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 interest 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.

The Noncommutative geometry, Analysis and Groups (NAG) seminar series


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