82 research outputs found
Combinatorics of embeddings
We offer the following explanation of the statement of the Kuratowski graph
planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas
intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to
every cell there corresponds a unique cell with the complementary set of
vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere;
there exist precisely two 3-dimensional dichotomial cell complexes, and their
1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their
1-skeleta all but one graphs of the Petersen family.
In higher dimensions n>2, we observe that in order to characterize those
compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited
minors", it suffices to establish finiteness of the list of all (n-1)-connected
n-dimensional finite cell complexes that do not embed in S^{2n} yet all their
proper subcomplexes and proper cell-like combinatorial quotients embed there.
Our main result is that this list contains the n-skeleta of (2n+1)-dimensional
dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell
complexes include (apart from the three joins of the i-skeleta of
(2i+2)-simplices) at least ten non-simplicial complexes.Comment: 49 pages, 1 figure. Minor improvements in v2 (subsection 4.C on
transforms of dichotomial spheres reworked to include more details;
subsection 2.D "Algorithmic issues" added, etc
Probability, Trees and Algorithms
The subject of this workshop were probabilistic aspects of algorithms for fundamental problems such as sorting, searching, selecting of and within data, random permutations, algorithms based on combinatorial trees or search trees, continuous limits of random trees and random graphs as well as random geometric graphs. The deeper understanding of the complexity of such algorithms and of shape characteristics of large discrete structures require probabilistic models and an asymptotic analysis of random discrete structures. The talks of this workshop focused on probabilistic, combinatorial and analytic techniques to study asymptotic properties of large random combinatorial structures
Loop Quantum Gravity: An Inside View
This is a (relatively) non -- technical summary of the status of the quantum
dynamics in Loop Quantum Gravity (LQG). We explain in detail the historical
evolution of the subject and why the results obtained so far are non --
trivial. The present text can be viewed in part as a response to an article by
Nicolai, Peeters and Zamaklar [hep-th/0501114]. We also explain why certain no
go conclusions drawn from a mathematically correct calculation in a recent
paper by Helling et al [hep-th/0409182] are physically incorrect.Comment: 58 pages, no figure
Spectral correspondences for finite graphs without dead ends
We compare the spectral properties of two kinds of linear operators
characterizing the (classical) geodesic flow and its quantization on connected
locally finite graphs without dead ends. The first kind are transfer operators
acting on vector spaces associated with the set of non backtracking paths in
the graphs. The second kind of operators are averaging operators acting on
vector spaces associated with the space of vertices of the graph. The choice of
vector spaces reflects regularity properties. Our main results are
correspondences between classical and quantum spectral objects as well as some
automatic regularity properties for eigenfunctions of transfer operators.Comment: 37 pages, 2 figure
Teichmüller Space (Classical and Quantum)
This is a short report on the conference “Teichmüller Space (Classical and Quantum) ” held in Oberwolfach from May 28th to June 3rd, 2006
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
Introduction to Modern Canonical Quantum General Relativity
This is an introduction to the by now fifteen years old research field of
canonical quantum general relativity, sometimes called "loop quantum gravity".
The term "modern" in the title refers to the fact that the quantum theory is
based on formulating classical general relativity as a theory of connections
rather than metrics as compared to in original version due to Arnowitt, Deser
and Misner. Canonical quantum general relativity is an attempt to define a
mathematically rigorous, non-perturbative, background independent theory of
Lorentzian quantum gravity in four spacetime dimensions in the continuum. The
approach is minimal in that one simply analyzes the logical consequences of
combining the principles of general relativity with the principles of quantum
mechanics. The requirement to preserve background independence has lead to new,
fascinating mathematical structures which one does not see in perturbative
approaches, e.g. a fundamental discreteness of spacetime seems to be a
prediction of the theory providing a first substantial evidence for a theory in
which the gravitational field acts as a natural UV cut-off. An effort has been
made to provide a self-contained exposition of a restricted amount of material
at the appropriate level of rigour which at the same time is accessible to
graduate students with only basic knowledge of general relativity and quantum
field theory on Minkowski space.Comment: 301 pages, Latex; based in part on the author's Habilitation Thesis
"Mathematische Formulierung der Quanten-Einstein-Gleichungen", University of
Potsdam, Potsdam, Germany, January 2000; submitted to the on-line journal
Living Reviews; subject to being updated on at least a bi-annual basi
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