Associate Professor James McCoy

Associate Professor James McCoy

Associate Professor

School of Mathematical and Physical Sciences

Career Summary

Biography

James McCoy completed his PhD in pure mathematics in 2002 under the supervision of Klaus Ecker at Monash University. The title of his thesis was "The surface area preserving mean curvature flow".  After completing post doctoral positions at the Australian National University with Ben Andrews and Neil Trudinger, in 2005 James took up an ARC Postdoctoral Fellowship as part of a joint ARC grant with Ben Andrews. In July that year he moved to the University of Wollongong, combining a lecturing position with his ARC research grant. In January 2018 he moved to the University of Newcastle.

James is a 'geometric analyst' whose main research interests are questions related to curvature driven heat-type flows of hypersurfaces. Such questions include long time existence of globally constrained curvature flows, formation of singularities (where the curvature becomes unbounded) in curvature flows and extension of classical flow solutions beyond singularities using a 'surgery' procedure. Such analysis relies heavily on techniques from differential geometry, partial differential equations (second and higher even order elliptic and parabolic equations) and topology.


Qualifications

  • Doctor of Philosophy, Monash University
  • Bachelor of Science (Honours), Monash University

Keywords

  • partial differential equations
  • differential geometry
  • mathematical analysis
  • geometric evolution problems

Languages

  • English (Mother)

Fields of Research

Code Description Percentage
010110 Partial Differential Equations 60
010203 Calculus of Variations, Systems Theory and Control Theory 10
010102 Algebraic and Differential Geometry 30

Professional Experience

UON Appointment

Title Organisation / Department
Associate Professor University of Newcastle
School of Mathematical and Physical Sciences
Australia

Professional appointment

Dates Title Organisation / Department
1/07/2005 - 14/01/2018 Lecturer-Associate Professor University of Wollongong
Australia
1/01/2005 - 31/12/2008 ARC Postdoctoral Fellow University of Wollongong
Australia
1/04/2002 - 31/12/2004 Research Associate Australian National University
Centre for Mathematics and its Applications (CMA)
Australia

Thesis Examinations

Year Level Discipline Thesis
2010 PHD Mathematics Mean curvature flow of graphs with free boundaries
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Publications

For publications that are currently unpublished or in-press, details are shown in italics.


Chapter (3 outputs)

Year Citation Altmetrics Link
2018 McCoy JA, Wheeler G, Wu Y, 'A sixth order curvature flow of plane curves with boundary conditions', Elliptic Partial Differential Equations of Second Order: Celebrating 40 Years of Gilbarg and Trudinger¿s Book, Springer, Unknown (2018)
2016 Edwards M, Gerhardt-Bourke A, McCoy J, Wheeler G, Wheeler VM, 'The mechanics of ribbons and möbius bands', The Mechanics of Ribbons and Möbius Bands 191-212 (2016)

© Springer Science+Business Media Dordrecht 2015. In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This reduces... [more]

© Springer Science+Business Media Dordrecht 2015. In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This reduces to studying the curve diffusion flow for the profile curve of the ribbon.We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, translators, and rotators; and (3) an explicit parametrisation of a shrinking figure eight curve.

DOI 10.1007/978-94-017-7300-3
2011 Mccoy J, 'Mathematical modelling of helical protein structure', Protein Structure 105-122 (2011)

We review recent work on modelling helical parts of protein structure mathematically. We use classical calculus of variations and consider mathematical energies dependent on the c... [more]

We review recent work on modelling helical parts of protein structure mathematically. We use classical calculus of variations and consider mathematical energies dependent on the curvature and torsion of the protein backbone curve. We demonstrate classes of mathematical energies for which a given helix, or all helices, are extremisers. A future goal is to find within these energies some which permit less regular protein backbone curves, as also observed in nature. As a step in this direction we incorporate a new function into the model, dependent on position along the backbone curve, which permits some allowance for variability in structure due to different amino acid components, side chains, or obstacles. © 2011 by Nova Science Publishers, Inc. All rights reserved.


