Mathematical research
Coarse geometry
A metric space is a set equipped with a function measuring the distance between two points. Topology arises from the study of metric spaces by looking at when points are close together. It is a framework where limits, continuity, and the shape of objects are studied.
Coarse geometry arises from the study of metric spaces by looking at when points are far apart. Small scale structure does not matter in coarse geometry; indeed, every space of a finite size is equivalent to a single point as far as coarse geometry is concerned. All that matters is the large scale geometry of infinitely large spaces.
There are a number of tools available to analyse topological spaces. Some of my research in coarse geometry focuses on finding analogues of these tools for coarse geometry. Many of the results obtained are very similar to those of topology. This statement is less of a surprise than it appears; many of the ideas of topology and coarse geometry can be put into the same abstract algebraic framework.
I am also interested in results on descent which, broadly speaking, map results in coarse geometry to results in topology, and in applications of coarse geometry to index theory and the study of positive scalar curvature manifolds.
Categories of operators
A C*-algebra is a closed algebra of bounded linear operators from a given Hilbert space to itself. C*-algebras are of importance in quantum physics, where observable quantities are defined to be operators on an appropriate Hilbert space. Any commutative C*-algebra is isomorphic to the algebra of continuous functions from some given topological space to the complex numbers. Consequently, the theory of C*-algebras can be looked at as some kind of non-commutative geometry. This idea has many applications in geometry, analysis, and physics.
My research concerns objects similar to C*-algebras called C*-categories. A C*-category can be defined to be a closed subcategory of a category consisting of a collection of Hilbert spaces and bounded linear operators between them. C*-categories are natural generalisations of C*-algebras, and most of the elementary theory of C*-algebras can be extended without too much difficulty to the theory of C*-categories.
In many cases a C*-algebra associated to a given geometric structure is defined by arbitrarily choosing a Hilbert space satisfying certain criteria and considering operators on that Hilbert space possessing properties determined by the geometry. In such cases it is natural to define a C*-category rather than a C*-algebra by considering all suitable Hilbert spaces at once.
This technique enables problems to be solved where one encounters difficulties because of an arbitrary choice of C*-algebra. For example, it is possible to use homotopy-theoretic machinery to characterise the analytic assembly map and Baum-Connes assembly map by considering the K-theory of C*-categories. C*-categories also feature in the most natural formulation of some the basic ideas in coarse geometry.
More recent work involves looking at categories of unbounded linear operators as a tool to eliminate difficulties that arise, for example in quantum physics, when looking at bounded linear operators, or from the approach that comes from looking at algebras rather than categories of such operators.
Articles
The following articles are in PDF or DVI format.
Email p.mitchener@sheffield.ac.uk for any enquiries.
Categories of unbounded operators
We introduce the concept of an LK*-algebroid, which is defined axiomatically. The main example of an LK*-algebroid is the category of all subspaces of a Hilbert space and closed (not necessarily bounded) linear operators. We prove that for any LK*-algebroid there is a faithful functor that respects its structure and maps it into this main example.
Descent and the KH-assembly map
We show that a general notion of descent in coarse geometry can be applied to the study of injectivity of the KH-assembly map. We also show that the coarse assembly map is injective in general for finite coarse $CW$-complexes.
The general notion of descent in coarse geometry
We introduce the notion of a functor on coarse spaces being coarsely excisive - a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an associated coarse assembly map from the topologically exicisive functor to the coarsely excisive functor.
We conjecture that this coarse assembly map is an isomorphism for uniformly contractible spaces with bounded geometry, and show that the coarse isomorphism conjecture, along with some mild technical conditions, implies that a correspoding equivariant assembly map is injective. Particular instances of this equivariant assembly map are the maps in the Farrell-Jones conjecture, and in the Baum-Connes conjecture.
The Baum-Connes assembly map as a boundary map
In a recent article, John Roe proved by a direct computation that the Baum-Connes assembly map can be expressed as a boundary map associated to a certain short exact sequence of C*-algebras. In this article, we deduce the same result as a corollary of my characterisation of the Baum-Connes assembly map in the article C*-categories, groupoid actions, equivariant KK-theory, and the Baum-Connes conjecture (DVI, 138KB).
A primer on some methods in homotopy theory
This article is an exposition of the theory of simplicial sets and spaces, classifying spaces, homotopy colimits, the plus construction and the group completion theorem. The techniques are applied at the end of the article to compare constructions in algebraic K-theory.
C*-categories, groupoid actions, equivariant KK-theory, and the Baum-Connes conjecture
In this article we give a characterisation of the Baum-Connes assembly map with coefficients. The technical tools needed are the K-theory of C*-categories, and a new version of equivariant KK-theory in the world of groupoids.
Journal of Functional Analysis, volume 214 (2004), pages 1-39.
Symmetric Waldhausen K-theory spectra of topological categories
We show how to use Waldhausen's K-theory machine to define symmetric K-theory spectra associated to certain topological categories. The K-theory spectra of C*-categories and algebraic K-theory spectra arise as special cases. As an application we give a new approach to a homotopy-theoretic description of the analytic assembly map.
Coarse homology theories
In this paper we develop an axiomatic approach to coarse homology theories. We prove a uniqueness result concerning coarse homology theories on the category of 'coarse CW-complexes'. This result is used to prove a version of the coarse Baum-Connes conjecture for such spaces.
Algebraic and Geometric Topology, volume 1 (2001), pages 271-297.
Addendum to 'Coarse homology theories'
This paper contains corrections to two mistakes in the article 'Coarse homology theories' along with further discussion.
Algebraic and Geometric Topology, volume 3 (2003), pages 1089-1101.
A brief review of the theory of symmetric spectra
This article is a very short summary of some aspects of the theory of symmetric spectra that I find useful in my work.
KK-theory of C*-categories and the analytic assembly map
We define KK-theory spectra associated to C*-categories and look at certain instances of the Kasparov product at this level. This machinery is used to give a description of the analytic assembly map as a natural map of spectra.
K-theory, volume 26 (2002), pages 307-344.
Symmetric K-theory spectra of C*-categories
We define K-theory spectra associated to graded C*-categories and show that the exterior product of K-theory groups can be expressed in terms of the smash product of symmetric spectra.
K-theory, volume 24 (2001), pages 157-201.
C*-categories
The purpose of this paper is to give a detailed study of the basic theory of C*-categories. The study includes some examples of C*-categories that occur naturally in geometric applications, such as groupoid C*-categories, and C*-categories associated to structures in coarse geometry. We conclude the paper with a brief survey of Hilbert modules over C*-categories.
Proceedings of the London Mathematical Society, volume 84 (2002), pages 375-404.
