5 research outputs found
A cellular topological field theory
We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah-Segal-type gluing formula with respect to composition of cobordisms
The Complexity of Drawing Graphs on Few Lines and Few Planes
It is well known that any graph admits a crossing-free straight-line drawing
in and that any planar graph admits the same even in
. For a graph and , let denote
the minimum number of lines in that together can cover all edges
of a drawing of . For , must be planar. We investigate the
complexity of computing these parameters and obtain the following hardness and
algorithmic results.
- For , we prove that deciding whether for a
given graph and integer is -complete.
- Since , deciding is NP-hard for . On the positive side, we show that the problem
is fixed-parameter tractable with respect to .
- Since , both and
are computable in polynomial space. On the negative side, we show
that drawings that are optimal with respect to or
sometimes require irrational coordinates.
- Let be the minimum number of planes in needed
to cover a straight-line drawing of a graph . We prove that deciding whether
is NP-hard for any fixed . Hence, the problem is
not fixed-parameter tractable with respect to unless
Remarks on Chern-Simons invariants
The perturbative Chern-Simons theory is studied in a finite-dimensional
version or assuming that the propagator satisfies certain properties (as is the
case, e.g., with the propagator defined by Axelrod and Singer). It turns out
that the effective BV action is a function on cohomology (with shifted degrees)
that solves the quantum master equation and is defined modulo certain canonical
transformations that can be characterized completely. Out of it one obtains
invariants.Comment: 33 pages; minor corrections, new appendices with technical details,
new references, new example; to appear in Commun. Math. Phys
A Dynamic Setup for Elementary Geometry
In this article we survey the theoretical background that is required to build a consistent and continuous setup of dynamic elementary geometry. Unlike i
Some Algorithmic Problems in Polytope Theory
Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact..