Imperial College London

Professor Mark Haskins FLSW

Faculty of Natural SciencesDepartment of Mathematics

Visiting Professor
 
 
 
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Contact

 

+44 (0)20 7594 8550m.haskins CV

 
 
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Location

 

668Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

21 results found

Foscolo L, Haskins M, 2017, New G(2)-holonomy cones and exotic nearly Kahler structures on S-6 and S-3 x S-3, ANNALS OF MATHEMATICS, Vol: 185, Pages: 59-130, ISSN: 0003-486X

JOURNAL ARTICLE

Haskins M, Hein H-J, Nordstroem J, 2015, ASYMPTOTICALLY CYLINDRICAL CALABI-YAU MANIFOLDS, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 101, Pages: 213-265, ISSN: 0022-040X

JOURNAL ARTICLE

Corti A, Haskins M, Nordstroem J, Pacini Tet al., 2015, G(2)-MANIFOLDS AND ASSOCIATIVE SUBMANIFOLDS VIA SEMI-FANO 3-FOLDS, DUKE MATHEMATICAL JOURNAL, Vol: 164, Pages: 1971-2092, ISSN: 0012-7094

JOURNAL ARTICLE

Foscolo L, Haskins M, 2015, New G2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres

There is a rich theory of so-called (strict) nearly Kaehler manifolds,almost-Hermitian manifolds generalising the famous almost complex structure onthe 6-sphere induced by octonionic multiplication. Nearly Kaehler 6-manifoldsplay a distinguished role both in the general structure theory and also becauseof their connection with singular spaces with holonomy group the compactexceptional Lie group G2: the metric cone over a Riemannian 6-manifold M hasholonomy contained in G2 if and only if M is a nearly Kaehler 6-manifold. A central problem in the field has been the absence of any completeinhomogeneous examples. We prove the existence of the first completeinhomogeneous nearly Kaehler 6-manifolds by proving the existence of at leastone cohomogeneity one nearly Kaehler structure on the 6-sphere and on theproduct of a pair of 3-spheres. We conjecture that these are the only simplyconnected (inhomogeneous) cohomogeneity one nearly Kaehler structures in sixdimensions.

WORKING PAPER

Degeratu A, Haskins M, Weiß H, 2015, Mini-Workshop: Singularities in $\mathrm G_2$-geometry, Oberwolfach Reports, Vol: 12, Pages: 449-488, ISSN: 1660-8933

JOURNAL ARTICLE

Corti A, Haskins M, Nordstroem J, Pacini Tet al., 2013, Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds, GEOMETRY & TOPOLOGY, Vol: 17, Pages: 1955-2059, ISSN: 1465-3060

JOURNAL ARTICLE

Haskins M, Kapouleas N, 2012, Closed twisted products and SO(p) X SO(q)-invariant special Lagrangian cones, COMMUNICATIONS IN ANALYSIS AND GEOMETRY, Vol: 20, Pages: 95-162, ISSN: 1019-8385

JOURNAL ARTICLE

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

We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL)cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues ofthe $\sorth{2}$-invariant SL cones constructed previously by MH and used in ourgluing constructions of higher genus SL cones in $\C^{3}$. We study in detailthe geometry of these $\sorth{p}\times \sorth{q}$-invariant SL cones, inpreparation for their application to our higher dimensional special Legendriangluing constructions. In particular the symmetries of these cones and theirasymptotics near the spherical limit are analysed. All $\sorth{p} \times\sorth{q}$-invariant SL cones arise from a more general construction ofindependent interest which we call the special Legendrian twisted productconstruction. Using this twisted product construction and simple variants of itwe can construct a constellation of new special Lagrangian and Hamiltonianstationary cones in $\C^{n}$. We prove the following theorems: A. there areinfinitely many topological types of special Lagrangian and Hamiltonianstationary cones in $\C^{n}$ for all $n\ge 4$, B. for $n\ge 4$ specialLagrangian and Hamiltonian stationary torus cones in $\C^{n}$ can occur incontinuous families of arbitrarily high dimension and C. for $n\ge 6$ there areinfinitely many topological types of special Lagrangian and Hamiltonianstationary cones in $\C^{n}$ that can occur in continuous families ofarbitrarily high dimension.

JOURNAL ARTICLE

Haskins M, Kapouleas N, 2008, Gluing Constructions of Special Lagrangian Cones, Handbook of geometric analysis, Editors: Ji, Publisher: International Press, Pages: 77-145, ISBN: 978-1-57146-130-8

We survey our recent work constructing new special Lagrangian cones in complex n-space for all n greater than 3 by gluing methods.

