Interactions of model theory and set theory with Banach space theory; isometric structure of Banach spaces
Lead Research Organisation:
University of East Anglia
Department Name: Mathematics
Abstract
Model theory is part of mathematical foundations that aims to discover common mathematical truths that hold in many different branches of mathematics.Classicaly it has been very successful in dealing with first order structures, such as groups, linear orders, Boolean algebras etc. An important notion missingon this list is that of a topological space, which cannot be seen as a first order structure. Therefore classical model theory is much less successful indealing with problems coming from analysis than those coming from algebra. Recent developments in model theory have brought about new tools that offerpotential to bridge this gap. In particular, a model theory of Banach spaces have been developed.The PI makes the point that it is now time to apply these tools to specific problems coming from Banach space theory and relevant to the Banach spacecommunity.For reasons described in the proposal, one of the most promising areas of Banach space theory where model theoretic methods could be used isthe isometric theory of various classes of Banach spaces. The proposal plans to match a leading expert on the isomorphic theory of Banach spaces,a leading expert on the model theory of Banach spaces and the PI, whose interests and expertise go into both of these areas. The plan is to havea one month long research meeting of all three participants when the research would be started and developed, followed by email collaborationand a visit by the PI to each of the other two participants to discuss dissemination, publication and future research plans.The proposal can be seen as a new step in the successful programme of applications of infinitary combinatorics to Banach space theory, asstarted by Gowers in his Fields Medal work. That programme has grown into an internationally active research direction combining set theoryand Banach space theory. The novelty of this proposal is that another part of mathematical logic is included in this mixed set theory/Banachgroup theory research perspective, by including the insights offered by recent top developments in model theory.It is hoped that the research supported by this grant will form a seed of a long term interaction and branch into further research directions.
Organisations
People |
ORCID iD |
| Mirna Dzamonja (Principal Investigator) |
Publications
Borodulin-Nadzieja P
(2013)
On the isomorphism problem for measures on Boolean algebras
in Journal of Mathematical Analysis and Applications
Dolinar G
(2013)
Forcing ? ? 1 with finite conditions
in Annals of Pure and Applied Logic
| Description | Graphs and directed graphs are simple and pervasive mathematical objects which are studied both for their potential applications and for their intrinsic mathematical interest. Often these two viewpoints interact, but the emphasis in this project was on the latter. A graph consists of a collection of points (called vertices), certain pairs of which are joined together (by edges). In a directed graph (or digraph), the edges have a direction on them. For example, one could have a graph where the vertices represent certain towns and the edges represent roads between them. In a directed graph the roads would be one-way. When studying a class of mathematical objects, it is often fruitful to focus on the objects in the class which possess a high degree of symmetry. Here we focused on highly arc transitive digraphs: ones where for any two directed paths of the same length in the digraph there is a symmetry of the digraph which moves one path to the other. In particular, the descendant set of a vertex, that is the collection vertices which can be reached by a directed path starting at the vertex, will look the same for all vertices in the digraph and will also have a high degree of symmetry. In general there is no hope of describing all highly arc transitive digraphs, though there are many interesting open questions which can be asked about them. However, there is a natural subclass where recent work suggest a possibility of being able to classify the descendant sets, and doing this was one of the main aims (and outcomes) of the project. This is where the group of symmetries of the digraph is also primitive on the set of vertices, and where there are only finitely many directed edges coming out of each vertex. In this case, the descendant sets look approximately like a finitely-branching tree, but this is only a crude approximation, as if looking at the structure from a distance, and our work gives a much better understanding of the real picture here. The project also studied properties of the groups of symmetries of highly arc transitive digraphs and investigated generalizations which assumed less arc transitivity. In more technical terms, the main outcomes of the work done were: the construction of new examples of highly arc transitive digraphs and primitive groups; a rather complete description of the descendant sets in highly arc transitive digraphs of finite out-valency; a classification of certain descendant homogeneous digraphs and the start of a study the group-theoretic structure of the resulting automorphism groups, in particular, their normal subgroup structure. The main beneficiaries of the project will be other mathematicians, particularly those interested in combinatorics, group theory and logic. Some of the publications which resulted from the project and some slides from talks given on the results of the project can be found on the PI's webpage. |
| Sectors | Creative Economy |
| Description | Ministerstwo Nauki i Szkolnictwa Wyzszeg |
| Amount | £20,000 (GBP) |
| Funding ID | NN201 418939 (2010-2013) |
| Organisation | Government of Poland |
| Department | Ministry of Science and Higher Education |
| Sector | Public |
| Country | Poland |
| Start | |