McCoy, James A.

Top Publications

Year	Title
2013	A classification theorem for Helfrich surfaces Published in Mathematische Annalen
2013	Non-collapsing in fully nonlinear curvature flows Published in Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
2015	Convexity estimates for surfaces moving by curvature functions Published in Journal of Differential Geometry
2013	Contracting convex hypersurfaces by curvature Published in Calculus of Variations and Partial Differential Equations
2011	Self-similar solutions of fully nonlinear curvature flows Published in Annali della Scuola Normale Superiore di Pisa: Classe di Scienze
2014	Convexity estimates for hypersurfaces moving by convex curvature functions Published in Analysis and PDE

Research Areas

0101 - PURE MATHEMATICS

Research Overview

I am a 'geometric analyst' whose main research interests are questions related to curvature driven heat-type flows of hypersurfaces. I am interested in second order and higher order nonlinear evolution equations (and corresponding elliptic problems). Questions of interest include short- and long-time existence, formation of singularities (where the curvature becomes unbounded) and their classifications and extension of classical flow solutions beyond singularities using a 'surgery' procedure. Such analysis relies heavily on techniques from differential geometry, partial differential equations, functional analysis and topology.

Available as Research Supervisor

Head Of

Institute for Mathematics and its Applications Director 2014 - 2017

Member Of

Australian Mathematical Society Member 2001 -
Professionals Australia Member 1999 -

Has Collaborator

Selected Publications

Journal Article

Year	Title
2020	A rigidity theorem for ideal surfaces with flat boundary Published In Annals of Global Analysis and Geometry Volume 57 Pages 1 - 13 ISSN 0232-704X Altmetric
2019	Evolution of closed curves by length-constrained curve diffusion Published In Proceedings of the American Mathematical Society Volume 147 Pages 3493 - 3506 ISSN 1088-6826 Altmetric
2018	Willmore energy for joining of carbon nanostructures Published In Philosophical Magazine Volume Online First Pages 1 - 14 ISSN 1478-6435 Altmetric
2017	Curvature contraction flows in the sphere Published In Proceedings of the American Mathematical Society Volume Online First Pages 1 - 14 ISSN 1088-6826 Altmetric
2017	Curvature contraction of convex hypersurfaces by nonsmooth speeds Published In Journal fuer die Reine und Angewandte Mathematik: Crelle's journal Volume 2017 Pages 169 - 190 ISSN 0075-4102 Altmetric
2017	More mixed volume preserving curvature flows Published In Journal of Geometric Analysis Volume Online First Pages 1 - 26 ISSN 1050-6926 Altmetric
2017	The geometric triharmonic heat flow of immersed surfaces near spheres Published In Nonlinear Analysis Theory Methods and Applications Volume 161 Pages 44 - 86 ISSN 0362-546X Altmetric
2016	Contraction of convex surfaces by nonsmooth functions of curvature Published In Communications in Partial Differential Equations Volume 41 Pages 1089 - 1107 ISSN 0360-5302 Altmetric
2016	Finite time singularities for the locally constrained Willmore flow of surfaces Published In Communications in Analysis and Geometry Volume 24 Pages 843 - 886 ISSN 1019-8385 Altmetric
2015	Convexity estimates for surfaces moving by curvature functions Published In Journal of Differential Geometry Volume 99 Pages 47 - 75 ISSN 0022-040X
2015	Expansion of co-compact convex spacelike hypersurfaces in Minkowski space by their curvature Published In Indiana University Mathematics Journal Volume 64 Pages 635 - 662 ISSN 0022-2518 Altmetric
2015	The shrinking figure eight and other solitons for the curve diffusion flow Published In Journal of Elasticity Volume 119 Pages 191 - 211 ISSN 0374-3535 Altmetric
2014	Convexity estimates for hypersurfaces moving by convex curvature functions Published In Analysis and PDE Volume 7 Pages 407 - 433 ISSN 1948-206X Altmetric
2014	Fully nonlinear curvature flow of axially symmetric hypersurfaces Published In No DEA: Nonlinear Differential Equations and Applications Volume 22 Pages 325 - 343 ISSN 1420-9004 Altmetric
2014	Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions Published In Annali di Matematica Pura ed Applicata Volume 193 Pages 1443 - 1455 ISSN 0373-3114 Altmetric
2013	A classification theorem for Helfrich surfaces Published In Mathematische Annalen Volume 357 Pages 1485 - 1508 ISSN 0025-5831 Altmetric
2013	Contracting convex hypersurfaces by curvature Published In Calculus of Variations and Partial Differential Equations Volume 47 Pages 611 - 665 ISSN 0944-2669 Altmetric
2013	Non-collapsing in fully nonlinear curvature flows Published In Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire Volume 30 Pages 23 - 32 ISSN 0294-1449 Altmetric
2012	Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature Published In Transactions of the American Mathematical Society Volume 364 Pages 3427 - 3447 ISSN 1088-6850 Altmetric
2011	Lifespan theorem for constrained surface diffusion flows Published In Mathematische Zeitschrift Volume 269 Pages 147 - 178 ISSN 0025-5874 Altmetric
2011	Self-similar solutions of fully nonlinear curvature flows Published In Annali della Scuola Normale Superiore di Pisa: Classe di Scienze Volume 10 Pages 317 - 333 ISSN 0391-173X Altmetric
2009	A new class of fully nonlinear curvature flows Published In East Journal on Approximations Volume 15 Pages 349 - 373 ISSN 1310-6236
2008	Energy density functions for protein structures Published In Quarterly Journal of Mechanics and Applied Mathematics Volume 61 Pages 431 - 452 ISSN 0033-5614 Altmetric
2008	Helices for mathematical modelling of proteins, nucleic acids and polymers Published In Journal of Mathematical Analysis and Applications Volume 347 Pages 255 - 265 ISSN 0022-247X Altmetric
2007	Bernstein properties of solutions to some higher order equations Published In Differential and Integral Equations: an international journal for theory and applications Volume 20 Pages 1153 - 1166 ISSN 0893-4983

