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Florian Hanisch (Potsdam University)
James McCoy (Royal Melbourne Institute of Technology)
Myles Workman (National Taiwan Normal University)
| Time | Speaker/Event |
|---|---|
| 11:00 | Florian Hanisch |
| 12:00 | Lunch |
| 13:30 | Myles Workman |
| 14:30 | Afternoon Tea |
| 15:00 | James McCoy |
Speaker: Florian Hanisch
Title: Relative Traces in Obstacle Scattering
Abstract:
In obstacle scattering, one is interested in properties of the Laplacian $\Delta$ on the complement of a compact set $\mathcal{O}$ (the obstacles) of Euclidean space under suitable boundary conditions. It may be compared with the free Laplacian $\Delta_0$, defined on all of $\mathbb{R}^d$ without the presence of obstacles. For functions $f$ satisfying restrictive assumptions, it is known that differences $f(\Delta) - f(\Delta_0)$ are trace class operators and traces are given by integrals of the Krein spectral shift function associated with $\mathcal{O}$. We will discuss a relative version of this result. Assuming that $\mathcal{O}$ has two connected components, we look at the setting where both obstacles are present relative to the situation, where one of them has been removed. The former is described by the operator $\Delta$; let $\Delta_1$ and $\Delta_2$ denote the Laplacians after removal of an obstacle. We show that the operator $f(\Delta) - f(\Delta_1) - f(\Delta_2) + f(\Delta_0)$ is now trace class for a much larger class of functions $f$. This is important for physical applications when the choice $f(x) = \sqrt{x}$ corresponds to relative (Casimir) energy densities.
Speaker: Myles Workman
Title: Parabolic Rectifiability of the Brakke Flow
Abstract: The Brakke flow is a weak, measure theoretic solution to the mean curvature flow. The goal of this talk is to motivate the formulation of the flow as a single Radon measure over the space-time product.
First, after introducing and motivating the Brakke flow, we will discuss the existence of such a canonical space-time measure, and how this measure characterises the flow. Moreover we will show that important geometric quantities along the flow, the mean curvature vector, the density, and the tangent map, are all measurable with respect to our space-time measure.
Secondly, we will prove that the support of this space-time measure (which can be thought of as the space-time track of the flow), is parabolically rectifiable. An immediate consequence of this is the existence almost everywhere of static planar tangent flows, and the almost everywhere equality of various densities for the flow, i.e. Gausssian density, parabolic density, and the density of time slices.
This is all joint work with Y.T. Liu.
Speaker: James McCoy
Title: TBA
Abstract: TBA