Journal article (27 outputs)

Year Citation Altmetrics Link
2018 McCoy JA, 'Curvature contraction flows in the sphere', Proceedings of the American Mathematical Society, 146 1243-1256 (2018)

© 2017 American Mathematical Society. We show that convex surfaces in an ambient three-sphere contract to round points in finite time under fully nonlinear, degree one homogeneous... [more]

© 2017 American Mathematical Society. We show that convex surfaces in an ambient three-sphere contract to round points in finite time under fully nonlinear, degree one homogeneous curvature flows, with no concavity condition on the speed. The result extends to convex axially symmetric hypersurfaces of ð ¿ ¿ ¿ n+1 . Using a different pinching function we also obtain the analogous results for contraction by Gauss curvature.

DOI 10.1090/proc/13831
2018 Sripaturad P, Alshamarri N, Thamwattana N, McCoy JA, Baowan D, 'Willmore energy for joining of carbon nanostructures', PHILOSOPHICAL MAGAZINE, (2018)
DOI 10.1080/14786435.2018.1442029
2017 McCoy J, Parkins S, Wheeler G, 'The geometric triharmonic heat flow of immersed surfaces near spheres', Nonlinear Analysis, Theory, Methods and Applications, 161 44-86 (2017)

© 2017 Elsevier Ltd We consider closed immersed surfaces in R 3 evolving by the geometric triharmonic heat flow. Using local energy estimates, we prove interior estimates and a po... [more]

© 2017 Elsevier Ltd We consider closed immersed surfaces in R 3 evolving by the geometric triharmonic heat flow. Using local energy estimates, we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in L 2 . We further use an e-regularity type result to prove a gap lemma for stationary solutions. Using a monotonicity argument, we then prove that a blowup of the flow approaching a singular time is asymptotic to a non-umbilic embedded stationary surface. This allows us to conclude that any solution with initial L 2 -norm of the tracefree curvature tensor smaller than an absolute positive constant converges exponentially fast to a round sphere with radius equal to 3V 0 /4p3, where V 0 denotes the signed enclosed volume of the initial data.

DOI 10.1016/j.na.2017.05.016
2017 McCoy JA, 'More Mixed Volume Preserving Curvature Flows', Journal of Geometric Analysis, 27 3140-3165 (2017)

© 2017, Mathematica Josephina, Inc. We extend the results of McCoy (Calc Var Partial Differ Equ 24:131¿154, 2005) to include several new cases where convex surfaces evolve to sphe... [more]

© 2017, Mathematica Josephina, Inc. We extend the results of McCoy (Calc Var Partial Differ Equ 24:131¿154, 2005) to include several new cases where convex surfaces evolve to spheres under mixed volume preserving curvature flows, using recent results for unconstrained curvature flows and new regularity arguments in the constrained flow setting. We include results for speeds that are degree 1 homogeneous in the principal curvatures and indicate how, with sufficient curvature pinching conditions on the initial hypersurfaces, some results may be extended to speed homogeneous of degree a > 1. In particular, these extensions require lower speed bounds that are obtained here without using estimates for equations of porous medium type, in contrast to most previous work.

DOI 10.1007/s12220-017-9799-y
2017 Andrews B, Holder A, McCoy J, Wheeler G, Wheeler VM, Williams G, 'Curvature contraction of convex hypersurfaces by nonsmooth speeds', Journal fur die Reine und Angewandte Mathematik, 2017 169-190 (2017)
DOI 10.1515/crelle-2014-0087
Citations Scopus - 1
2016 Andrews B, McCoy J, 'Contraction of convex surfaces by nonsmooth functions of curvature', Communications in Partial Differential Equations, 41 1089-1107 (2016)

© 2016, Copyright © Taylor & Francis Group, LLC. ABSTRACT: We consider the motion of convex surfaces with normal speed given by arbitrary strictly monotone, homogeneous degr... [more]

© 2016, Copyright © Taylor & Francis Group, LLC. ABSTRACT: We consider the motion of convex surfaces with normal speed given by arbitrary strictly monotone, homogeneous degree one functions of the principal curvatures (with no further smoothness assumptions). We prove that such processes deform arbitrary uniformly convex initial surfaces to points in finite time, with spherical limiting shape. This result was known previously only for smooth speeds. The crucial new ingredient in the argument, used to prove convergence of the rescaled surfaces to a sphere without requiring smoothness of the speed, is a surprising hidden divergence form structure in the evolution of certain curvature quantities.

