2,483 research outputs found
Differential Invariants of Conformal and Projective Surfaces
We show that, for both the conformal and projective groups, all the
differential invariants of a generic surface in three-dimensional space can be
written as combinations of the invariant derivatives of a single differential
invariant. The proof is based on the equivariant method of moving frames.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Self-affine Manifolds
This paper studies closed 3-manifolds which are the attractors of a system of
finitely many affine contractions that tile . Such attractors are
called self-affine tiles. Effective characterization and recognition theorems
for these 3-manifolds as well as theoretical generalizations of these results
to higher dimensions are established. The methods developed build a bridge
linking geometric topology with iterated function systems and their attractors.
A method to model self-affine tiles by simple iterative systems is developed
in order to study their topology. The model is functorial in the sense that
there is an easily computable map that induces isomorphisms between the natural
subdivisions of the attractor of the model and the self-affine tile. It has
many beneficial qualities including ease of computation allowing one to
determine topological properties of the attractor of the model such as
connectedness and whether it is a manifold. The induced map between the
attractor of the model and the self-affine tile is a quotient map and can be
checked in certain cases to be monotone or cell-like. Deep theorems from
geometric topology are applied to characterize and develop algorithms to
recognize when a self-affine tile is a topological or generalized manifold in
all dimensions. These new tools are used to check that several self-affine
tiles in the literature are 3-balls. An example of a wild 3-dimensional
self-affine tile is given whose boundary is a topological 2-sphere but which is
not itself a 3-ball. The paper describes how any 3-dimensional handlebody can
be given the structure of a self-affine 3-manifold. It is conjectured that
every self-affine tile which is a manifold is a handlebody.Comment: 40 pages, 13 figures, 2 table
The Polyhedron-Hitting Problem
We consider polyhedral versions of Kannan and Lipton's Orbit Problem (STOC
'80 and JACM '86)---determining whether a target polyhedron V may be reached
from a starting point x under repeated applications of a linear transformation
A in an ambient vector space Q^m. In the context of program verification, very
similar reachability questions were also considered and left open by Lee and
Yannakakis in (STOC '92). We present what amounts to a complete
characterisation of the decidability landscape for the Polyhedron-Hitting
Problem, expressed as a function of the dimension m of the ambient space,
together with the dimension of the polyhedral target V: more precisely, for
each pair of dimensions, we either establish decidability, or show hardness for
longstanding number-theoretic open problems
2-frieze patterns and the cluster structure of the space of polygons
We study the space of 2-frieze patterns generalizing that of the classical
Coxeter-Conway frieze patterns. The geometric realization of this space is the
space of n-gons (in the projective plane and in 3-dimensional vector space)
which is a close relative of the moduli space of genus 0 curves with n marked
points. We show that the space of 2-frieze patterns is a cluster manifold and
study its algebraic and arithmetic properties.Comment: 36 pages, 4 figures, minor changes from the previous versions;
references added (v2), section 5.5 added (v3), picture added (v4
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