Chapter

Year	Title
2016	The shrinking figure eight and other solitons for the curve diffusion flow Published In The Mechanics of Ribbons and Möbius Bands Publisher Switzerland: Springer Netherlands Pages 191 - 211 Altmetric
2011	Mathematical modelling of helical protein structure Published In Protein Structure Publisher Nova Science Publishers Pages 105 - 121

Conference Paper

Year	Title
2015	Modelling dynamic bushfire spread: perspectives from the theory of curvature flow Published In MODSIM2015, 21st International Congress on Modelling and Simulation Pages 319 - 325
2013	Curvature flows and barriers in fire front modelling Published In 20th International Congress on Modelling and Simulation Pages 297 - 303
2013	Modelling fire line merging using plane curature flow Published In 20th International Congress on Modelling and Simulation Pages 256 - 262

Featured In

Research & Innovation News Issue Two 2015 Newsletter

Investigator On

Year	Title
2018	Parabolic methods for elliptic boundary value problems Awarded by: Australian Research Council (ARC) Funding Scheme: Discovery Projects
2015	Higher order curvature flow of curves and hypersurfaces Awarded by: Australian Research Council (ARC) Funding Scheme: Discovery Projects
	missing activity

Other Research Activities

ARC Discovery Project "New Directions in Geometric Evolution Equations" Chief Investigator 2012 - 2014
ARC DP Singularities and surgery in curvature flows Chief Investigator and Australian Postdoctoral Fellow 2005 - 2008

Available as Research Supervisor

Potential Supervision Topics

I am available to supervise theses in second or higher order elliptic or parabolic equations specifically related to geometric analysis and/or geometric flow problems.

