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Mark Haskins

Professor of Mathematics
Mathematics
120 Science Drive, 117 Physics Building, Campus Box 90320, Durham, NC 27708-0320
187 Physics Building, 120 Science Drive, West Campus, Durham, NC 27708-0320

Selected Publications


Cohomogeneity-one solitons in Laplacian flow: local, smoothly-closing and steady solitons

Journal Article · December 16, 2021 We initiate a systematic study of cohomogeneity-one solitons in Bryant's Laplacian flow of closed G_2-structures on a 7-manifold, motivated by the problem of understanding finite-time singularities of that flow. Here we focus on solitons with symmetry grou ... Link to item Cite

Complete noncompact g2-manifolds from asymptotically conical calabi-yau 3-folds

Journal Article Duke Mathematical Journal · October 15, 2021 Featured Publication We develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically G2-manifolds, that is, Riemannian 7- manifolds .M;g/ whose holonomy group is the compact exceptional Lie group G2. Our construction gives t ... Full text Open Access Cite

Infinitely many new families of complete cohomogeneity one G2-manifolds: G2analogues of the Taub-NUT and Eguchi-Hanson spaces

Journal Article Journal of the European Mathematical Society · January 1, 2021 Featured Publication We construct infinitely many new 1-parameter families of simply connected complete non-compact G2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the na ... Full text Cite

New G2-holonomy cones and exotic nearly Kahler structures on S6 and S3 x S3

Journal Article Annals of Mathematics · January 1, 2017 Featured Publication There is a rich theory of so-called (strict) nearly Kahler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kahler 6-manifolds play a distinguished role both ... Full text Open Access Cite

Mini-Workshop: Singularities in $\mathrm G_2$-geometry

Journal Article Oberwolfach Reports · December 4, 2015 Full text Cite

Asymptotically cylindrical Calabi-Yau manifolds

Journal Article Journal of Differential Geometry · October 1, 2015 Let M be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to [0, ∞) x X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M; in particular we sho ... Full text Open Access Cite

G2-Manifolds and associative submanifolds via semi-fano 3-folds

Journal Article Duke Mathematical Journal · January 1, 2015 Featured Publication We construct many new topological types of compact G2-manifolds, that is, Riemannian 7-manifolds with holonomy group G2. To achieve this we extend the twisted connected sum construction first developed by Kovalev and apply it to the large class of asymptot ... Full text Cite

Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds

Journal Article Geometry and Topology · July 15, 2013 We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few h ... Full text Cite

The geometry of SO(p) × SO(q)-invariant special Lagrangian cones

Journal Article Communications in Analysis and Geometry · January 1, 2013 SO(p) × SO(q)-invariant special Lagrangian cones in ℂp+q(equivalently, SO(p) × SO(q)-invariant special Legendrians in S2(p+q)-1) are an important family of special Lagrangians (SL) whose basic features were studied in our previous paper [13]. In some ways, ... Full text Cite

Closed twisted products and SO(p) × SO(q)-invariant special Lagrangian cones

Journal Article Communications in Analysis and Geometry · January 1, 2012 We study a construction we call the twisted product; in this construction higher dimensional special Lagrangian (SL) and Hamiltonian stationary cones in Cp+q (equivalently special Legendrian or contact stationary submanifolds in S2(p+q)?1) are constructed ... Full text Cite

Twisted products and $SO(p)\times SO(q)$-invariant special Lagrangian cones

Journal Article · May 9, 2010 We construct $\sorth{p} \times \sorth{q}$-invariant special Lagrangian (SL) cones in $\C^{p+q}$. These SL cones are natural higher-dimensional analogues of the $\sorth{2}$-invariant SL cones constructed previously by MH and used in our gluing constructions ... Link to item Cite

Special Lagrangian cones with higher genus links

Journal Article Inventiones Mathematicae · February 1, 2007 Featured Publication For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in ℂ3 over a closed Riemann surface of genus g, using a geometric PDE gluing method. © Springer-Verlag 2007. ... Full text Cite

Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem

Journal Article Geometry and Topology · October 21, 2006 We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/nonexistence of such smoothings of special Lagrangian cones is an important component of the cu ... Full text Cite

The geometric complexity of special Lagrangian T2-cones

Journal Article Inventiones Mathematicae · July 29, 2004 Featured Publication We prove a number of results on the geometric complexity of special Lagrangian (SLG) T2-cones in ℂ3. Every SLG T 3-cone has a fundamental integer invariant, its spectral curve genus. We prove that the spectral curve genus of an SLG T2-cone gives a lower bo ... Full text Cite

Special Lagrangian cones

Journal Article American Journal of Mathematics · January 1, 2004 We study special Lagrangian cones in ℂn with isolated singularities especially the case n = 3. Our main result constructs an infinite family of special Lagrangian cones in ℂ3 each of which has a toroidal link. We obtain a detailed geometric description of ... Full text Cite

The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps

Journal Article Journal of Mathematical Physics · August 1, 2003 The most fruitful approach to studying low energy soliton dynamics in field theories of Bogomol'nyi type is the geodesic approximation of Manton. In the case of vortices and monopoles, Stuart has obtained rigorous estimates of the errors in this approximat ... Full text Cite

Breather initial profiles in chains of weakly coupled anharmonic oscillators

Journal Article Physics Letters, Section A: General, Atomic and Solid State Physics · July 15, 2002 A systematic correlation between the initial profile of discrete breathers and their frequency is described. The context is that of a very weakly harmonically coupled chain of softly anharmonic oscillators. The results are structurally stable, that is, rob ... Full text Cite

Breathers in the weakly coupled topological discrete sine-Gordon system

Journal Article Nonlinearity · November 1, 1998 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 ... Full text Cite