Associate Professor James McCoy
Associate Professor
School of Mathematical and Physical Sciences
 Email:james.mccoy@newcastle.edu.au
 Phone:(02) 4033 9633
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 heattype 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)
Professional Experience
UON Appointment
Title  Organisation / Department 

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

1/12/2019  31/3/2020  Visiting Professor and Excellence Chair in Pure Mathematics  Okinawa Institute of Science and Technology Japan 
15/1/2018  31/12/2020  Honourary Principal Fellow  University of Wollongong School of Mathematics and Applied Statistics Australia 
Professional appointment
Dates  Title  Organisation / Department 

1/7/2005  14/1/2018  LecturerAssociate Professor  University of Wollongong Australia 
1/1/2005  31/12/2008  ARC Postdoctoral Fellow  University of Wollongong Australia 
1/4/2002  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 
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 
Publications
For publications that are currently unpublished or inpress, 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, GerhardtBourke A, McCoy J, Wheeler G, Wheeler VM, 'The mechanics of ribbons and möbius bands', The Mechanics of Ribbons and Möbius Bands 191212 (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.


2011 
Mccoy J, 'Mathematical modelling of helical protein structure', Protein Structure 105122 (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 (35 outputs)
Year  Citation  Altmetrics  Link  

2021 
McCoy J, Wheeler G, Wu Y, 'High order curvature flows of plane curves with generalised Neumann boundary conditions', Advances in Calculus of Variations, (2021) © 2021 Walter de Gruyter GmbH, Berlin/Boston 2021. We consider the parabolic polyharmonic diffusion and the L2{L^{2}}gradient flow for the square integral of the mth arclength d... [more] © 2021 Walter de Gruyter GmbH, Berlin/Boston 2021. We consider the parabolic polyharmonic diffusion and the L2{L^{2}}gradient flow for the square integral of the mth arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L2, then the evolving curve converges exponentially in the C8 topology to a straight horizontal line segment. The same behaviour is shown for the L2gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.


2021  McCoy J, 'Contraction of convex hypersurfaces by nonhomogeneous functions of curvature', Journal of Evolution Equations, (2021)  
2020 
McCoy J, Wheeler G, 'A rigidity theorem for ideal surfaces with flat boundary', Annals of Global Analysis and Geometry, 57 (2020) [C1]


2020  Andrews B, McCoy J, Wheeler G, Wheeler V, 'Closed ideal planar curves', Geometry and Topology, 24 10191049 (2020) [C1]  
2020 
McCoy JA, 'Contracting Selfsimilar Solutions of Nonhomogeneous Curvature Flows', The Journal of Geometric Analysis, (2020)


2020 
Mccoy J, Wheeler G, Wu Y, 'A sixth order flow of plane curves with boundary conditions', Tohoku Mathematical Journal, 72 379393 (2020) [C1] © 2020 Tohoku University, Mathematical Institute. All rights reserved. For smallenergy initial regular planar curves with generalised Neumann boundary conditions, we consider the... [more] © 2020 Tohoku University, Mathematical Institute. All rights reserved. For smallenergy initial regular planar curves with generalised Neumann boundary conditions, we consider the steepestdescent gradient flow for the L2norm of the derivative of curvature with respect to arc length. We show that such curves between parallel lines converge exponentially in the C8topology in infinite time to straight lines.


2019 
McCoy J, Wheeler G, Wu Y, 'Evolution of closed curves by lengthconstrained curve diffusion', Proceedings of the American Mathematical Society, 147 34933506 (2019) [C1]


2018 
Sripaturad P, Alshamarri NA, Thamwattana N, McCoy JA, Baowan D, 'Willmore energy for joining of carbon nanostructures', PHILOSOPHICAL MAGAZINE, 98 15111524 (2018) [C1]


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 307339 (2018) [C1]


2016 
Andrews B, McCoy J, 'Contraction of convex surfaces by nonsmooth functions of curvature', Communications in Partial Differential Equations, 41 10891107 (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.


2016 
Mccoy J, Wheeler G, 'Finite time singularities for the locally constrained willmore flow of surfaces', Communications in Analysis and Geometry, 24 843886 (2016) In this paper we study the steepest descent L2gradient flow of the functional W¿1,¿2 , which is the the sum of the Willmore energy, ¿1weighted surface area, and ¿2weighted encl... [more] In this paper we study the steepest descent L2gradient flow of the functional W¿1,¿2 , which is the the sum of the Willmore energy, ¿1weighted surface area, and ¿2weighted enclosed volume, for surfaces immersed in R3. This coincides with the Helfrich functional with zero 'spontaneous curvature'. Our first results are a concentrationcompactness alternative and interior estimates for the flow. For initial data with small energy, we prove preservation of embeddedness, and by directly estimating the EulerLagrange 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.


2015 
Ben A, Langford M, McCoy J, 'CONVEXITY ESTIMATES FOR SURFACES MOVING BY CURVATURE FUNCTIONS', JOURNAL OF DIFFERENTIAL GEOMETRY, 99 4775 (2015)


2015 
Andrews B, Chen X, Fang H, McCoy J, 'Expansion of CoCompact Convex Spacelike Hypersurfaces in Minkowski Space by their Curvature', INDIANA UNIVERSITY MATHEMATICS JOURNAL, 64 635662 (2015)


2015 
Edwards M, GerhardtBourke A, McCoy J, Wheeler G, Wheeler VM, 'The Shrinking Figure Eight and Other Solitons for the Curve Diffusion Flow', JOURNAL OF ELASTICITY, 119 191211 (2015)


2015 
McCoy JA, Mofarreh FYY, Wheeler VM, 'Fully nonlinear curvature flow of axially symmetric hypersurfaces', NODEANONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 22 325343 (2015)


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 14431455 (2014) © 2013, Fondazione Annali di Matematica Pura ed Applicata and SpringerVerlag 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 SpringerVerlag 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, degreeone 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 curvaturedependent speeds.


2014 
Andrews B, Langford M, McCoy J, 'CONVEXITY ESTIMATES FOR HYPERSURFACES MOVING BY CONVEX CURVATURE FUNCTIONS', ANALYSIS & PDE, 7 407433 (2014)


2013 
McCoy J, Wheeler G, 'A classification theorem for Helfrich surfaces', Mathematische Annalen, 357 14851508 (2013) In this paper we study the functional W¿1,¿2,, which is the sum of the Willmore energy, ¿1weighted surface area, and ¿2weighted 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, ¿1weighted surface area, and ¿2weighted 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 nonexistence of critical points of the functional satisfying the energy condition for which the surface area and enclosed volume are positively weighted. © 2013 SpringerVerlag Berlin Heidelberg.


2013 
Andrews B, Langford M, McCoy J, 'Noncollapsing in fully nonlinear curvature flows', ANNALES DE L INSTITUT HENRI POINCAREANALYSE NON LINEAIRE, 30 2332 (2013)


2013 
Andrews B, McCoy J, Zheng Y, 'Contracting convex hypersurfaces by curvature', CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 47 611665 (2013)


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 34273447 (2012)


2011 
McCoy J, Wheeler G, Williams G, 'Lifespan theorem for constrained surface diffusion flows', MATHEMATISCHE ZEITSCHRIFT, 269 147178 (2011)


2011 
McCoy JA, 'Selfsimilar solutions of fully nonlinear curvature flows', ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISACLASSE DI SCIENZE, 10 317333 (2011)


2008 
Mccoy J, 'Helices for mathematical modelling of proteins, nucleic acids and polymers', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 347 255265 (2008)


2008 
Thamwattana N, McCoy JA, Hill JM, 'Energy density functions for protein structures', QUARTERLY JOURNAL OF MECHANICS AND APPLIED MATHEMATICS, 61 431451 (2008)


2007 
McCoy JA, 'A Bernstein property of solutions to a class of prescribed affine mean curvature equations', ANNALS OF GLOBAL ANALYSIS AND GEOMETRY, 32 147165 (2007)


2007 
McCoy JA, 'BERNSTEIN PROPERTIES OF SOLUTIONS TO SOME HIGHERORDER EQUATIONS', DIFFERENTIAL AND INTEGRAL EQUATIONS, 20 11531166 (2007)


2005 
McCoy JA, 'Mixed volume preserving curvature flows', CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 24 131154 (2005)


2004 
McCoy JA, 'The mixed volume preserving mean curvature flow', MATHEMATISCHE ZEITSCHRIFT, 246 155166 (2004)


Show 32 more journal articles 
Conference (7 outputs)
Year  Citation  Altmetrics  Link  

2020 
McCoy J, Wheeler G, 'A Rigidity Theorem for Ideal Surfaces with Flat Boundary', 2018 MATRIX Annals, Creswick, Victoria (2020) [E1]


2019 
McCoy J, Wheeler G, Wu Y, 'A Sixth Order Curvature Flow of Plane Curves with Boundary
Conditions.', MATRIX, Cresswick, Victoria, Australia (2019)


2017 
Dyer T, Thamwattana N, Cox B, McCoy J, 'ICTAM 2016', Intercalation of carbon nanotubes into a graphene sheet, Montreal, Canada (2017)


2015  Wheeler VM, 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 VM, 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)


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


Show 4 more conferences 
Grants and Funding
Summary
Number of grants  16 

Total funding  $1,812,082 
Click on a grant title below to expand the full details for that specific grant.
20191 grants / $76,050
Visiting Professor in Mathematics$76,050
Under this scheme I conducted research in second and higher order nonlinear parabolic partial differential equations in geometry and also organised a specialist minisymposium with over 30 guests from Australia, China, Japan, Korea, Taiwan, the US and UK, see
https://groups.oist.jp/mathprog/yyysymposium
Funding body: Okinawa Institute of Science and Technology
Funding body  Okinawa Institute of Science and Technology 

