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

18 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

There is a rich theory of so-called (strict) nearly K¨ahler manifolds,almost-Hermitian manifolds generalising the famous almost complex structureon the 6-sphere induced by octonionic multiplication. Nearly K¨ahler6-manifolds play a distinguished role both in the general structure theoryand also because of their connection with singular spaces with holonomygroup the compact exceptional Lie group G2: the metric cone over a Riemannian6-manifold M has holonomy contained in G2 if and only if M isa nearly K¨ahler 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 K¨ahler 6-manifolds by proving the existence of atleast one cohomogeneity one nearly K¨ahler structure on the 6-sphere andon the product of a pair of 3-spheres. We conjecture that these are theonly simply connected (inhomogeneous) cohomogeneity one nearly K¨ahlerstructures in six dimensions.

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)-manifold and associative submanifolds via semi-fano 3-folds, Duke Mathematical Journal, Vol: 164, Pages: 1971-2092, ISSN: 0012-7094

We construct many new topological types of compact G2G2-manifolds, that is, Riemannian 77-manifolds with holonomy group G2G2. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptotically cylindrical Calabi–Yau 33-folds built from semi-Fano 33-folds constructed previously by the authors. In many cases we determine the diffeomorphism type of the underlying smooth 77-manifolds completely; we find that many 22-connected 77-manifolds can be realized as twisted connected sums in a variety of ways, raising questions about the global structure of the moduli space of G2G2-metrics. Many of the G2G2-manifolds we construct contain compact rigid associative 33-folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/calibrated submanifolds) approach to defining deformation invariants of G2G2-metrics. By varying the semi-Fanos used to build different G2G2-metrics on the same 77-manifold we can change the number of rigid associative 33-folds we produce.

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, Nordström 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

We prove the existence of asymptotically cylindrical (ACyl) Calabi–Yau 3–foldsstarting with (almost) any deformation family of smooth weak Fano 3–folds. Thisallow us to exhibit hundreds of thousands of new ACyl Calabi–Yau 3–folds; previouslyonly a few hundred ACyl Calabi–Yau 3–folds were known. We pay particularattention to a subclass of weak Fano 3–folds that we call semi-Fano 3–folds. SemiFano3–folds satisfy stronger cohomology vanishing theorems and enjoy certaintopological properties not satisfied by general weak Fano 3–folds, but are far morenumerous than genuine Fano 3–folds. Also, unlike Fanos they often contain P1s withnormal bundle O.1/˚ O.1/,giving rise to compact rigid holomorphic curves inthe associated ACyl Calabi–Yau 3–folds.We introduce some general methods to compute the basic topological invariants ofACyl Calabi–Yau 3–folds constructed from semi-Fano 3–folds, and study a smallnumber of representative examples in detail. Similar methods allow the computationof the topology in many other examples.All the features of the ACyl Calabi–Yau 3–folds studied here find application in [17]where we construct many new compact G2 –manifolds using Kovalev’s twistedconnected sum construction. ACyl Calabi–Yau 3–folds constructed from semi-Fano3–folds are particularly well-adapted for this purpose.

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, Hein H-J, Nordström J, 2012, Asymptotically cylindrical Calabi-Yau manifolds

Report

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 and Topology

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, 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 with higher genus links, AM J MATH, Vol: 126, Pages: 845-871, ISSN: 0002-9327

We study special Lagrangian cones in C-n with isolated singularities especially the case n = 3. Our main result constructs an infinite family of special Lagrangian cones in C-3 each of which has a toroidal link. We obtain a detailed geometric description of these tori. We prove a regularity result for special Lagrangian cones in C-3 with a spherical link-any such cone must be a plane. We also construct a one-parameter family of asymptotically conical special Lagrangian submanifolds from any special Lagrangian cone.

Journal article

Haskins M, 2004, The geometric complexity of special Lagrangian <i>T</i><SUP>2</SUP>-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, 1998, Breathers in the weakly coupled topological discrete sine-Gordon system, Nonlinearity, Vol: 11, Pages: 1651-1671

Existence of breather (spatially localized, time periodic, oscillatory) solutions of the topological discrete sine-Gordon (TDSG) system, in the regime of weak coupling, is proved. The novelty of this result is that, unlike the systems previously considered in studies of discrete breathers, the TDSG system does not decouple into independent oscillator units in the weak coupling limit. The results of a systematic numerical study of these breathers are presented, including breather initial profiles and a portrait of their domain of existence in the frequency-coupling parameter space. It is found that the breathers are uniformly qualitatively different from those found in conventional spatially discrete systems.

Journal article

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