Mazzeo's proposed research focuses on a number of themes related to geometric analysis on singular and noncompact spaces. He is studying various types of curvature equations, both on compact stratified spaces and on complete manifolds with asymptotically regular geometries, via both elliptic and parabolic methods. Particular topics here include constant curvature and Einstein metrics with prescribed singular structure, for example with conic points or edges, or for certain problems even on stratified spaces of arbitrary depth, and also the development of Ricci flow techniques on such spaces. He will also conduct research in several parts of spectral geometry on singular spaces, including the study of analytic torsion on manifolds with edges and on smooth manifolds degenerating conically, and on more classical spectral problems on the space of polygons. Other parts of this project include the analysis of a class of degenerate parabolic problems on piecewise smooth, e.g. polyhedral, domains, arising from the Wright-Fisher model in population genetics. He is also investigating the regularity theory for a nonlinear Dirichlet-to-Neumann operator which arises in the study of properly embedded minimal surfaces in hyperbolic space. Finally, he is also analyzing the singular solutions of a class of semilinear Toda-like elliptic systems, which has direct application to some newly introduced string field theories.
In general terms, Mazzeo's research is driven by the central tenet that certain types of singular spaces -- specifically the ones known as stratified spaces -- arise just as naturally as smooth manifolds, which are the most common objects of study in geometry, and both classes of spaces should be considered as comparably important. However, the foundations of geometric analysis on singular spaces are still in a relatively primitive state, and Mazzeo's work is aimed at developing techniques which are meant to be broadly applicable to many natural geometric and analytic problems, both linear and nonlinear, on such spaces. This work is guided by a close examination of many particular problems of recognized importance, arising from both commonly studied questions in pure mathematics and from problems emerging at the interface of mathematics and physics. The expectation is that these natural problems will drive the formulation of the general theory so as to make it accessible and useful, and in turn, this new set of techniques should help answer many problems of interest in these established fields.