Project Team  James McCoy 
Scheme  Visiting Mathematics Professor Program 
Role  Lead 
Funding Start  2019 
Funding Finish  2020 
GNo  
Type Of Funding  External 
Category  EXTE 
UON  N 
20182 grants / $506,332
Parabolic methods for elliptic boundary value problems$311,642
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, ValentinaMira 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 
20122 grants / $347,562
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 
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 
20021 grants / $6,000
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  2002 
Funding Finish  2002 
GNo  
Type Of Funding  Internal 
Category  INTE 
UON  N 
Research Supervision
Number of supervisions
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 higherorder 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 
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 15111524 (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 307339 (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 VM, 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 VM, McCoy JA, 'Modelling fire line merging using plane curvature flow', 20TH INTERNATIONAL CONGRESS ON MODELLING AND SIMULATION (MODSIM2013), Adelaide, AUSTRALIA (2013)
Wheeler VM, 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 431451 (2008)
Mccoy J, 'Helices for mathematical modelling of proteins, nucleic acids and polymers', JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 347 255265 (2008)
Mccoy J, 'Mathematical modelling of helical protein structure', Protein Structure 105122 (2011)
Higher order elliptic and parabolic geometric partial differential equations 2007 
This is a longterm ongoing project covering one of my main research areas: geometricallyinspired 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 147165 (2007)
McCoy JA, 'BERNSTEIN PROPERTIES OF SOLUTIONS TO SOME HIGHERORDER EQUATIONS', DIFFERENTIAL AND INTEGRAL EQUATIONS, 20 11531166 (2007)
McCoy J, Wheeler G, Williams G, 'Lifespan theorem for constrained surface diffusion flows', MATHEMATISCHE ZEITSCHRIFT, 269 147178 (2011)
McCoy J, Wheeler G, 'A classification theorem for Helfrich surfaces', Mathematische Annalen, 357 14851508 (2013)
Edwards M, GerhardtBourke A, McCoy J, Wheeler G, Wheeler VM, 'The Shrinking Figure Eight and Other Solitons for the Curve Diffusion Flow', JOURNAL OF ELASTICITY, 119 191211 (2015)
Mccoy J, Wheeler G, 'Finite time singularities for the locally constrained willmore flow of surfaces', Communications in Analysis and Geometry, 24 843886 (2016)
McCoy J, Parkins S, Wheeler G, 'The geometric triharmonic heat flow of immersed surfaces near spheres', NONLINEAR ANALYSISTHEORY METHODS & APPLICATIONS, 161 4486 (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, 'A sixth order curvature flow with boundary conditions', Tohuko Mathematical Journal, 72 379393 (2020) [C1]
McCoy J, Wheeler G, 'A rigidity theorem for ideal surfaces with flat boundary', Annals of Global Analysis and Geometry, 57 (2020) [C1]
Students
Program  Research Title 

PhD University of Wollongong 
A selection of higherorder 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, 7 730 (2003)
McCoy JA, 'The mixed volume preserving mean curvature flow', MATHEMATISCHE ZEITSCHRIFT, 246 155166 (2004)
McCoy JA, 'Mixed volume preserving curvature flows', CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 24 131154 (2005)
McCoy JA, 'A new class of fully nonlinear curvature flows', East Journal on Approximations, 15 349373 (2009)
McCoy JA, 'Selfsimilar solutions of fully nonlinear curvature flows', ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISACLASSE DI SCIENZE, 10 317333 (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 34273447 (2012)
Andrews B, Langford M, McCoy J, 'Noncollapsing in fully nonlinear curvature flows', ANNALES DE L INSTITUT HENRI POINCAREANALYSE NON LINEAIRE, 30 2332 (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 14431455 (2014)
Andrews B, Langford M, McCoy J, 'CONVEXITY ESTIMATES FOR HYPERSURFACES MOVING BY CONVEX CURVATURE FUNCTIONS', ANALYSIS & PDE, 7 407433 (2014)
McCoy JA, Mofarreh FYY, Wheeler VM, 'Fully nonlinear curvature flow of axially symmetric hypersurfaces', NODEANONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS, 22 325343 (2015)
Ben A, Langford M, McCoy J, 'CONVEXITY ESTIMATES FOR SURFACES MOVING BY CURVATURE FUNCTIONS', JOURNAL OF DIFFERENTIAL GEOMETRY, 99 4775 (2015)
Andrews B, Chen X, Fang H, McCoy J, 'Expansion of CoCompact Convex Spacelike Hypersurfaces in Minkowski Space by their Curvature', INDIANA UNIVERSITY MATHEMATICS JOURNAL, 64 635662 (2015)
Andrews B, McCoy J, 'Contraction of convex surfaces by nonsmooth functions of curvature', Communications in Partial Differential Equations, 41 10891107 (2016)
McCoy JA, 'More Mixed Volume Preserving Curvature Flows', JOURNAL OF GEOMETRIC ANALYSIS, 27 31403165 (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 169190 (2017)
McCoy JA, 'CURVATURE CONTRACTION FLOWS IN THE SPHERE', PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 146 12431256 (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
College of Engineering, Science and Environment
Contact Details
james.mccoy@newcastle.edu.au  
Phone  (02) 4033 9633 
Office
Room  SR266 

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