Advisees

Graduate Advising Relationship

Degree	Research Title	Advisee
Doctor of Philosophy	Gradient Flow of the Dirichlet Energy for the Mean Curvature of Closed Surfaces	Fang, Yanqin
Doctor of Philosophy	Gradient Flow of the Dirichlet Energy for the Curvature of Plane Curves	Wu, Yuhan

Education And Training

Ph.D. in Mathematics, Monash University, Department of Mathematics, The surface area preserving mean curvature flow 1999 - 2002
B Science (Hon) in Mathematics, Monash University, Department of Mathematics, The Ricci flow on surfaces 1995 - 1998
Centre for Mathematics and its Applications 2002 - 2005

Credentials

Fellow of the Australian Mathematical Society, 2014

Awards And Honors

Australian Postdoctoral Fellowship, conferred by Australian Research Council (ARC), 2005

Teaching Overview

I have taught many undergraduate subjects at three Australian universities over the last 15 or so years, to mathematics students and as service subjects to engineering, physics, computer science and IT students. This includes first year single variable calculus, second year multivariable calculus, linear algebra, differential equations and complex analysis. At third year level I have taught calculus of variations, geometry of curves and surfaces and numerical analysis. I maintain an interest in teaching and learning theory of partial differential equations and regularly offer related undergraduate student projects.

Keywords

mathematics, pure mathematics, partial differential equations, differential geometry, geometric analysis, calculus of variations, geometric evolution equations, nonlinear partial differential equations, second order partial differential equations, higher order partial differential equations, elliptic partial differential equations, parabolic partial differential equations

Mailing Address

School of Mathematics and Applied Statistics

University of Wollongong

Wollongong

NSW

2522

Australia

Geographic Location

University of Wollongong

Located In Facility

Orcid Id

http://orcid.org/0000-0001-6053-5144 (confirmed)

Web Of Science Researcher Id

C-8727-2013

Scopus Id

20434234300

Top Publications

Year	Title
2013	A classification theorem for Helfrich surfaces Published in Mathematische Annalen
2013	Non-collapsing in fully nonlinear curvature flows Published in Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
2015	Convexity estimates for surfaces moving by curvature functions Published in Journal of Differential Geometry
2013	Contracting convex hypersurfaces by curvature Published in Calculus of Variations and Partial Differential Equations
2011	Self-similar solutions of fully nonlinear curvature flows Published in Annali della Scuola Normale Superiore di Pisa: Classe di Scienze
2014	Convexity estimates for hypersurfaces moving by convex curvature functions Published in Analysis and PDE

Research Areas

0101 - PURE MATHEMATICS

Research Overview

I am a 'geometric analyst' whose main research interests are questions related to curvature driven heat-type flows of hypersurfaces. I am interested in second order and higher order nonlinear evolution equations (and corresponding elliptic problems). Questions of interest include short- and long-time existence, formation of singularities (where the curvature becomes unbounded) and their classifications and extension of classical flow solutions beyond singularities using a 'surgery' procedure. Such analysis relies heavily on techniques from differential geometry, partial differential equations, functional analysis and topology.

Head Of

Institute for Mathematics and its Applications Director 2014 - 2017

Member Of

Australian Mathematical Society Member 2001 -
Professionals Australia Member 1999 -

Has Collaborator

Selected Publications

Journal Article

Year	Title
2020	A rigidity theorem for ideal surfaces with flat boundary Published In Annals of Global Analysis and Geometry Volume 57 Pages 1 - 13 ISSN 0232-704X Altmetric
2019	Evolution of closed curves by length-constrained curve diffusion Published In Proceedings of the American Mathematical Society Volume 147 Pages 3493 - 3506 ISSN 1088-6826 Altmetric
2018	Willmore energy for joining of carbon nanostructures Published In Philosophical Magazine Volume Online First Pages 1 - 14 ISSN 1478-6435 Altmetric
2017	Curvature contraction flows in the sphere Published In Proceedings of the American Mathematical Society Volume Online First Pages 1 - 14 ISSN 1088-6826 Altmetric
2017	Curvature contraction of convex hypersurfaces by nonsmooth speeds Published In Journal fuer die Reine und Angewandte Mathematik: Crelle's journal Volume 2017 Pages 169 - 190 ISSN 0075-4102 Altmetric
2017	More mixed volume preserving curvature flows Published In Journal of Geometric Analysis Volume Online First Pages 1 - 26 ISSN 1050-6926 Altmetric
2017	The geometric triharmonic heat flow of immersed surfaces near spheres Published In Nonlinear Analysis Theory Methods and Applications Volume 161 Pages 44 - 86 ISSN 0362-546X Altmetric
2016	Contraction of convex surfaces by nonsmooth functions of curvature Published In Communications in Partial Differential Equations Volume 41 Pages 1089 - 1107 ISSN 0360-5302 Altmetric
2016	Finite time singularities for the locally constrained Willmore flow of surfaces Published In Communications in Analysis and Geometry Volume 24 Pages 843 - 886 ISSN 1019-8385 Altmetric
2015	Convexity estimates for surfaces moving by curvature functions Published In Journal of Differential Geometry Volume 99 Pages 47 - 75 ISSN 0022-040X
2015	Expansion of co-compact convex spacelike hypersurfaces in Minkowski space by their curvature Published In Indiana University Mathematics Journal Volume 64 Pages 635 - 662 ISSN 0022-2518 Altmetric
2015	The shrinking figure eight and other solitons for the curve diffusion flow Published In Journal of Elasticity Volume 119 Pages 191 - 211 ISSN 0374-3535 Altmetric
2014	Convexity estimates for hypersurfaces moving by convex curvature functions Published In Analysis and PDE Volume 7 Pages 407 - 433 ISSN 1948-206X Altmetric
2014	Fully nonlinear curvature flow of axially symmetric hypersurfaces Published In No DEA: Nonlinear Differential Equations and Applications Volume 22 Pages 325 - 343 ISSN 1420-9004 Altmetric
2014	Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions Published In Annali di Matematica Pura ed Applicata Volume 193 Pages 1443 - 1455 ISSN 0373-3114 Altmetric
2013	A classification theorem for Helfrich surfaces Published In Mathematische Annalen Volume 357 Pages 1485 - 1508 ISSN 0025-5831 Altmetric
2013	Contracting convex hypersurfaces by curvature Published In Calculus of Variations and Partial Differential Equations Volume 47 Pages 611 - 665 ISSN 0944-2669 Altmetric
2013	Non-collapsing in fully nonlinear curvature flows Published In Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire Volume 30 Pages 23 - 32 ISSN 0294-1449 Altmetric
2012	Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature Published In Transactions of the American Mathematical Society Volume 364 Pages 3427 - 3447 ISSN 1088-6850 Altmetric
2011	Lifespan theorem for constrained surface diffusion flows Published In Mathematische Zeitschrift Volume 269 Pages 147 - 178 ISSN 0025-5874 Altmetric
2011	Self-similar solutions of fully nonlinear curvature flows Published In Annali della Scuola Normale Superiore di Pisa: Classe di Scienze Volume 10 Pages 317 - 333 ISSN 0391-173X Altmetric
2009	A new class of fully nonlinear curvature flows Published In East Journal on Approximations Volume 15 Pages 349 - 373 ISSN 1310-6236
2008	Energy density functions for protein structures Published In Quarterly Journal of Mechanics and Applied Mathematics Volume 61 Pages 431 - 452 ISSN 0033-5614 Altmetric
2008	Helices for mathematical modelling of proteins, nucleic acids and polymers Published In Journal of Mathematical Analysis and Applications Volume 347 Pages 255 - 265 ISSN 0022-247X Altmetric
2007	Bernstein properties of solutions to some higher order equations Published In Differential and Integral Equations: an international journal for theory and applications Volume 20 Pages 1153 - 1166 ISSN 0893-4983

Chapter

Year	Title
2016	The shrinking figure eight and other solitons for the curve diffusion flow Published In The Mechanics of Ribbons and Möbius Bands Publisher Switzerland: Springer Netherlands Pages 191 - 211 Altmetric
2011	Mathematical modelling of helical protein structure Published In Protein Structure Publisher Nova Science Publishers Pages 105 - 121

Conference Paper

Year	Title
2015	Modelling dynamic bushfire spread: perspectives from the theory of curvature flow Published In MODSIM2015, 21st International Congress on Modelling and Simulation Pages 319 - 325
2013	Curvature flows and barriers in fire front modelling Published In 20th International Congress on Modelling and Simulation Pages 297 - 303
2013	Modelling fire line merging using plane curature flow Published In 20th International Congress on Modelling and Simulation Pages 256 - 262

Featured In

Research & Innovation News Issue Two 2015 Newsletter

Investigator On

Year	Title
2018	Parabolic methods for elliptic boundary value problems Awarded by: Australian Research Council (ARC) Funding Scheme: Discovery Projects
2015	Higher order curvature flow of curves and hypersurfaces Awarded by: Australian Research Council (ARC) Funding Scheme: Discovery Projects
	missing activity

Other Research Activities

ARC Discovery Project "New Directions in Geometric Evolution Equations" Chief Investigator 2012 - 2014
ARC DP Singularities and surgery in curvature flows Chief Investigator and Australian Postdoctoral Fellow 2005 - 2008

Potential Supervision Topics

I am available to supervise theses in second or higher order elliptic or parabolic equations specifically related to geometric analysis and/or geometric flow problems.

Advisees

Graduate Advising Relationship

Degree	Research Title	Advisee
Doctor of Philosophy	Gradient Flow of the Dirichlet Energy for the Mean Curvature of Closed Surfaces	Fang, Yanqin
Doctor of Philosophy	Gradient Flow of the Dirichlet Energy for the Curvature of Plane Curves	Wu, Yuhan

Education And Training

Ph.D. in Mathematics, Monash University, Department of Mathematics, The surface area preserving mean curvature flow 1999 - 2002
B Science (Hon) in Mathematics, Monash University, Department of Mathematics, The Ricci flow on surfaces 1995 - 1998
Centre for Mathematics and its Applications 2002 - 2005

Credentials

Fellow of the Australian Mathematical Society, 2014

Awards And Honors

Australian Postdoctoral Fellowship, conferred by Australian Research Council (ARC), 2005

Teaching Overview

I have taught many undergraduate subjects at three Australian universities over the last 15 or so years, to mathematics students and as service subjects to engineering, physics, computer science and IT students. This includes first year single variable calculus, second year multivariable calculus, linear algebra, differential equations and complex analysis. At third year level I have taught calculus of variations, geometry of curves and surfaces and numerical analysis. I maintain an interest in teaching and learning theory of partial differential equations and regularly offer related undergraduate student projects.

Keywords

mathematics, pure mathematics, partial differential equations, differential geometry, geometric analysis, calculus of variations, geometric evolution equations, nonlinear partial differential equations, second order partial differential equations, higher order partial differential equations, elliptic partial differential equations, parabolic partial differential equations

Mailing Address

School of Mathematics and Applied Statistics

University of Wollongong

Wollongong

NSW

2522

Australia

Geographic Location

University of Wollongong

Located In Facility

Orcid Id

http://orcid.org/0000-0001-6053-5144 (confirmed)

Web Of Science Researcher Id

C-8727-2013

Scopus Id

20434234300

Overview

Top Publications

Research Areas

Research Overview

Available as Research Supervisor

Affiliation

Head Of

Member Of

Has Collaborator

Publications

Selected Publications

Journal Article

Chapter

Conference Paper

Featured In

Funding

Investigator On

Other Research Activities

Supervision

Available as Research Supervisor

Potential Supervision Topics

Advisees

Graduate Advising Relationship

Background

Education And Training

Credentials

Awards And Honors

Teaching Overview

Keywords

Contact

Mailing Address

Geographic Location

Located In Facility

Identity

Orcid Id

Web Of Science Researcher Id

Scopus Id

Overview

Top Publications

Research Areas

Research Overview

Affiliation

Head Of

Member Of

Has Collaborator

Publications

Selected Publications

Journal Article

Chapter

Conference Paper

Featured In

Funding

Investigator On

Other Research Activities

Supervision

Potential Supervision Topics

Advisees

Graduate Advising Relationship

Background

Education And Training

Credentials

Awards And Honors

Teaching Overview

Keywords

Contact

Mailing Address

Geographic Location

Located In Facility

Identity

Orcid Id

Web Of Science Researcher Id

Scopus Id

Visualisations

Research Areas