Mathematics

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. 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.

C*-categories

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 centres on 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.

Articles

The following articles are in DVI format. You can reach me by e-mail at mitch@uni-math.gwdg.de if you have any enquiries.

• KK-theory spectra for C*-categories and discrete groupoid C*-algebras
  In this paper we refine a version of bivariant K-theory developed by Cuntz to define symmetric spectra representing the KK-theory of C*-categories and discrete groupoid C*-algebras. In both cases, the Kasparov product can be expressed as a smash product of spectra.
  
• Algebraic K-theory Spectra and Factorisations of Analytic Assembly Maps
  In this article we use existing machinery to define connective K-theory spectra associated to topological ringoids. Algebraic K-theory of discrete ringoids, and the analytic K-theory of Banach categories are obtained as special cases. As an application, we show how the analytic assembly maps featuring in the Novikov and Baum-Connes conjectures can be factorised into composites of assembly maps resembling those appearing in algebraic K-theory and maps coming from completions of certain topological ringoids into Banach categories. These factorisations are proved by using existing characterisations of assembly maps along with our unified picture of algebraic and analytic K-theory.
• 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.
• 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

In this article 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

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