McCoy, James

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

Selected Publications

Journal Article

Year	Title
2021	A Length-Constrained Ideal Curve Flow Published In QUARTERLY JOURNAL OF MATHEMATICS
2021	Contracting Self-similar Solutions of Nonhomogeneous Curvature Flows Published In Journal of Geometric Analysis
2021	Contraction of convex hypersurfaces by nonhomogeneous functions of curvature Published In Journal of Evolution Equations
2021	High order curvature flows of plane curves with generalised Neumann boundary conditions Published In Advances in Calculus of Variations
2020	A Sixth Order Curvature Flow of Plane Curves with Boundary Conditions
2020	Evolution of closed curves by length-constrained curve diffusion Published In Proceedings of the American Mathematical Society
2020	A rigidity theorem for ideal surfaces with flat boundary Published In Annals of Global Analysis and Geometry
2020	A sixth order flow of plane curves with boundary conditions Published In Tohoku Mathematical Journal
2020	Closed ideal planar curves Published In Geometry and Topology
2020	Contraction of convex hypersurfaces by nonhomogeneous functions of curvature
2020	The length-constrained ideal curve flow
2019	A Sixth Order Curvature Flow of Plane Curves with Boundary Conditions
2019	Evolution of closed curves by length-constrained curve diffusion Published In Proceedings of the American Mathematical Society
2018	Modelling joining of various carbon nanostructures using calculus of variations Published In Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms
2018	Willmore energy for joining of carbon nanostructures Published In Philosophical Magazine
2017	Curvature contraction flows in the sphere Published In Proceedings of the American Mathematical Society
2017	Curvature contraction of convex hypersurfaces by nonsmooth speeds Published In Journal fuer die Reine und Angewandte Mathematik: Crelle's journal
2017	More mixed volume preserving curvature flows Published In Journal of Geometric Analysis
2017	The geometric triharmonic heat flow of immersed surfaces near spheres Published In Nonlinear Analysis Theory Methods and Applications
2016	Contraction of convex surfaces by nonsmooth functions of curvature Published In Communications in Partial Differential Equations
2016	Finite time singularities for the locally constrained Willmore flow of surfaces Published In Communications in Analysis and Geometry
2015	Convexity estimates for surfaces moving by curvature functions Published In Journal of Differential Geometry
2015	Expansion of co-compact convex spacelike hypersurfaces in Minkowski space by their curvature Published In Indiana University Mathematics Journal
2015	The shrinking figure eight and other solitons for the curve diffusion flow Published In Journal of Elasticity
2014	Convexity estimates for hypersurfaces moving by convex curvature functions Published In Analysis and PDE
2014	Fully nonlinear curvature flow of axially symmetric hypersurfaces Published In No DEA: Nonlinear Differential Equations and Applications
2014	Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions Published In Annali di Matematica Pura ed Applicata
2013	A classification theorem for Helfrich surfaces Published In Mathematische Annalen
2013	Contracting convex hypersurfaces by curvature Published In Calculus of Variations and Partial Differential Equations
2013	Non-collapsing in fully nonlinear curvature flows Published In Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
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
2011	Lifespan theorem for constrained surface diffusion flows Published In Mathematische Zeitschrift
2011	Self-similar solutions of fully nonlinear curvature flows Published In Annali della Scuola Normale - Classe di Scienze
2011	Self-similar solutions of fully nonlinear curvature flows Published In Annali della Scuola Normale Superiore di Pisa: Classe di Scienze
2009	A new class of fully nonlinear curvature flows Published In East Journal on Approximations
2008	Energy density functions for protein structures Published In Quarterly Journal of Mechanics and Applied Mathematics
2008	Helices for mathematical modelling of proteins, nucleic acids and polymers Published In Journal of Mathematical Analysis and Applications
2007	Bernstein properties of solutions to some higher order equations Published In Differential and Integral Equations: an international journal for theory and applications

Chapter

Year	Title
2020	A Rigidity Theorem for Ideal Surfaces with Flat Boundary Published In 2018 MATRIX Annals Publisher Springer International Publishing
2016	The mechanics of ribbons and möbius bands Published In The Mechanics of Ribbons and Möbius Bands Publisher Springer Netherlands
2016	The shrinking figure eight and other solitons for the curve diffusion flow Published In The Mechanics of Ribbons and Möbius Bands Publisher Switzerland
2011	Mathematical modelling of helical protein structure Published In Protein Structure

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
2013	Curvature flows and barriers in fire front modelling Published In 20th International Congress on Modelling and Simulation
2013	Modelling fire line merging using plane curature flow Published In 20th International Congress on Modelling and Simulation

