Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
Mark Walsh's research deals with the relationship between curvature and topology on smooth manifolds. This is a rich subject combining techniques from Geometry, Topology and Analysis. In particular, Walsh is interested in metrics of positive scalar curvature (psc-metrics). Although the problem of whether or not a particular smooth manifold admits a psc-metric has been studied extensively, relatively little is known about the topology of the space of psc-metrics on a given manifold, or it's corresponding moduli spaces. Much of Walsh's work involves the creation of tools for constructing interesting families of psc-metrics. This has helped show, for example, the non-triviality of certain higher homotopy groups of the moduli space of psc-metrics.
Research has been focussed on positively and non-negatively curved Riemannian manifolds, which admit "large" isometric group actions, where "large" can be defined in a number of ways. The existence of an isometric group action G on a metric space X leads to information about the space itself and can be used both as a tool to identify the space and as a means to improve the metric on that space. More recently she has been studying isometric group actions in these two contexts, namely, as a tool to identify both Riemannian manifolds and Alexandrov spaces with a lower curvature bound and as a tool to improve the metric on a Riemannian manifold with a G-invariant metric.
Dr. Jeffres' field of investigation is Riemannian and complex differential geometry. In particular, she is interested in special metrics on manifolds that are noncompact or which have singularities. Special metrics are those for which some curvature is constant. Since the curvature is an expression in the derivatives of the metric, setting this equal to a constant yields a partial differential equation. Therefore, solution of these problems frequently also involves some PDE techniques. Related to this, Dr. Jeffres is also interested in the heat operator.