Principal Investigator: Peter B. Gilkey
Gilkey will study time dependent processes which are controlled by the heat equation and the short time asymptotics which arise thereby. The first project involves the heat content asymptotics, moving from the static setting to the setting where the data (metric, internal heat sources, boundary conditions) are time dependent. Previous work always assumed smooth boundary conditions; in this project the boundary conditions are discontinuous. The second project involves studying the asymptotics of the short time expansion of the fundamental solution of the heat equation. Most previous studies have involved static geometries, but it is natural to study these asymptotic expansions for time dependent geometries for quite general operators of Laplace type and for quite general homogeneous boundary conditions. This investigation will involve expanding the usual calculus of pseudo differential operators to establish the existence of the required short time asymptotics as well as determining how the time dependent variation of the metric and the coupling constants for Neumann boundary conditions influences the short time asymptotics of the heat kernel.
There are many physical settings where boundary conditions are discontinuous. For example, a body floating in ice water satisfies Dirichlet boundary conditions on the part immersed in the water and Neumann boundary conditions on the remainder (assuming as a first approximation that there is no heat transfer from the air to the body); the boundary conditions are discontinuous along the water/air interface. Understanding additional boundary contributions coming from the water/air interface are likely to be of great physical importance, and links the differential geometry of the situation to the physical underpinnings of the subject. There are also many conditions where the geometry is not static; the Universe is expanding to cite one example. Boundary conditions in physical settings may vary with time; for example, the temperature outside a building varies with the time of day as well as with the season; consequently the associated heat flow is not well modeled by a static setting. Understanding the heat flow in this setting has obvious physical applications. The heat equation asymptotics and heat content asymptotics have proven to be of central importance in theoretical physics. For example, these asymptotics play an important role in the renormalization of field theory in curved space and also in string and membrane theory, and they are related to zeta function renormalization.
