Mathematical Physics

Asymptotically conical (AC) Calabi-Yau manifolds are Ricci-at Kahler manifolds that resemble a Ricci-at Kahler cone at infinity. I will describe an existence theorem for AC Calabi-Yau manifolds which, in particular, yields a refinement of an existence theorem of Tian and Yau for such manifolds. I will also discuss some examples. This is ongoing work with Hans-Joachim Hein.

I will discuss some joint work with K. Uhlenbeck. There is a general method for constructing soliton hierarchies from a splitting of Lie algebras. We explain how formal scattering and inverse scattering, Hamiltonian structures, commuting conservation laws, Backlund transformations, tau functions, and Virasoro actions on tau functions can all be constructed in a uni ed way from such splittings.

When considering flat unitary bundles on a punctured Riemann surface, it is often convenient to have a space that includes all possible holonomies around the punctures; such a space is provided by the extended moduli space of Jeffrey. On the other hand, there are certain inconveniences, in particular no clear link to complex geometry via a Narasimhan-Seshadri type theorem. It turns out that the situation can be remedied quite nicely by considering bundles with framings taking values in a Grassmannian. Analogs for general structure groups, and in particular links with recent work of Martens and Thaddeus, will also be discussed. (This is joint work with U. Bhosle and I. Biswas.)