Imperial College London

Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries.

The presence of additives, which may or may not be surface-active, can have a dramatic influence on interfacial flows. The presence of surfactants alters the interfacial tension and drives Marangoni flow that leads to fingering instabilities in drops spreading on ultra-thin films. Surfactants also play a major role in coating flows, foam drainage, jet breakup and may be responsible for the so-called ``super-spreading" of drops on hydrophobic substrates. The addition of surface-inactive nano-particles to thin films and drops also influences the interfacial dynamics and has recently been shown to accelerate spreading and to modify the boiling characteristics of nanofluids. These findings have been attributed to the structural component of the disjoining pressure resulting from the ordered layering of nanoparticles in the region near the contact line. In this talk, we present a collection of results which demonstrate that the above-mentioned effects of surfactants and nano-particles can be captured using long-wave models.

I will show that generating functions for certain non-compact Calabi-Yau 3-folds are modular forms. This is joint work with Hiroshi Iritani.

I will show that generating functions for certain non-compact

Calabi-Yau 3-folds are modular forms. This is joint work with Hiroshi

Iritani.

I will describe some more of the deformation theory necessary for the first talk. This leads to a number of natural questions and counterexamples. This talk requires a strong stomach, or a fanatical devotion to symmetric obstruction theories.

The Katz-Klemm-Vafa formula is a conjecture expressing Gromov-Witten invariants of K3 surfaces in terms of modular forms. In genus 0 it reduces to the (proved) Yau-Zaslow formula. I will explain how the correspondence between stable pairs and Gromov-Witten theory for toric 3-folds (proved by Maulik-Oblomkov-Okounkov-Pandharipande), some calculations with stable pairs (due to Kawai-Yoshioka) and some deformation theory lead to a proof of the KKV formula.
(This is joint work with Davesh Maulik and Rahul Pandharipande. Only they understand the actual formulae. People who like modular forms are not encouraged to come to this talk.)

This talk will be largely speculative. First we consider the formal properties that could be expected of a "topological field theory" in 6+1 dimensions defined by $G_2$ instantons. We explain that this could lead to holomorphic bundles over moduli spaces of Calabi-Yau 3-folds whose ranks are the DT-invariants. We also discuss in more detail the compactness problem for $G_2$ instantons and associative submanifolds.

The talk will be held in Room 408, Imperial College Maths Department, Huxley Building, 180 Queen’s Gate, London.