DOI 10.1080/03605302.2016.1202266
2016 Mccoy J, Wheeler G, 'Finite time singularities for the locally constrained willmore flow of surfaces', Communications in Analysis and Geometry, 24 843-886 (2016)

In this paper we study the steepest descent L 2 -gradient flow of the functional W ¿1 , ¿2 , which is the the sum of the Willmore energy, ¿1 -weighted surface area, and ¿2 -weight... [more]

In this paper we study the steepest descent L 2 -gradient flow of the functional W ¿1 , ¿2 , which is the the sum of the Willmore energy, ¿1 -weighted surface area, and ¿2 -weighted enclosed volume, for surfaces immersed in R 3 . This coincides with the Helfrich functional with zero 'spontaneous curvature'. Our first results are a concentration-compactness alternative and interior estimates for the flow. For initial data with small energy, we prove preservation of embeddedness, and by directly estimating the Euler-Lagrange operator from below in L 2 we obtain that the maximal time of existence is finite. Combining this result with the analysis of a suitable blowup allows us to show that for such initial data the flow contracts to a round point in finite time.

DOI 10.4310/CAG.2016.v24.n4.a7
Citations Scopus - 1
2015 Ben A, Langford M, McCoy J, 'CONVEXITY ESTIMATES FOR SURFACES MOVING BY CURVATURE FUNCTIONS', JOURNAL OF DIFFERENTIAL GEOMETRY, 99 47-75 (2015)
Citations Scopus - 5
2015 Andrews B, Chen X, Fang H, McCoy J, 'Expansion of Co-Compact Convex Spacelike Hypersurfaces in Minkowski Space by their Curvature', INDIANA UNIVERSITY MATHEMATICS JOURNAL, 64 635-662 (2015)
DOI 10.1512/iumj.2015.64.5485
2015 Edwards M, Gerhardt-Bourke A, McCoy J, Wheeler G, Wheeler V-M, 'The Shrinking Figure Eight and Other Solitons for the Curve Diffusion Flow', JOURNAL OF ELASTICITY, 119 191-211 (2015)
DOI 10.1007/s10659-014-9502-5
2015 McCoy JA, Mofarreh FYY, Wheeler V-M, 'Fully nonlinear curvature flow of axially symmetric hypersurfaces', NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 22 325-343 (2015)
DOI 10.1007/s00030-014-0287-9
Citations Scopus - 2Web of Science - 1
2014 Andrews B, Langford M, McCoy J, 'CONVEXITY ESTIMATES FOR HYPERSURFACES MOVING BY CONVEX CURVATURE FUNCTIONS', ANALYSIS & PDE, 7 407-433 (2014)
DOI 10.2140/apde.2014.7.407
Citations Scopus - 7Web of Science - 6
2013 McCoy J, Wheeler G, 'A classification theorem for Helfrich surfaces', Mathematische Annalen, 357 1485-1508 (2013)

In this paper we study the functional W ¿1,¿2, , which is the sum of the Willmore energy, ¿ 1 -weighted surface area, and ¿ 2 -weighted volume, for surfaces immersed in R 3 . This... [more]

In this paper we study the functional W ¿1,¿2, , which is the sum of the Willmore energy, ¿ 1 -weighted surface area, and ¿ 2 -weighted volume, for surfaces immersed in R 3 . This coincides with the Helfrich functional with zero 'spontaneous curvature'. Our main result is a complete classification of all smooth immersed critical points of the functional with ¿ 1 =0 and small L 2 norm of tracefree curvature, with no assumption on the growth of the curvature in L 2 at infinity. This not only improves the gap lemma due to Kuwert and Schätzle for Willmore surfaces immersed in R 3 but also implies the non-existence of critical points of the functional satisfying the energy condition for which the surface area and enclosed volume are positively weighted. © 2013 Springer-Verlag Berlin Heidelberg.

