Mark Walsh

The topology of a shape is that which is maintained by continuous deformation, i.e. stretching, shrinking but not tearing. A ``topological" shape may take many geometric forms. For example, while a sphere is usually thought of as round, we may alter its shape in various ways and, provided our alterations are continuous and don't tear or puncture the sphere, we still maintain the topological condition of being a sphere. A torus (surface of a bagel) is topologically distinct from a sphere as there is no continuous way of turning one shape into the other. Such a transformation requires cutting or tearing: i.e. something discontinuous.

A large part of modern geometry concerns the problem of finding a ``good" geometric structure on a topological shape, given a plethora of possibilities.  The term ``good" is highly subjective. More broadly however, one may be interested in geometries with a particular property, concerning symmetry or curvature perhaps. Given a geometric constraint, say positive curvature, the problem is to find examples of topological shapes which admit such geometries and to understand what the topological obstructions are in the ones that do not. We know for example that the round geometry is just one of many positive curvature geometries on the sphere. In the standard ``bagel shaped" torus, the inner part has negative curvature. It is a famous theorem of Mathematics that no amount of continuous deformation can give the torus everywhere positive curvature. Thus, because of its topology, the torus can not admit a positive curvature geometry.

One case of this problem is in deciding which smooth manifolds (a particular type of mathematical shape) admit Riemannian metrics (geometric structures) of positive scalar curvature (psc-metrics). This question has attracted a good deal of attention over the years and a number of significant classification results have been achieved. For manifolds which admit psc-metrics there is a related problem, of which far less is known, and which motivates my work. This problem takes the form of the following question.

What is the topology of the space of psc-metrics on a given manifold?

 In other words, what is the shape of the space of geometric structures which satisfy the positive scalar curvature condition. This is a highly complicated infinite dimensional space. This problem is analogous to that of trying to understand the shape of all configurations of a robot arm. The arm itself is a $3$-dimensional object, but the space of all configurations may have many more dimensions, depending on factors such as the number of hinges on the arm. One may think of a path through this space of psc-metrics as an ``animation" of the manifold over time, gradually morphing it from one shape to another, but at every stage satisfying the curvature constraint. One may ask if given two such geometries, it is possible to continuously deform one into the other while maintaining positivity of the scalar curvature at every stage. In other words, is the space of psc-metrics path-connected? More generally, what can be said about so-called higher dimensional connectedness? Finally, what about the analogous questions for other curvature notions such as the Ricci curvature?

In recent years significant strides have been made in answering these questions and it is clear that there is a growing interest in this subject. My work forms part of those efforts.