39 research outputs found
Uniform locales and their constructive aspects
Much work has been done on generalising results about uniform spaces to the
pointfree context. However, this has almost exclusively been done using
classical logic, whereas much of the utility of the pointfree approach lies in
its constructive theory, which can be interpreted in many different toposes.
Johnstone has advocated for the development of a constructive theory of uniform
locales and wrote a short paper on the basic constructive theory via covering
uniformities, but he never followed this up with a discussion of entourage
uniformities or completions.
We present a more extensive constructive development of uniform locales,
including both entourage and covering approaches, their equivalence,
completions and some applications to metric locales and localic algebra.
Some aspects of our presentation might also be of interest even to
classically minded pointfree topologists. These include the definition and
manipulation of entourage uniformities using the internal logic of the
geometric hyperdoctrine of open sublocales and the emphasis on pre-uniform
locales. The latter leads to a description of the completion as the uniform
reflection of the pre-uniform locale of Cauchy filters and a new result
concerning the completion of pre-uniform localic rings, which can be used to
easily lift addition and multiplication on to (or
) in the pointfree setting.Comment: 27 pages, minor edit
Characterizations of model sets by dynamical systems
It is shown how regular model sets can be characterized in terms of
regularity properties of their associated dynamical systems. The proof proceeds
in two steps. First, we characterize regular model sets in terms of a certain
map and then relate the properties of to ones of the underlying
dynamical system. As a by-product, we can show that regular model sets are, in
a suitable sense, as close to periodic sets as possible among repetitive
aperiodic sets.Comment: 41 pages, revised versio
The bicompletion of the Hausdorff quasi-uniformity
We study conditions under which the Hausdorff quasi-uniformity of a quasi-uniform space on the set of the nonempty subsets of is bicomplete.
Indeed we present an explicit method to construct the bicompletion of the
-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It
is used to find a characterization of those quasi-uniform -spaces
for which the Hausdorff quasi-uniformity
of their bicompletion
on
is bicomplete
Pattern-Equivariant Homology of Finite Local Complexity Patterns
This thesis establishes a generalised setting with which to unify the study
of finite local complexity (FLC) patterns. The abstract notion of a "pattern"
is introduced, which may be seen as an analogue of the space group of
isometries preserving a tiling but where, instead, one considers partial
isometries preserving portions of it. These inverse semigroups of partial
transformations are the suitable analogue of the space group for patterns with
FLC but few global symmetries. In a similar vein we introduce the notion of a
\emph{collage}, a system of equivalence relations on the ambient space of a
pattern, which we show is capable of generalising many constructions applicable
to the study of FLC tilings and Delone sets, such as the expression of the
tiling space as an inverse limit of approximants.
An invariant is constructed for our abstract patterns, the so called
pattern-equivariant (PE) homology. These homology groups are defined using
infinite singular chains on the ambient space of the pattern, although we show
that one may define cellular versions which are isomorphic under suitable
conditions. For FLC tilings these cellular PE chains are analogous to the PE
cellular cochains \cite{Sadun1}. The PE homology and cohomology groups are
shown to be related through Poincar\'{e} duality.
An efficient and highly geometric method for the computation of the PE
homology groups for hierarchical tilings is presented. The rotationally
invariant PE homology groups are shown not to be a topological invariant for
the associated tiling space and seem to retain extra information about global
symmetries of tilings in the tiling space. We show how the PE homology groups
may be incorporated into a spectral sequence converging to the \v{C}ech
cohomology of the rigid hull of a tiling. These methods allow for a simple
computation of the \v{C}ech cohomology of the rigid hull of the Penrose
tilings.Comment: 159 pages, 8 figures, PhD thesi
Uniform sheaves and differential equations
Real blow-ups and more refined "zooms" play a key role in the analysis of
singularities of complex-analytic differential modules. They do not change the
underlying topology, but the uniform structure.
This suggests to revisit the cohomology theory of differential modules with
help of a suitable new notion of uniform sheaves based on the uniformity rather
than the topology.
We also investigate the -adic situation (in particular, an analog of real
blow-ups) from this uniform viewpoint
Dagger and Dilation in the Category of Von Neumann algebras
This doctoral thesis is a mathematical study of quantum computing,
concentrating on two related, but independent topics. First up are dilations,
covered in chapter 2. In chapter 3 "diamond, andthen, dagger" we turn to the
second topic: effectus theory. Both chapters, or rather parts, can be read
separately and feature a comprehensive introduction of their own
Functional transitive quasi-uniformities and their bicompletions
Bibliography: pages 111-117