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

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

Edwards, Maureen Associate Professor
Mofarreh, Fatemah Person
Thamwattana, Natalie Dr.
Wheeler, Glen Dr
Wheeler, V.-M Dr
Williams, Graham Professor

Selected Publications

Journal Article

Year	Title
2021	A Length-Constrained Ideal Curve Flow Published In QUARTERLY JOURNAL OF MATHEMATICS
2021	Contracting Self-similar Solutions of Nonhomogeneous Curvature Flows Published In Journal of Geometric Analysis
2021	Contraction of convex hypersurfaces by nonhomogeneous functions of curvature Published In Journal of Evolution Equations
2021	High order curvature flows of plane curves with generalised Neumann boundary conditions Published In Advances in Calculus of Variations
2020	A Sixth Order Curvature Flow of Plane Curves with Boundary Conditions
2020	Evolution of closed curves by length-constrained curve diffusion Published In Proceedings of the American Mathematical Society
2020	A rigidity theorem for ideal surfaces with flat boundary Published In Annals of Global Analysis and Geometry
2020	A sixth order flow of plane curves with boundary conditions Published In Tohoku Mathematical Journal
2020	Closed ideal planar curves Published In Geometry and Topology
2020	Contraction of convex hypersurfaces by nonhomogeneous functions of curvature
2020	The length-constrained ideal curve flow
2019	A Sixth Order Curvature Flow of Plane Curves with Boundary Conditions
2019	Evolution of closed curves by length-constrained curve diffusion Published In Proceedings of the American Mathematical Society
2018	Modelling joining of various carbon nanostructures using calculus of variations Published In Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms
2018	Willmore energy for joining of carbon nanostructures Published In Philosophical Magazine
2017	Curvature contraction flows in the sphere Published In Proceedings of the American Mathematical Society
2017	Curvature contraction of convex hypersurfaces by nonsmooth speeds Published In Journal fuer die Reine und Angewandte Mathematik: Crelle's journal
2017	More mixed volume preserving curvature flows Published In Journal of Geometric Analysis
2017	The geometric triharmonic heat flow of immersed surfaces near spheres Published In Nonlinear Analysis Theory Methods and Applications
2016	Contraction of convex surfaces by nonsmooth functions of curvature Published In Communications in Partial Differential Equations
2016	Finite time singularities for the locally constrained Willmore flow of surfaces Published In Communications in Analysis and Geometry
2015	Convexity estimates for surfaces moving by curvature functions Published In Journal of Differential Geometry
2015	Expansion of co-compact convex spacelike hypersurfaces in Minkowski space by their curvature Published In Indiana University Mathematics Journal
2015	The shrinking figure eight and other solitons for the curve diffusion flow Published In Journal of Elasticity
2014	Convexity estimates for hypersurfaces moving by convex curvature functions Published In Analysis and PDE
2014	Fully nonlinear curvature flow of axially symmetric hypersurfaces Published In No DEA: Nonlinear Differential Equations and Applications
2014	Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions Published In Annali di Matematica Pura ed Applicata
2013	A classification theorem for Helfrich surfaces Published In Mathematische Annalen
2013	Contracting convex hypersurfaces by curvature Published In Calculus of Variations and Partial Differential Equations
2013	Non-collapsing in fully nonlinear curvature flows Published In Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
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
2011	Lifespan theorem for constrained surface diffusion flows Published In Mathematische Zeitschrift
2011	Self-similar solutions of fully nonlinear curvature flows Published In Annali della Scuola Normale - Classe di Scienze
2011	Self-similar solutions of fully nonlinear curvature flows Published In Annali della Scuola Normale Superiore di Pisa: Classe di Scienze
2009	A new class of fully nonlinear curvature flows Published In East Journal on Approximations
2008	Energy density functions for protein structures Published In Quarterly Journal of Mechanics and Applied Mathematics
2008	Helices for mathematical modelling of proteins, nucleic acids and polymers Published In Journal of Mathematical Analysis and Applications
2007	Bernstein properties of solutions to some higher order equations Published In Differential and Integral Equations: an international journal for theory and applications

Chapter

Year	Title
2020	A Rigidity Theorem for Ideal Surfaces with Flat Boundary Published In 2018 MATRIX Annals Publisher Springer International Publishing
2016	The mechanics of ribbons and möbius bands Published In The Mechanics of Ribbons and Möbius Bands Publisher Springer Netherlands
2016	The shrinking figure eight and other solitons for the curve diffusion flow Published In The Mechanics of Ribbons and Möbius Bands Publisher Switzerland
2011	Mathematical modelling of helical protein structure Published In Protein Structure

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
2013	Curvature flows and barriers in fire front modelling Published In 20th International Congress on Modelling and Simulation
2013	Modelling fire line merging using plane curature flow Published In 20th International Congress on Modelling and Simulation

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

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