Australian National University

There has been a great deal of interest in understanding the link between the axiomatic descriptions of conformal field theory given by vertex operator algebras and conformal nets. In recent work, we establish an equivalence between certain vertex algebras and conformally-symmetric quantum field theories in the sense of Wightman. In this talk I will give an overview of these results and discuss some of the difficulties that arise, the functional analytic properties of vertex algebras, and some of the ideas for future work in this area.

This is joint work with James Tener and Yoh Tanimoto.

The Persistent Homology Transform, and the Euler Characteristic Transform are topological analogs of the Radon transform that can be used in statsistical shape analysis. In this talk I will consider an interesting variant called the Extended Persistent Homology Transform (XPHT) which replaces the normal persistent homology with extended persistent homology. We are particularly interested in the application of the XPHT to binary images. This paper outlines an algorithm for efficient calculation of the XPHT exploting relationships between the PHT of the boundary curves to the XPHT of the foreground.

We present a selection of recent results pertaining to Hessian

and Monge-Ampere equations, where the Hessian matrix is augmented by a

matrix valued lower order operator. Equations of this type arise in

conformal geometry, geometric optics and optimal transportation.In

particular we will discuss structure conditions, due to Ma,Wang and

myself, which imply the regularity of solutions.These conditions are a

refinement of a condition used originally by Pogorelev for general

equations of Monge-Ampere type in two variables and called strong

ellipticity by him.