I am interested in the application of boundaries in mathematical physics.
Boundaries in mathematics have typically been used to force certain nice theorems to hold. For this reason there has been much focus on compactifications and various types of completions, usually with respect to some form of expression of topological information (such as function spaces or distances).
Such a point of view fails to take into account the need to nicely express certain additional types of information present in mathematical physics. For example given a Lorentzian manifold the one point, the Stone-Čeck compactifications and Cauchy completions are unable to provide a boundary which is compatible with the manifolds causal structure.
Indeed there are many examples where one wishes to work with several different structures with different boundary behaviours such that the proper expression of this behaviour requires multiple boundaries and maps between them.
Such an approach has been used implicitly in the General Relativity community for many years. I am interested in the study of boundaries and their relationship to each other in the contexts hinted at above.
Oh, and of course such studies often have applications in pure maths as well….
Physical Predictions of General Relativity
My study of General Relativity (and more generally Lorentzian geometry) is driven by the desire to discover results of physical importance. My interests here lie in two areas, the application of global differential topological techniques and the application of numerical methods. In both case I am interested in the singularities of General Relativity.
At present my efforts in these areas are some what dislocated from each other. It is my firm belief, however, that there is potential for ideas from each side to influence the other. Thus I am currently looking for problems which are amenable to both approaches.
In the mean time I am applying global techniques to a study of the relations between Jacobi fields, gravitational collapse and the boundary structure of Lorentzian manifolds. Results in my thesis indicate that there is room for a new type of singularity theorem that places stricter conditions on the nature of gravitational collapse, while still maintain a strong physical justification.
On the numerical side I am discovering the joy of working with the Weyl tensor at conformal infinity. This work has several applications. One of the more important is the development of a better understanding of the relation between conformal infinity and initial data on a Cauchy surface. This will provide insight into the various boundary value formulations of General Relativity and how to use conformal techniques in numerical work.
Algebraic Approaches to Geometry
My first love was algebra my second was geometry. Naturally when the two meet I’m immediately enamoured.
Currently I’m doing some categorical thinking about partial boundaries on spaces. The idea here is that imposing a particular boundary on a space, in some sense, also imposes additional rigidity to the topological structure. A collection of partial boundaries, cleverly chosen, might give the same boundary information and (hopefully) not introduce this additional rigidity.
The more I read about -algebras the more interested I become, particularly once Dirac operators, Clifford algebras and spin structures are bought up. Eventually I wish to start a project or two in this area.