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; in July 2018 he became Head of Mathematics Discipline at 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.

James has refereed articles for over 30 international journals, see his record at publons.com

James has reviewed over 140 articles for Math Reviews of the American Mathematical Society, see ams.org/mathscinet and search for "McCoy, James" under "Reviewer".

James is an Associate Editor of the Journal of the Australian Mathematical Society.


Qualifications

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

Keywords

  • calculus of variations
  • curvature flow
  • differential geometry
  • geometric evolution problems
  • mathematical analysis
  • mathematical modelling
  • partial differential equations

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

Academic appointment

Dates Title Organisation / Department
15/01/2018 - 31/12/2020 Honourary Principal Fellow University of Wollongong
School of Mathematics and Applied Statistics
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
2019 PHD Mathematics Thesis
2014 PHD Mathematics Thesis
Examination details not publicly available.
2011 PHD Mathematics Thesis
Examination details not publicly available.
2010 PHD Mathematics Mean curvature flow of graphs with free boundaries
Edit

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)
DOI 10.1007/978-3-030-04161-8_16
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 (31 outputs)

Year Citation Altmetrics Link
2019 McCoy J, Wheeler G, 'A rigidity theorem for ideal surfaces with flat boundary', Annals of Global Analysis and Geometry, (2019)
DOI 10.1007/s10455-019-09685-6
2019 McCoy J, Wheeler G, Wu Y, 'Evolution of closed curves by length-constrained curve diffusion', Proceedings of the American Mathematical Society, 147 3493-3506 (2019)
DOI 10.1090/proc/14473
2019 McCoy J, Wheeler G, Wu Y, 'A sixth order curvature flow with boundary conditions', Tohuko Mathematical Journal, (2019)
2018 Sripaturad P, Alshamarri NA, Thamwattana N, McCoy JA, Baowan D, 'Willmore energy for joining of carbon nanostructures', PHILOSOPHICAL MAGAZINE, 98 1511-1524 (2018) [C1]
DOI 10.1080/14786435.2018.1442029
Citations Scopus - 1
Co-authors Natalie Thamwattana
2018 McCoy JA, 'CURVATURE CONTRACTION FLOWS IN THE SPHERE', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 146 1243-1256 (2018)
DOI 10.1090/proc/13831
Citations Scopus - 1Web of Science - 1
2018 Alshammari NA, Thamwattana N, McCoy JA, Baowan D, Cox BJ, Hill JM, 'Modelling joining of various carbon nanostructures using calculus of variations', Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 25 307-339 (2018) [C1]
Co-authors Natalie Thamwattana
2017 McCoy J, Parkins S, Wheeler G, 'The geometric triharmonic heat flow of immersed surfaces near spheres', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 161 44-86 (2017)
DOI 10.1016/j.na.2017.05.016
Citations Scopus - 3
2017 McCoy JA, 'More Mixed Volume Preserving Curvature Flows', JOURNAL OF GEOMETRIC ANALYSIS, 27 3140-3165 (2017)
DOI 10.1007/s12220-017-9799-y
Citations Scopus - 3Web of Science - 3
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 L2-gradient flow of the functional W¿1,¿2 , which is the the sum of the Willmore energy, ¿1-weighted surface area, and ¿2-weighted encl... [more]

In this paper we study the steepest descent L2-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 R3. 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 L2 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 - 3Web of Science - 2
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 - 7
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
Citations Scopus - 1
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
Citations Scopus - 1Web of Science - 1
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 - 3Web of Science - 2
2014 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 (2014)

