This is m1.
The overall, extended project, for which this was a pilot, aims
a) Strong Shape Theory is a branch of topology midway
between algebraic topology (homotopy theory) and geometric topology. It
handles the analysis of spaces that have singularities in their
topological structure (e.g. lack of local connectedness) . It has close
links with C\ast -algebra theory and, in particular, the
theory of asymptotic morphisms (Connes-Higson) and E -theory,
b) Finitary Approximations, initially developed to model
(quantum ) space-time, use closely related methods to Strong Shape
Theory, but have developed them more in the direction of differential
geometry);
c) Analysis on Fractals uses inner approximations
rather than the finitary or polyhedral outer approximations
used by the other theories, but is where considerable work on fractals per
se has been done, (cf. Stritchartz's article in the Notices Amer.
Math. Soc. via the link earlier on this page)
In the extended project, these theories will be used to attempt the generalisation of classical analytic, geometric and topological results for manifolds to each of the candidate classes of fractal spaces. The key intuition is that the linking maps between the approximations are as important for the possibility of good structure `in the limit' as are the approximations themselves. Although the extension of the results would be significant, their use here is, in the first instance, to act as an analytic tool to help in the identification of the optimal class for the status of `fractafold'.
Three interrelated candidate classes have been identified (i)
manifold-like compacta, (ii) spaces of manifold shape, (iii) Menger
manifolds These latter are spaces locally like a (possibly higher
dimensional version, mn of the)
Menger sponge, the hypercube analogue of the Sierpinski triangle,
obtained by deleting the middle cubelets of each face of the hypercube
together with the central cubelet. To get mn
you start with I{2n+1}.
To summarise: the extended project aims to identify a useful class of `fractafolds' and to develop as much as possible of their geometry and topology using analogues of classical theory.
For the initial pilot project, the aim will be to set the foundations for the main investigation. The pilot study will concentrate on two of the candidate classes and, within them, on their generic models.
Amongst the manifold-like compacta, probably the simplest class of examples are the solenoids. These are S1 -like and are realisable as inverse limits of well behaved systems of approximations. They are standard examples in dynamical systems theory, fractal geometry and shape theory.
The finitary approximations at different refinements are linked by refinement maps that are finitary covering maps of the circle and it is expected that this `good behaviour' will result in correspondingly good behaviour at the level of `differential forms'. (The general case of manifold-like compacta is likely to have many of these same features, but of course, with higher dimensional complications and interactions that will initially obscure the picture - hence the choice of this class of simple example.)
The behaviour of the other chosen candidate, Menger manifolds, clearly depends on the behaviour of the theory on the Menger sponges themselves, so these will be the second objects of study. Features of this class of spaces are, in some sense, complementary to those of the solenoids, so it is hoped and expected that combining the experience gained from these two `generic' classes will provide the research team with a wide range of behaviours that will be applicable to more general situations in the extended project.
Within the limited time of a pilot project, the investigation did
not attempt to attack all the aims of the extended project, even on the
`generic examples'. Rather it will concentrate on the de Rham complex
analogues and the development of an embryonic limiting differential
geometry applicable to these fractal spaces. This aspect will be
greatly helped by new results coming from finitary approximation
theory, that have become available only since the outline proposal was
submitted. Here there are some interesting variants that will be looked
at, these include non-commutative versions of de Rham based
differential forms, but also using fibre bundle based ideas leading,
perhaps, to versions of stacks, gerbes etc. as currently under
development by workers in mathematical physics (Baez, Breen-Messing,
MacKaay-Picken, etc.) This bundle approach should tie in very nicely
with the de Rham based version, but is also more `geometric' in some
sense. Some links with work on (topological) quantum field theories may
also be hidden here.
Added: June 2005.
The pilot project has now ended. Here is a version of the
report.
Fractafolds:
Report on the Pilot Project.
Summary
Report:
a)
Aims and
Objectives:
·
To investigate a class
of topological spaces, `fractafolds’, to see if they have a
well-behaved
geometry generalising classical differential geometry of manifolds and
can act
as a `test bed’ for the development of more general techniques of the
same
type;
·
To examine the general
questions of observation and refinement of observation
of
geometric and topological structure using ideas from (algebraic
topology), and
mathematical physics;
·
To look for potential
applications of these ideas in physics, mathematics and computer
science.
b)
Broad findings
and results
·
The methods of
constructing combinatorial and topological models of observational
phenomena
were clarified and greatly extended.
·
Several interesting new
candidates for the algebra underpinning the discrete differential
geometries
available were given but there was too little time to investigate these
in
full.
·
The links with a putative
Geometry of Information were explored.
c)
Types of publication resulting
·
Two 40 page articles
(Geometry of Information I and II) have been accepted (unrefereed)
for the online proceedings of the Dagstuhl workshop: Spatial
Representation: Discrete vs. Continuous Computational Models, (see the Dagstuhl website: http://drops.dagstuhl.de/portals/04351/);
·
One article (A Geometry
of Information) has been submitted for the published proceedings of the
Dagstuhl
workshop (special issue of Theoretical Computer Science);
·
An overview article (The
shape of space-time) was placed on the electronic Physics preprint
archive and
a revised version is being prepared for submission to a journal;
·
A third 40 page article
(Geometry of Information III) exists in draft form but needs quite
extensive
work before submission.
d)
Strengths and weakness of the
research.
