Victorian GAP Seminar 2026

Speakers:
Florian Hanisch (Potsdam University)
James McCoy (Royal Melbourne Institute of Technology)
Myles Workman (National Taiwan Normal University)

Schedule:
Time Speaker/Event
11:00 Florian Hanisch
12:00 Lunch
13:30 Myles Workman
14:30 Afternoon Tea
15:00 James McCoy

Titles and Abstracts:

Speaker: Florian Hanisch

Title: Relative Traces in Obstacle Scattering

Abstract: In obstacle scattering, one is interested in properties of the Laplacian Δ on the complement of a compact set 𝒪 (the obstacles) of Euclidean space under suitable boundary conditions. It may be compared with the free Laplacian Δ₀, defined on all of ℝᵈ without the presence of obstacles. For functions f satisfying restrictive assumptions, it is known that differences f(Δ) − f(Δ₀) are trace class operators and traces are given by integrals of the Krein spectral shift function associated with 𝒪.

We will discuss a relative version of this result. Assuming that 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 Δ; let Δ₁ and Δ₂ denote the Laplacians after removal of an obstacle. We show that the operator f(Δ) − f(Δ₁) − f(Δ₂) + f(Δ₀) is now trace class for a much larger class of functions f. This is important for physical applications when the choice f(x) = √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: Axially symmetric surface diffusion with generalised Neumann boundary conditions between planes

Abstract: The surface diffusion flow was introduced by Mullins in the 1950s to model thermal grooving in metals. Together with Nichols, Mullins then studied this flow in the setting of axially-symmetric surfaces given applications to blunting of field emission tips and spheroidization of cylindrical rods. Previous work by Lecrone and Simonett investigated axially symmetric hypersurfaces evolving by the surface diffusion flow, showing in particular that initial hypersurfaces C²,ᵅ-close to cylinders of radius r > 1 lead to long-time solutions that converge as t → ∞ to cylinders. We replace that condition with a geometric smallness condition, namely the initial axially-symmetric hypersurface has L² norm of the ‘axial curvature’ sufficiently small and is sufficiently far from the axis of rotation. We show the solution remains away from the axis of rotation and, in particular, not only is the smallness condition preserved, but it decays exponentially, as do enough curvature derivatives in L² to invoke a classical linearised stability argument at a finite time. We use the stability argument to obtain long time existence and exponential convergence to a cylinder. This is joint work with Mashniah Gazwani.