## Overview

### Differential geometry and geometric analysis.

- manifolds with exceptional holonomy especially manifolds with G
_{2}holonomy. - Riemannian collapse and manifolds with special and exceptional holonomy.
- calibrated geometries, particularly associative submanifolds of compact manifolds with G
_{2}holonomy and singularities of special Lagrangian submanifolds. - geometric PDE gluing constructions of calibrated submanifolds.
- applications of Geometric Measure Theory to calibrated geometry.
- regularity results for calibrated currents.
- differential geometric applications of integrable systems techniques.

### Evolution equations from mathematical physics.

- Solitons and breathers in spatially discrete models.
- Adiabatic/slow-motion limits of geometric nonlinear wave equations, particularly wave maps.

## Funding

**Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics.**July 2016 to 2020.**$8.5m**for collaboration as a whole, Imperial College budget**$1.73m.****EPSRC Grant EP/L001527/1.**Singular spaces of special and exceptional holonomy. July 2013-2015.**£253,271.****EPSRC Leadership Fellowship**EP/G007241/1.*Geometric Analysis and special Lagrangian geometry.*January 2009-2013.**£1,042,335.****EPSRC Small Grant**EP/J014206/1.*Gluing, Rigidity and Uniqueness Questions in Geometric Analysis.*(PI: Jason Lotay UCL, Co-I MH)**£9,634.****Leverhulme Trust Visiting Professorship**, F07058AR. Jan-June 2009. Funded six month visit to Imperial College of Professor Nicos Kapouleas, Brown University,**£41,200.**- Leverhulme lecture notes from Kapouleas' 2009 Leverhulme lecture series delivered via the Taught Course Centre.

## Collaborators

Dr Lorenzo Foscolo, Stony Brook University, Nearly Kaehler geometry and singular spaces of exceptional holonomy.

Dr Johannes Nordstrom, Bath University, Manifolds with exceptional holonomy

Hans-Joachim Hein, Imperial College London, noncompact Kahler Ricci-flat manifolds

Johannes Nordstrom, Imperial College London, Manifolds with exceptional holonomy, G_2 manifolds, calibrated submanifolds, associative submanifolds.

Tommaso Pacini, Scuola Normale Superiore, Pisa, Special Lagrangian geometry, calibrated submanifolds and manifolds with exceptional holonomy.

Corti, Alessio, Maths, Imperial College London

Martin Speight, University of Leeds

Nikolaos Kapouleas, Brown University, Providence RI, Gluing methods in special Lagrangian geometry.

## Guest Lectures

Recent progress in $G_2$ geometry, Perimeter Institute for Theoretical Physics, Waterloo, Canada, 2012

Gluing methods in calibrated geometry, Korean Institute for Advanced Study, Seoul, South Korea, 2012

Analytic methods in special Lagrangian geometry, Taida Institute for Mathematical Sciences, Department of Mathematics, National Taiwan University, Taipei, Taiwan, 2011

## Research Staff

Di Nezza,E

## Research Student Supervision

Foscolo,L, On moduli spaces of periodic monopoles and gravitational instantons

Morris,D, Nearly Kaehler geometry in 6 dimensions