The research
developed in a slightly unexpected
direction following contacts with a mixed group of mathematical
physicists and
computer scientists at
(i)
The various areas were
linked, but used different methods that, although clearly connected,
had not
been studied in an integrated manner and therefore needed detailed
comparison
if they were to be of use to us in the original problem of `fractafolds
and
their geometry’;
(ii)
Any methods we developed
or clarified in the project were likely to have a much wide potential
use than
originally envisaged.
The contacts at
Imperial resulted in an invitation to
the Dagstuhl workshop Spatial
Representation: Discrete vs. Continuous Computational Models and the further contacts made there further
strengthened and widened our view on the applicability of the methods
we were
using. Perhaps the most interesting result from this was the new
insight on how
observations of any system may
be organised geometrically and,
as the observations change, it may be possible to apply methods of
discrete
differential geometry to analyse the flow of information through the
resulting
`space’.
The project was a
Pilot project. We managed to do a
lot in 10 months, but the implications of work done were such that a
full
project was needed. It seems that funding was not available for this.
Detailed Report.
a)
Objectives
·
To study the structure
of a class of spaces, tentatively called `fractafolds’ in order to see
if tools
could be developed (or adapted from other contexts) analogous to
differential
geometry, to reveal deeper geometric structure.
·
To compare methods (Sorkin model, various nerve constructions,
…) that gave ways of studying the geometry of such fractal
spaces.
·
To develop and extend
existing discrete differential geometry, so as to apply to the models
above
mentioned.
b)
Research
Activity.
The project
initially studied the previous results and ideas of Zapatrin, Raptis and Mallios
on discrete
analogues of differential geometry and the ways of extracting geometric
information from `observations’ of a physical space, where, following Sorkin, observations corresponded to a family of
open sets
(potentially distinguishing points) The existing theory in both these
areas was
somewhat lacking in detailed examples and extended theory, so we spent
time
exploring various parts of these.
The motivation
behind the project was, in part, of a fairly philosophical nature. Quantum Physics predicts that space-time
should be `granular' as it should be based on `observations’, whilst
the
differential geometric models of space-time as used, say, by general
relativity, required `smooth spaces'.
Could a sufficiently rich differential geometric model of a more
`granular' nature be developed or was this impossible? The fractafold
notion
would provide a test bed to see if such a granular
differential geometry
might be possible. The `fractafolds'
would be idealised objects and, for various reasons, would be unlikely
to
provide models for a quantum theory of relativity,
however they would test the myth of the necessity of smoothness in a
way not
previously attempted.
We found in
fact that similar considerations occur in several other areas and we
quickly
realised that there was much work to do in that more general setting to
ensure
that these interconnections were fully understood and could be
exploited by us
with `spin-off’ both into and out of the main project focus area.
Discussions
with Dr. Raptis and Dr. J. Webster at
A further
motivation for the Pilot project study had been, from the start, that
the
potential for application of such a theory was very strong. Fractal spaces occur frequently as limiting
models, e.g. as strange attractors of dynamical systems, yet it is
extremely
difficult to get one’s hands on the geometry of such spaces. The idea of a fractafold would be a
well-controlled version of such a general system, more regular, but
still
fractal, a half way house between the well behaved nature of a manifold
and the
potentially wild one of a general fractal space. They
would thus be more amenable to study,
and would provide a `toy model’ for development of new concepts and new
tools.
The types of system that could be modelled are many and varied, ranging
from
physical systems to observations of state change in distributed
computing systems
and, possibly combining aspects of these two, biological and other
complex
systems in which many scales or hierarchical levels are interacting. A similar problem of approximation and
multiple scales of behaviour occurs in
computer
graphics and data visualisation, as they both use sampling of scanned
data, and
also with techniques for the numerical solution of partial differential
equations. Again, in each of these there
is a `multiscale’ aspect and polyhedral models are used as an
approximating
framework for the patches that give the final image or model for the
solution. If the image is varying in
time or the degree of resolution is also changing, an analysis of that
evolution can require methods very similar to those we were examining. If a solution is changing rapidly near a
`singularity, the scale and grid is changed and again there are
similarities in
the methods used.
The focus of
the Project did change towards the more general Geometry of
Information
as it would have been silly to ignore that our methods could be used
for that
wider field. The results however were still extremely relevant to the
original
focus of `fractafolds’ as that aspect was the limiting case as the
scale of the
observations became finer and finer. In the event and in the limited
time
available (10 months) the majority of the effort was needed in studying
the
`static’, `single scale’ case and the comparison map (two related
scales, one
finer than the other) rather than the limiting case, although some
examples of
that were explored.
c)
Conclusions
and Achievements.
The Pilot
project has been extremely successful.