© 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 - 3Web of Science - 2
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 - 11Web of Science - 8
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 R3. This coincid... [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 R3. 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 L2 norm of tracefree curvature, with no assumption on the growth of the curvature in L2 at infinity. This not only improves the gap lemma due to Kuwert and Schätzle for Willmore surfaces immersed in R3 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 - 14Web of Science - 10
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 - 22Web of Science - 22
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 - 29Web of Science - 28
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 - 31Web of Science - 27
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 - 9Web of Science - 9
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 - 11Web of Science - 9
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 - 6Web 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 - 27Web of Science - 21
Co-authors Natalie Thamwattana
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 - 6Web of Science - 6
2007 McCoy JA, 'BERNSTEIN PROPERTIES OF SOLUTIONS TO SOME HIGHER-ORDER EQUATIONS', DIFFERENTIAL AND INTEGRAL EQUATIONS, 20 1153-1166 (2007)
Citations Web of Science - 4
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 - 38Web of Science - 42
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 - 28Web of Science - 27
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 28 more journal articles

Conference (4 outputs)

Year Citation Altmetrics Link
2017 Dyer T, Thamwattana N, Cox B, McCoy J, 'ICTAM 2016', Intercalation of carbon nanotubes into a graphene sheet, Montreal, Canada (2017)
Co-authors Natalie Thamwattana
2015 Wheeler V-M, Wheeler GE, McCoy JA, Sharples JJ, 'Modelling dynamic bushfire spread: perspectives from the theory of curvature flow', 21ST INTERNATIONAL CONGRESS ON MODELLING AND SIMULATION (MODSIM2015), Gold Coast, AUSTRALIA (2015)
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
2013 Sharples JJ, Towers IN, Wheeler G, Wheeler V-M, McCoy JA, 'Modelling fire line merging using plane curvature flow', 20TH INTERNATIONAL CONGRESS ON MODELLING AND SIMULATION (MODSIM2013), Adelaide, AUSTRALIA (2013)
Citations Web of Science - 7
Show 1 more conference
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Grants and Funding

Summary

Number of grants 15
Total funding $1,730,566

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


20182 grants / $500,866

Parabolic methods for elliptic boundary value problems$306,176

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$194,690

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

20153 grants / $463,300

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

Geometric analysis and curvature flow$10,000

Funding body: University of Wollongong

Funding body University of Wollongong
Project Team

James McCoy, Glen Wheeler, Valentina-Mira Wheeler

Scheme International Links Grant Scheme
Role Lead
Funding Start 2015
Funding Finish 2015
GNo
Type Of Funding Internal
Category INTE
UON N

Study Leave Assistance Grant$2,500

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme Study Leave Assistance Grant
Role Lead
Funding Start 2015
Funding Finish 2015
GNo
Type Of Funding Internal
Category INTE
UON N

20131 grants / $10,000

Curvature flow of surfaces with constraints and boundaries$10,000

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme Near miss grant
Role Lead
Funding Start 2013
Funding Finish 2013
GNo
Type Of Funding Internal
Category INTE
UON N

20123 grants / $353,562

New directions in geometric evolution equations$343,562

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

The surface area preserving mean curvature flow$6,000

Funding body: Monash University

Funding body Monash University
Scheme Postgraduate Publications Award
Role Lead
Funding Start 2012
Funding Finish 2012
GNo
Type Of Funding Internal
Category INTE
UON N

Study leave assistance grant$4,000

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme Study Leave Assistance Grant
Role Lead
Funding Start 2012
Funding Finish 2012
GNo
Type Of Funding Internal
Category INTE
UON N

20111 grants / $2,500

Singularities in fully nonlinear curvature flow$2,500

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme Faculty of Informatics Research Development Scheme
Role Lead
Funding Start 2011
Funding Finish 2011
GNo
Type Of Funding Internal
Category INTE
UON N

20101 grants / $6,000

Flow of convex hypersurfaces to spheres by high powers of curvature$6,000

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme University Research Council Small Grant Scheme
Role Lead
Funding Start 2010
Funding Finish 2010
GNo
Type Of Funding Internal
Category INTE
UON N

20091 grants / $4,000

Study Leave Assistance Grant$4,000

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme Study Leave Assistance Grant
Role Lead
Funding Start 2009
Funding Finish 2009
GNo
Type Of Funding Internal
Category INTE
UON N