DOI 10.1007/s00208-013-0944-z
Citations Scopus - 9
2013 Andrews B, Langford M, McCoy J, 'Non-collapsing in fully non-linear curvature flows', ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, 30 23-32 (2013)
DOI 10.1016/j.anihpc.2012.05.003
Citations Scopus - 20Web of Science - 19
2013 McCoy JA, Mofarreh FYY, Williams GH, 'Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions', Annali di Matematica Pura ed Applicata, 193 1443-1455 (2013)

© 2013, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg. Inspired by earlier results on the quasilinear mean curvature flow, and recent inv... [more]

© 2013, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg. Inspired by earlier results on the quasilinear mean curvature flow, and recent investigations of fully nonlinear curvature flow of closed hypersurfaces which are not convex, we consider contraction of axially symmetric hypersurfaces by convex, degree-one homogeneous fully nonlinear functions of curvature. With a natural class of Neumann boundary conditions, we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Some results continue to hold in the cases of mixed Neumann¿Dirichlet boundary conditions and more general curvature-dependent speeds.

DOI 10.1007/s10231-013-0337-7
Citations Scopus - 1
2013 Andrews B, McCoy J, Zheng Y, 'Contracting convex hypersurfaces by curvature', CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 47 611-665 (2013)
DOI 10.1007/s00526-012-0530-3
Citations Scopus - 21Web of Science - 20
2012 Andrews B, McCoy J, 'CONVEX HYPERSURFACES WITH PINCHED PRINCIPAL CURVATURES AND FLOW OF CONVEX HYPERSURFACES BY HIGH POWERS OF CURVATURE', TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 364 3427-3447 (2012)
DOI 10.1090/S0002-9947-2012-05375-X
Citations Scopus - 24Web of Science - 21
2011 McCoy J, Wheeler G, Williams G, 'Lifespan theorem for constrained surface diffusion flows', MATHEMATISCHE ZEITSCHRIFT, 269 147-178 (2011)
DOI 10.1007/s00209-010-0720-7
Citations Scopus - 7Web of Science - 8
2011 McCoy JA, 'Self-similar solutions of fully nonlinear curvature flows', ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE, 10 317-333 (2011)
Citations Scopus - 9Web of Science - 7
2009 McCoy JA, 'A new class of fully nonlinear curvature flows', East Journal on Approximations, 15 349-373 (2009)
2008 Mccoy J, 'Helices for mathematical modelling of proteins, nucleic acids and polymers', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 347 255-265 (2008)
DOI 10.1016/j.jmaa.2008.05.094
Citations Scopus - 5Web of Science - 5
2008 Thamwattana N, McCoy JA, Hill JM, 'Energy density functions for protein structures', QUARTERLY JOURNAL OF MECHANICS AND APPLIED MATHEMATICS, 61 431-451 (2008)
DOI 10.1093/qjmam/hbn012
Citations Scopus - 24Web of Science - 20
2007 McCoy JA, 'A Bernstein property of solutions to a class of prescribed affine mean curvature equations', ANNALS OF GLOBAL ANALYSIS AND GEOMETRY, 32 147-165 (2007)
DOI 10.1007/s10455-006-9051-7
Citations Scopus - 4Web of Science - 4
2007 McCoy JA, 'Bernstein properties of solutions to some higher-order equations', Differential and Integral Equations, 20 1153-1166 (2007)
2005 McCoy JA, 'Mixed volume preserving curvature flows', CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 24 131-154 (2005)
DOI 10.1007/s00526-004-0316-3
Citations Scopus - 35Web of Science - 37
2004 McCoy JA, 'The mixed volume preserving mean curvature flow', MATHEMATISCHE ZEITSCHRIFT, 246 155-166 (2004)
DOI 10.1007/s00209-003-0592-1
Citations Scopus - 25Web of Science - 24
2003 McCoy J, 'The surface area preserving mean curvature flow', Asian Journal of Mathematics, 7 7-30 (2003)
DOI 10.4310/AJM.2003.v7.n1.a2
Show 24 more journal articles