We have a much richer and deeper appreciation of the problems
involved
and several partial results that are suggestive of the way to attack
the
overall situation. The initial thrust was to understand the possible
tools
needed for a `differential geometry of simplicial complexes’ as these
objects
would be used to approximate the fractafolds and more particularly, the
solenoids and the Menger cubes that had been singled out for attention
in this
initial phase. Dr. Gratus concentrated on a discrete differential
geometric
idea due to R. Zapatrin, whilst I initially investigated commutative
differential forms as used by Whitney and Thom.
The Zapatrin model is non-commutative, so would not give a
`usual’
differential geometry, but, on the other hand, is nearer to the modern
non-commutative geometric approaches to differential geometry, and
theoretical
physics due to Connes, Kontsevich, etc.
It would also seem to be related to some constructions in
computational
visualisation of flows and discrete Morse theory. Further detailed
investigation has shown that this model has a very rich structure that
to a
large extent has not been studied, and, in part, not even noticed
before. The high points to note are:
(i) the
model is easy to use, but is very rich;
(ii) applying the model via
finite approximations to solenoids and Menger cubes (in low dimensions)
reveals
evidence of phenomena in the limit related to the history of each point
considered;
(iii) the Zapatrin model
supports not only a differential graded algebra structure, but
coalgebra
structures of interest also from a combinatorial viewpoint;
(iv) a
generalisation of the Zapatrin model applicable
to graded partially ordered sets is linked with constructions of
objects called
`pseudocomplexes’ by Lazariou, in the context of the Kontsevich theory of A∞ and differential
graded categories as a model for D-branes in quantum physics;
(v) there is interest from
computer scientists in the potential of these models for (a) modelling
evolving
spatial representations and (b) studying models of sets of states in a
distributed system. In both cases, the
use of fractafolds as `toy models’ has aroused interest.
These models seem to have a
lot more structure than was previously
thought. Some related structures are
known from other subject areas (combinatorics and deformation theory,
in
particular), but the combination of them in this one model seems
important. In particular, we believe the
fibration-like structures encountered in both Menger manifolds and
solenoids
seem to interact well with these models but in a surprising way. We
have high
hopes for the success of the future research in this area and also for
some
interesting `spin-off’ to other areas of mathematics, computer science
and
physics.
As a result of the visit to
Dagstuhl mentioned earlier, we looked again
at the minimal structures that would lead to the situations that we had
been
modelling. The results were very
encouraging. The basic situation could
be viewed as a collection of `objects’ and a collection of `attributes’
that
they may or may not satisfy. In the Physical situation the attributes
were
observational values of some test. In the spatial context they were
open sets
in some open cover. The organisation of
the observations that we had been studying was in this new perspective
an
example of a structure known in Theoretical Computer Science as a
d)
Publications and
dissemination.
Two
papers appeared in the (unrefereed)
proceedings of
the Dagstuhl workshop:
(i)
J.
Gratus and T. Porter: A geometry of
information, I:
Nerves, posets and differential forms. In
Proceedings
Dagstuhl workshop 04351 (see
the
Dagstuhl website: http://drops.dagstuhl.de/portals/04351/);
(ii) J. Gratus and T. Porter:
A
geometry of information, II: Sorkin models, and biextensional
collapses. In Proceedings Dagstuhl workshop 04351 (see the Dagstuhl
website: http://drops.dagstuhl.de/portals/04351/).
A preprint ((iii) below) was
made available via the ArXiv
website early on in the planning stage of the project. A revised
version of
this is being prepared for submission to a journal:
(iii)
T. Porter, What `shape' is space-time?,
available from the ArXiv
gr-qc/0210075.
An article, (iv), has been prepared and
submitted for the refereed proceedings of the Dagstuhl workshop. This
is a
shortened version of (i) and (ii) above
omitting
certain parts of the exposition and some detailed proofs and results. It is hoped that this will appear in the
journal, Theoretical Computer Science.
(iv)
J. Gratus and T.
Porter: A
geometry of information (preprint available from T. Porter).
The material, (v), on discrete
analogues of
differential forms is in a draft version, which still needs some more
work:
(v) J.
Gratus and T. Porter, A geometry of information,
III: Differential graded algebras for
discrete situations: the MRZ calculus.
e)
Future Plans.
Several
areas
have been opened up by this project.
These include
·
Topological data
analysis,
·
Spatial representation
and visualisation,
·
Geometric organisation
of information,
·
Discrete analogues of
differential geometry.
It is clear that each
interacts with various areas of Physics or
Computer Science/A.I., but that often the interrelationships are
unknown to
workers in the various fields. Preparation of new grant proposals to
explore
these application areas with our methods is underway.
Further work on the third part
of the main group of papers is needed. Since
the end of the grant period, T.P. has
spent much time consolidating the results and preparing them for
publication.
That process of consolidation has interacted with research interests of
colleagues in the
g) Key Phrases:
Geometry of
Information; Spatial Representation; Fractal analogues of manifolds; Discrete differential geometry; Observations.
I would like to thank the
Trust for their support.
T. Porter