20071 grants / $2,500

Helfrich flow of small initial energy surfaces into spheres$2,500

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme Faculty of Informatics Research Development Scheme
Role Lead
Funding Start 2007
Funding Finish 2007
GNo
Type Of Funding Internal
Category INTE
UON N

20061 grants / $8,000

Curvature contraction of nonsmooth, weakly convex hypersurfaces into spheres$8,000

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme University Research Council Small Grant Scheme
Role Lead
Funding Start 2006
Funding Finish 2006
GNo
Type Of Funding Internal
Category INTE
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
Current2

Current Supervision

Commenced Level of Study Research Title Program Supervisor Type
2019 PhD Higher order curvature flow of surfaces Mathematics, University of Wollongong Consultant Supervisor
2017 PhD Higher order curvature flow of planar curves Mathematics, University of Wollongong Consultant Supervisor

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
Edit

Research Projects

Modelling carbon nanostructures 2015 - 2018

In this project we used calculus of variations to model the joining of various carbon nanostructures.

Publications

Sripaturad P, Alshamarri NA, Thamwattana N, McCoy JA, Baowan D, 'Willmore energy for joining of carbon nanostructures', PHILOSOPHICAL MAGAZINE, 98 1511-1524 (2018) [C1]

Alshammari NA, Thamwattana N, McCoy JA, Baowan D, Cox BJ, Hill JM, 'Modelling joining of various carbon nanostructures using calculus of variations', Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 25 307-339 (2018) [C1]

Students

Program Research Title
PhD
University of Wollongong
Mathematical modelling in nanotechnology

Fire front modelling using curvature flow 2013 - 2015

In this project we modelled fire line merging using flow of planar curves by their curvature, a suitable model for uniform fuel over flat terrain (eg grass fires).  We also introduced an evolving surface model that could take into account height dependence from varying topography and fuel properties.

Publications

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)

Sharples JJ, Towers IN, Wheeler G, Wheeler V-M, McCoy JA, 'Modelling fire line merging using plane curvature flow', 20TH INTERNATIONAL CONGRESS ON MODELLING AND SIMULATION (MODSIM2013), Adelaide, AUSTRALIA (2013)

Wheeler V-M, Wheeler GE, McCoy JA, Sharples JJ, 'Modelling dynamic bushfire spread: perspectives from the theory of curvature flow', 21ST INTERNATIONAL CONGRESS ON MODELLING AND SIMULATION (MODSIM2015), Gold Coast, AUSTRALIA (2015)


Modelling protein structure 2007 - 2008

In this project we used calculus of variations to find mathematical energies that would support the regular helical structures commonly seen in proteins.

Publications

Thamwattana N, McCoy JA, Hill JM, 'Energy density functions for protein structures', QUARTERLY JOURNAL OF MECHANICS AND APPLIED MATHEMATICS, 61 431-451 (2008)

Mccoy J, 'Helices for mathematical modelling of proteins, nucleic acids and polymers', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 347 255-265 (2008)

Mccoy J, 'Mathematical modelling of helical protein structure', Protein Structure 105-122 (2011)


Higher order elliptic and parabolic geometric partial differential equations 2007 -

This is a long-term ongoing project covering one of my main research areas: geometrically-inspired higher even order elliptic and parabolic partial differential equations, many eminating from geometric problems in the calculus of variations.  I have supervised PhD students in this field and postdoctoral fellows who have gone on to be successful academic mathematicians in their own rights or found lucrative positions working in industry.

We have considered various scenarios of static (elliptic) and evolving (parabolic) curves, surfaces and hypersurfaces, with or without boundaries.  Much ongoing work and collaborations continue.

Applications of this work include improving understanding of the Helfrich model for biomembranes such as the red blood cell.