Conference (3 outputs)

Year Citation Altmetrics Link
2015 Wheeler V-M, Wheeler G, McCoy JA, Sharples JJ, 'Modelling bushfires using curvature flow', Gold Coast (2015)
2013 Sharples JJ, Towers I, Wheeler G, Wheeler V-M, McCoy JA, 'Modelling fire line merging using plane curvature flow', Adelaide (2013)
2013 Wheeler V-M, McCoy JA, Wheeler GE, Sharples JJ, 'Curvature flows and barriers in fire front modelling', 20TH INTERNATIONAL CONGRESS ON MODELLING AND SIMULATION (MODSIM2013), Adelaide, AUSTRALIA (2013)
Citations Web of Science - 4
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Grants and Funding

Summary

Number of grants 5
Total funding $1,626,116

Click on a grant title below to expand the full details for that specific grant.


20182 grants / $475,478

Parabolic methods for elliptic boundary value problems$297,478

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Project Team Associate Professor James McCoy
Scheme Discovery Projects
Role Lead
Funding Start 2018
Funding Finish 2020
GNo G1800524
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON Y

Higher order curvature flow of curves and hypersurfaces$178,000

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Project Team Associate Professor James McCoy, Dr Glen Wheeler, Professor Ben Andrews
Scheme Discovery Projects
Role Lead
Funding Start 2018
Funding Finish 2018
GNo G1800523
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON Y

20151 grants / $450,800

Higher order curvature flow of curves and hypersurfaces$450,800

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Project Team

James McCoy, Glen Wheeler, Ben Andrews

Scheme Discovery Projects
Role Lead
Funding Start 2015
Funding Finish 2018
GNo
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON N

20121 grants / $320,000

New directions in geometric evolution equations$320,000

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Project Team

James McCoy, Ben Andrews

Scheme Discovery Projects
Role Lead
Funding Start 2012
Funding Finish 2014
GNo
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON N

20051 grants / $379,838

Singularities and surgery in curvature flows$379,838

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Project Team

Ben Andrews, James McCoy

Scheme Discovery Projects
Role Investigator
Funding Start 2005
Funding Finish 2008
GNo
Type Of Funding Aust Competitive - Commonwealth
Category 1CS
UON N
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Research Supervision

Number of supervisions

Completed4
Current0

Past Supervision

Year Level of Study Research Title Program Supervisor Type
2017 PhD A selection of higher-order parabolic curvature flows Mathematics, University of Wollongong Principal Supervisor
2015 PhD Fully nonlinear curvature flow of axially symmetric hypersurfaces Mathematics, University of Wollongong Principal Supervisor
2015 PhD Mathematical modelling in nanotechnology Mathematical Sciences, University of Wollongong Principal Supervisor
2009 PhD Fourth order geometric evolution equations Mathematics, University of Wollongong Principal Supervisor
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Research Opportunities

PhD scholarship

A scholarship is available for a domestic student to undertake a PhD in mathematics related to my own research.

Scholarship

Priority Research Centre Computer Assisted Research Mathematics and its Applications

1/01/2018 - 31/12/2018

Contact

Associate Professor James McCoy
University of Newcastle
School of Mathematical and Physical Sciences
james.mccoy@newcastle.edu.au

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Associate Professor James McCoy

Position

Associate Professor
School of Mathematical and Physical Sciences
Faculty of Science

Contact Details

Email james.mccoy@newcastle.edu.au
Phone (02) 4033 9633

Office

Room V236
Building Mathematics Building
Location Callaghan
University Drive
Callaghan, NSW 2308
Australia
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