## Summary

Professor Mark Haskins FLSW is a Professor in Pure Mathematics at Imperial College London. His main research interests lie in differential geometry and geometric analysis; several of his research interests also lie at the intersection between geometry and theoretical physics. Since July 2016 Professor Haskins has been the Deputy Collaboration Director and member of the Steering Committee for the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics. The Collaboration, funded by an $8.5 million grant from the Simons Foundation, is a large-scale international collaboration, directed by Professor Robert Bryant at Duke University. The Collaboration will advance the theory and applications of spaces with special holonomy and the geometric structures—calibrated submanifolds and instantons—associated with them, particularly in the two exceptional cases: spaces with holonomy G_{2} or Spin(7) in 7 or 8 dimensions, respectively.

Haskins joined Imperial College in 2004 as a postdoctoral research fellow working with Professor Sir Simon Donaldson FRS. In 2005 he was appointed Lecturer in Pure Mathematics at Imperial, and promoted to Reader in 2007. From 2009 to 2013 Professor Haskins was a Leadership Fellow of the Engineering and Physical Sciences Research Council (EPSRC) pursuing his research project on Geometric Analysis and Special Lagrangian geometry. In 2013 Haskins was promoted to Professor of Pure Mathematics. From 2013 to 2015 he was an EPSRC Developing Leaders Fellow working on his research project Singular spaces of special and exceptional holonomy. In 2014 he was elected Fellow of the Learned Society of Wales. In 2016 he was a Research Professor at the MSRI programme in Differential Geometry.

## Current Research

Currently his main research interests centre around the geometry of manifolds and singular spaces with special and exceptional holonomy and the geometric objects associated with these spaces — calibrated submanifolds and generalised instantons (generalised anti-self-dual connections). He has made important contributions to the theory of singular special Lagrangian n-folds, compact manifolds with G_{2 }holonomy, associative 3-folds in manifolds with G_{2 }holonomy, noncompact Calabi-Yau manifolds and nearly Kaehler 6-manifolds.

Recently, Haskins and former PhD student Dr Lorenzo Foscolo solved a well-known long-standing (since 1968) foundational problem in nearly Kaehler geometry: do there exist any complete inhomogeneous nearly Kaehler 6-manifolds? They proved that the 6-sphere S^{6 }and the product of a pair of 3-spheres S^{3 }×S^{3} both admit complete inhomogeneous nearly Kaehler structures. Their work will appear in Annals of Mathematics in early 2017.

At present his work is focused on the phenomenon of Riemannian collapse within the context of spaces with special or exceptional holonomy, especially the construction of families of G_{2} holonomy metrics on 7 manifolds that collapse in the limit to 6 dimensional Calabi-Yau spaces. In theoretical physics, M theory is an 11 dimensional physical theory, while String Theories are 10 dimensional theories. To obtain real-world physics in 4 dimensions, M theory must be compactified on a 7 dimensional manifold, while in String theory the compactification space is 6 dimensional. In the simplest cases, supersymmetry forces these spaces to be G_{2} manifolds in the case of M theory, and Calabi-Yau manifolds in the case of String Theories. In various limits it is expected that M theory reduces to a String Theory. In one of these limits the geometric underpinning of the connection between M theory in 11 dimensions and String Theory in 10 dimensions is the collapse of G_{2} holonomy metrics on a 7 manifold to a Calabi-Yau metric on a 6 manifold.

## Postdoctoral position available

A 3-year postdoctoral position at Imperial funded by the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics is currently available. The postdoc will work directly with Professor Haskins on projects in the broad areas of the Collaboration, and within Professor Haskins’ research interests. Further details of the position, including instructions on how to apply, can be found here.

The application deadline is 8 December 2016. (You must apply through the Imperial College job website as described in the adverts above.)

For additional information please email Professor Haskins.

## Research Meetings and Schools

Haskins has organised a number of recent meetings and research schools in geometry and geometric analysis, especially on the subject of the geometry of special or exceptional holonomy spaces.