Grants

Higher order curvature flow of curves and hypersurfaces

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Scheme Discovery Projects

Publications

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)

McCoy JA, 'BERNSTEIN PROPERTIES OF SOLUTIONS TO SOME HIGHER-ORDER EQUATIONS', DIFFERENTIAL AND INTEGRAL EQUATIONS, 20 1153-1166 (2007)

McCoy J, Wheeler G, Williams G, 'Lifespan theorem for constrained surface diffusion flows', MATHEMATISCHE ZEITSCHRIFT, 269 147-178 (2011)

McCoy J, Wheeler G, 'A classification theorem for Helfrich surfaces', Mathematische Annalen, 357 1485-1508 (2013)

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)

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

McCoy J, Parkins S, Wheeler G, 'The geometric triharmonic heat flow of immersed surfaces near spheres', NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 161 44-86 (2017)

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)

McCoy J, Wheeler G, Wu Y, 'Evolution of closed curves by length-constrained curve diffusion', Proceedings of the American Mathematical Society, 147 3493-3506 (2019)

McCoy J, Wheeler G, Wu Y, 'A sixth order curvature flow with boundary conditions', Tohuko Mathematical Journal, (2019)

McCoy J, Wheeler G, 'A rigidity theorem for ideal surfaces with flat boundary', Annals of Global Analysis and Geometry, (2019)

Students

Program Research Title
PhD
University of Wollongong
A selection of higher-order parabolic curvature flows
PhD
University of Wollongong
Fourth order geometric evolution equations

Second order fully nonlinear elliptic and parabolic partial differential equations related to curvature flow of hypersurfaces 1999 -

I have worked on second order fully nonlinear curvature flow and associated elliptic problems since I was a PhD student.  I continue to work in this area.  I have supervised PhD students in this field.

I have worked in particular on globally constrained curvature flows and unconstrained contraction flows.  I have considered with collaborators nonsmooth speeds and characterised singular behaviour.

Grants

Singularities and surgery in curvature flows

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Scheme Discovery Projects

New directions in geometric evolution equations

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Scheme Discovery Projects

Curvature flow of surfaces with constraints and boundaries

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme Near miss grant

Parabolic methods for elliptic boundary value problems

Funding body: ARC (Australian Research Council)

Funding body ARC (Australian Research Council)
Project Team Associate Professor James McCoy
Scheme Discovery Projects

Flow of convex hypersurfaces to spheres by high powers of curvature

Funding body: University of Wollongong

Funding body University of Wollongong
Scheme University Research Council Small Grant Scheme

Publications

McCoy J, 'The surface area preserving mean curvature flow', Asian Journal of Mathematics, 7 7-30 (2003)

McCoy JA, 'The mixed volume preserving mean curvature flow', MATHEMATISCHE ZEITSCHRIFT, 246 155-166 (2004)

McCoy JA, 'Mixed volume preserving curvature flows', CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 24 131-154 (2005)

McCoy JA, 'A new class of fully nonlinear curvature flows', East Journal on Approximations, 15 349-373 (2009)

McCoy JA, 'Self-similar solutions of fully nonlinear curvature flows', ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE, 10 317-333 (2011)

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)

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)

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 (2014)

Andrews B, Langford M, McCoy J, 'CONVEXITY ESTIMATES FOR HYPERSURFACES MOVING BY CONVEX CURVATURE FUNCTIONS', ANALYSIS & PDE, 7 407-433 (2014)

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)

Ben A, Langford M, McCoy J, 'CONVEXITY ESTIMATES FOR SURFACES MOVING BY CURVATURE FUNCTIONS', JOURNAL OF DIFFERENTIAL GEOMETRY, 99 47-75 (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)

Andrews B, McCoy J, 'Contraction of convex surfaces by nonsmooth functions of curvature', Communications in Partial Differential Equations, 41 1089-1107 (2016)

McCoy JA, 'More Mixed Volume Preserving Curvature Flows', JOURNAL OF GEOMETRIC ANALYSIS, 27 3140-3165 (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)

McCoy JA, 'CURVATURE CONTRACTION FLOWS IN THE SPHERE', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 146 1243-1256 (2018)

Students

Program Research Title
PhD
University of Wollongong
Fully nonlinear curvature flow of axially symmetric hypersurfaces

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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 SR266
Building SR Building
Location Callaghan
University Drive
Callaghan, NSW 2308
Australia
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