BOOK CHAPTER

Haskins M, Kapouleas N, 2007, Special Lagrangian cones with higher genus links, INVENTIONES MATHEMATICAE, Vol: 167, Pages: 223-294, ISSN: 0020-9910

JOURNAL ARTICLE

Haskins M, Pacini T, 2006, Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem, Geometry & Topology, Vol: 10, Pages: 1453-1521, ISSN: 1465-3060

JOURNAL ARTICLE

Haskins M, Pacini T, 2006, Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem, GEOMETRY & TOPOLOGY, Vol: 10, Pages: 1453-1521, ISSN: 1364-0380

JOURNAL ARTICLE

Haskins M, 2005, Special Lagrangian T^2-cones via spectral curves and spectral geometry, Fukuoka, Japan, Integrable Systems, Geometry and Visualization, Publisher: Kyushu University, Pages: 13-28

CONFERENCE PAPER

Haskins M, 2004, Special Lagrangian cones, AMERICAN JOURNAL OF MATHEMATICS, Vol: 126, Pages: 845-871, ISSN: 0002-9327

JOURNAL ARTICLE

Haskins M, 2004, The geometric complexity of special Lagrangian T-2-cones, INVENTIONES MATHEMATICAE, Vol: 157, Pages: 11-70, ISSN: 0020-9910

JOURNAL ARTICLE

Haskins M, Speight JM, 2003, The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps, JOURNAL OF MATHEMATICAL PHYSICS, Vol: 44, Pages: 3470-3494, ISSN: 0022-2488

JOURNAL ARTICLE

Haskins M, Speight JM, 2002, Breather initial profiles in chains of weakly coupled anharmonic oscillators, PHYSICS LETTERS A, Vol: 299, Pages: 549-557, ISSN: 0375-9601

JOURNAL ARTICLE

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

A systematic correlation between the initial profile of discrete breathersand their frequency is described. The context is that of a very weaklyharmonically coupled chain of softly anharmonic oscillators. The results arestructurally stable, that is, robust under changes of the on-site potential andare illustrated numerically for several standard choices. A precise genericitytheorem for the results is proved.

JOURNAL ARTICLE

Haskins M, Speight JM, 1998, Breathers in the weakly coupled topological discrete sine-Gordon system, NONLINEARITY, Vol: 11, Pages: 1651-1671, ISSN: 0951-7715

JOURNAL ARTICLE

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

We construct infinitely many new 1-parameter families of simply connectedcomplete noncompact G_2-manifolds with controlled geometry at infinity. Thegeneric member of each family has so-called asymptotically locally conical(ALC) geometry. However, the nature of the asymptotic geometry changes at twospecial parameter values: at one special value we obtain a unique member ofeach family with asymptotically conical (AC) geometry; on approach to the otherspecial parameter value the family of metrics collapses to an AC Calabi-Yau3-fold. Our infinitely many new diffeomorphism types of AC G_2-manifolds areparticularly noteworthy: previously the three examples constructed by Bryantand Salamon in 1989 furnished the only known simply connected AC G_2-manifolds. We also construct a closely related conically singular G_2 holonomy space:away from a single isolated conical singularity, where the geometry becomesasymptotic to the G_2-cone over the standard nearly Kaehler structure on theproduct of a pair of 3-spheres, the metric is smooth and it has ALC geometry atinfinity. We argue that this conically singular ALC G_2-space is the naturalG_2 analogue of the Taub-NUT metric in 4-dimensional hyperKaehler geometry andthat our new AC G_2-metrics are all analogues of the Eguchi-Hanson metric, thesimplest ALE hyperKaehler manifold. Like the Taub-NUT and Eguchi-Hansonmetrics, all our examples are cohomogeneity one, i.e. they admit an isometricLie group action whose generic orbit has codimension one.

JOURNAL ARTICLE

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

We develop a powerful new analytic method to construct complete non-compactG2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is thecompact exceptional Lie group G2. Our construction starts with a completenon-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle Mover B satisfying a necessary topological condition. Our method then produces a1-parameter family of circle-invariant complete G2-metrics on M that collapsesto the original Calabi-Yau metric on the base B as the parameter converges to0. The G2-metrics we construct have controlled asymptotic geometry at infinity,so-called asymptotically locally conical (ALC) metrics, and are the naturalhigher-dimensional analogues of the ALF metrics that are well known in4-dimensional hyperk\"ahler geometry. We give two illustrations of the strengthof our method. Firstly we use it to construct infinitely many diffeomorphismtypes of complete non-compact simply connected G2-manifolds; previously only ahandful of such diffeomorphism types was known. Secondly we use it to prove theexistence of continuous families of complete non-compact G2-metrics ofarbitrarily high dimension; previously only rigid or 1-parameter families ofcomplete non-compact G2-metrics were known.

JOURNAL ARTICLE

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