62 research outputs found
Relative Ruan and Gromov-Taubes Invariants of Symplectic 4-Manifolds
We define relative Ruan invariants that count embedded connected symplectic
submanifolds which contact a fixed stable symplectic hypersurface V in a
symplectic 4-manifold (X,w) at prescribed points with prescribed contact orders
(in addition to insertions on X\V) for stable V. We obtain invariants of the
deformation class of (X,V,w). Two large issues must be tackled to define such
invariants: (1) Curves lying in the hypersurface V and (2) genericity results
for almost complex structures constrained to make V pseudo-holomorphic (or
almost complex). Moreover, these invariants are refined to take into account
rim tori decompositions. In the latter part of the paper, we extend the
definition to disconnected submanifolds and construct relative Gromov-Taubes
invariants
Gauge transformations and symmetries of integrable systems
We analyze several integrable systems in zero-curvature form within the
framework of invariant gauge theory. In the Drienfeld-Sokolov gauge
we derive a two-parameter family of nonlinear evolution equations which as
special cases include the Kortweg-de Vries (KdV) and Harry Dym equations. We
find residual gauge transformations which lead to infinintesimal symmetries of
this family of equations. For KdV and Harry Dym equations we find an infinite
hierarchy of such symmetry transformations, and we investigate their relation
with local conservation laws, constants of the motion and the bi-Hamiltonian
structure of the equations. Applying successive gauge transformatinos of Miura
type we obtain a sequence of gauge equivalent integrable systems, among them
the modified KdV and Calogero KdV equations.Comment: 18pages, no figure Journal versio
Tzitz\'eica transformation is a dressing action
We classify the simplest rational elements in a twisted loop group, and prove
that dressing actions of them on proper indefinite affine spheres give the
classical Tzitz\'eica transformation and its dual. We also give the group point
of view of the Permutability Theorem, construct complex Tzitz\'eica
transformations, and discuss the group structure for these transformations
Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice
`Umbral calculus' deals with representations of the canonical commutation
relations. We present a short exposition of it and discuss how this calculus
can be used to discretize continuum models and to construct representations of
Lie algebras on a lattice. Related ideas appeared in recent publications and we
show that the examples treated there are special cases of umbral calculus. This
observation then suggests various generalizations of these examples. A special
umbral representation of the canonical commutation relations given in terms of
the position and momentum operator on a lattice is investigated in detail.Comment: 19 pages, Late
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