### Research Meetings

- In January 2017 he is organising a meeting on
**Collapse, adiabatic limits and special holonomy,**at Imperial College as part of the Simons Collaboration on Special Holonomy. - In September 2016 he was the main organiser of the Inaugural Meeting of the Simons Collaboration on Special Holonomy, hosted by the Simons Center for Geometry and Physics, at Stony Brook.
- In February 2015 he coorganised the week-long Oberwolfach miniworkshop Singularities in G
_{2}geometry. - In summer 2014 he coorganised, with Sir Simon Donaldson and Professor Dietmar Salamon, a 6-week long program G
_{2}manifolds at the Simons Center for Geometry and Physics. The successful proposal for a Simons Collaboration on Special Holonomy grew out of this activity.

### Research Schools

Haskins has also organised research schools aimed at early career researchers in geometry.

- In summer 2014 he was the main organiser for the London Mathematical Society/Clay Mathematics Institute Research School, An Invitation to Geometry and Topology via G
_{2}. - In summer 2013 he co-organised a 1-week summer school and a 1-week research workshop Ricci curvature: limit spaces and Kaehler geometry at ICMS Edinburgh. Speakers included: Jeff Cheeger, Sir Simon Donaldson, John Lott, Aaron Naber, Gang Tian and Burkhard Wilking.

## RECENT Invited Lectures and Presentations

### Talks in 2016

*2-part lecture: An Introduction to Ricci-flat spaces and metrics with special and exceptional holonomy,*MSRI, Introductory workshop: Modern Riemannian Geometry, Jan 2016.*Southern California Geometric Analysis conference*, University of California Irvine, Jan 2016.*Bay Area Differential Geometry seminar,*Stanford University, Feb 2016.*Differential Geometry in the Large, conference in honour of Wolfang Meyer's 80th birthday,*Florence, July 2016.*Inaugural meeting of the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics,*Simons Center for Geometry and Physics, Sept 2016.*Geometric flows and the geometry of space-time,*University of Hamburg, Sept 2016.*First Mathematics Colloquium,*Basque Centre for Applied Mathematics and

### Talks in 2015.

*London Geometry and Topology seminar*, Imperial College London, Jan 2015.*Department Colloquium,*University Of Warwick, Mar 2015.*Seminaire d'analyse et geometrie*, Institut de Mathematiques, Jussieu, Paris, Apr 2015.*Geometry and Analysis seminar,*Oxford Mathematical Institute, May 2015.*Oberseminar Differentialgeometrie*, Westfalische Wilhems-Universitat Munster, Nov 2015.*Department of Mathematics and Physics Kolloquium,*Leibniz Universitat Hannover, Nov 2015.

## Further Information

### Geometry-related seminars at Imperial.

I organised the London Geometry and Topology seminar from 2005 to 2013. Professor Paolo Cascini now organises the seminar. About the seminar.

In 2011 Professor Andre Neves and I started the Geometry and Analysis seminar with assistance of several postdocs in geometry.

### London Mathematical Society Journals.

From January 2011 to 2016 I was the editorial adviser responsible for Differential Geometry and Geometric Analysis for the three London Mathematical Society journals: Bulletin of the LMS, Journal of the LMS and Proceedings of the LMS. Dr Felixe Schulze has now taken over as editorial adviser in this area.

The *Proceedings*, *Journal* and *Bulletin* of the London Mathematical Society are among the world's leading mathematical research journals. Although they share a common Editorial Advisory Board, a paper should be submitted directly to one of the journals.

How to submit a paper to one of these LMS journals

## Publications

### Journals

Foscolo L, Haskins M, NordstrÃ¶m J, Infinitely many new families of complete cohomogeneity one G_2-manifolds: G_2 analogues of the Taub-NUT and Eguchi-Hanson spaces

Haskins M, Speight JM, 2002, Breather initial profiles in chains of weakly coupled anharmonic oscillators, *Phys. Lett. A.*, Vol:299, Pages:549-557

Haskins M, Kapouleas N, 2010, Twisted products and $SO(p)\times SO(q)$-invariant special Lagrangian cones

Foscolo L, Haskins M, NordstrÃ¶m J, Complete non-compact G2-manifolds from asymptotically conical Calabi-Yau 3-folds