of this note is to show that when extending such concepts to a multidimensional setting, a richer structure arises than in the one-dimensional case
Quasi-classical approximation in vortex filament dynamics. Integrable systems, gradient catastrophe and flutter
Quasiclassical approximation in the intrinsic description of the vortex
filament dynamics is discussed. Within this approximation the governing
equations are given by elliptic system of quasi-linear PDEs of the first order.
Dispersionless Da Rios system and dispersionless Hirota equation are among
them. They describe motion of vortex filament with slow varying curvature and
torsion without or with axial flow. Gradient catastrophe for governing
equations is studied. It is shown that geometrically this catastrophe manifests
as a fast oscillation of a filament curve around the rectifying plane which
resembles the flutter of airfoils. Analytically it is the elliptic umbilic
singularity in the terminology of the catastrophe theory. It is demonstrated
that its double scaling regularization is governed by the Painleve' I equation.Comment: 25 pages, 5 figures, minor typos correcte
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