Research in Mathematics

Research in Mathematics

Research in Mathematics

Research in the Department of Mathematics at Macquarie University is recognised nationally and internationally, with groupings of mathematicians working in analysis, category theory, fluid mechanics, nonlinear dynamics, number theory, optimisation and waves. The Department attracts a substantial amount of external research funding and hosts a steady stream of international visitors. 

The Department's research is supported by a number of vibrant seminars series, attracting both domestic and international speakers. In addition the Department's Colloquium, aimed at a general mathematical audience, features talks by visitors to the Department, both international and domestic, as well as the occasional talk by members of the department.

Our international collaborations are many and varied. Members of the Department work with colleagues at many of the other leading Australian Mathematics Departments as well as from many high profile international mathematics departments, including Universit√© de recherche Paris Sciences et Lettres, Universit√© de Bordeaux, Sun Yat-sen University, Central China Normal University, University of Birmingham, University of Wroclaw, INRIA (France), University of Bundeswehr, University of Bayreuth, University of Brest, Penn State University, University of Manchester amongst many others.

Our major programs of research are detailed below.

Analysis

The research interests of the Analysis Group span functional analysis, harmonic analysis, Lie groups, symmetric spaces, partial differential equations and control theory. Some of the problems studied are mathematical formulations of physical situations and the solutions provided have applications to real world problems. For example, wavelet analysis is recognized as a significant advance in signal processing and compression. Similarly, answers to control theoretic questions at the quantum level are essential to realize the potential of the coming generation of computing devices. Members of the Analysis Group enjoy extensive research interaction and collaboration, nationally and internationally, and attract many overseas visitors to Macquarie University. They also receive many invitations to be speakers and main speakers at national and international conferences and workshops.

Group Members:

  • Professor Xuan Duong: Singular integrals, functional calculi, Hardy spaces and partial differential equations, harmonic analysis and partial differential equations.
  • Dr Ji Li:
  • Dr Adam Sikora: Harmonic analysis on Lie groups and symmetric spaces.
  • Honorary Associate Dr Bonnington Clarke: Hyperbolic partial differential equations, control theory, boundary controllability and observability of linear systems of partial differential equations, hyperbolic differential boundary equations.
  • Honorary Associate Dr Christopher Meaney: Harmonic analysis on Lie groups and symmetric spaces.

HDR Students:

  • Joshua Peate (PhD, Macquarie University):
  • Anthony Wong (PhD, Macquarie University):

Research Highlights:

5 Recent Representative Publications:

Category Theory

The Australian Category Seminar meets weekly at Macquarie University. The seminar includes staff and students from Macquarie University as well as frequent visitors. The Centre of Australian Category Theory (CoACT) was formed to provide the environment for the top international centre for higher-dimensional category theory and a major international centre for general category theory. It continues to pursue vigorously research into those parts of mathematics and computer science which find natural expression and advancement in terms of category theory and to train research students of the highest quality. It pursues the development of algebra of widespread applicability for the synthesis and analysis of systems and processes in fields as diverse as physics and computer science, and also mathematics itself. Although having operated as a coherent group since the founding of the ongoing Australian Category Seminar in 1971, CoACT was formally established in 1999.

Group Members:

The group also includes members from the Department of Computing

  • Professor Michael Johnson: Category theory, applications to computer science, homotopy theory and universal algebra.
  • Professor Dominic Verity: Higher-dimensional category theory and its applications to simplicial/cubical geometries, homotopy theory, non-abelian cohomology, category theory and sheaf representations of rings.

HDR Students:

Supervised by Michael Batanin:-

  • Remy Tuyeras (PhD, Macquarie University 2016):  Sketches in Higher Category Theory

Supervised by Ross Street

  • Alexander Campbell (PhD, Macquarie University 2016): A Higher Categorical Approach to Giraud's Non-Abelian Cohomology
  • Branko Nikolic (PhD):
  • James Dolan (PhD):
  • Hardy Hulley (P/T PhD):

Supervised by Steve Lack

  • Christopher Nguyen (PhD):
  • Poon Leung (PhD):
  • Ramon Abud Alcala (PhD):

Supervised by Richard Garner

  • Edoardo Lanari (PhD):
  • Daniel Lin (PhD):
  • Charles Walker (PhD):

Research Highlights:

5 Recent Representative Publications:

Computational Mathematics

Group Members:

  • Dr Stuart Hawkins: boundary integral equations, surface integral equations, boundary element methods, high order methods, preconditioning and numerical linear algebra and simulation of electromagnetic and acoustic wave scattering,

Fluid Mechanics

Research in Fluid Mechanics is focussed around the fundamental question of how, and when, fluid flows transition from a laminar to a turbulent state. Members of the Department working in this area bring the bear the three threads of scientific discovery, theory, computation and experiment, to understand the complex dynamics of a wide range of fluid flows.

Group Members

  • Professor Jim Denier: transition to turbulence in incompressible, compressible and hypersonic flows; boundary-layer flows, impulsively driven flows, fetal haemodynamics, hydrodynamic stability, unsteady separation, the effect of slip on boundary-layer flows, shape optimisation for transition control.
  • Dr Sophie Calabretto: rotating flows
  • Dr Chris Lustri (from Nov.): water waves
  • Dr Adam Tunney: hydrodynamic stability of compressible and hypersonic flows.

HDR Students:

  • Mr Eric Bretshcer (PhD, Ext. - University of Auckland): Minimum resistance boat hulls
  • Mr Jordan Belperio (M. Phil., Ext. - University of Adelaide) : Flow through coils with elliptical cross-section

Research Highlights:

5 Recent Representative Publications:

  • Tunney, A.P., Denier, J.P., Mattner, T.W. and Cater, J.E. A new inviscid mode of instability in compressible boundary-layer flows, Journal of Fluid Mechanics, 785, 301-323 (2015).
  • Calabretto, S.A.W., Levy, B., Denier, J.P. and Mattner, T.W. The unsteady flow due to an impulsively rotated sphere, Proc. Roy. Soc. Lond. Series A., 471, 20150299 (2015)
  • Calabretto, S.A.W., Mattner, T.W. and Denier, J.P. The effect of seam imperfections in the spin-up of a fluid-filled torus, Journal of Fluid Mechanics, 767, 240-253 (2015)
  • Wade, S., Binder, B.J., Mattner, T.W. and Denier, J.P. On the free-surface flow of very steep forced solitary waves, Journal of Fluid Mechanics, 739, 1–21 (2014)
  • Jewell, N. and Denier, J.P. The decay of the flow in the end region of a suddenly blocked pipe, Journal of Fluid Mechanics, 730, 533–558 (2013)

Waves and Nonlinear Dynamics

Research in Waves and Nonlinear Dynamics is focussed on wave scattering and imaging, nonlinear dynamics, mechanics and geometry, and the work spans the areas of theoretical modelling, computational modelling and applications, including industrial applications as well as the theoretical development of the mathematical framework of large classes of models. Particular interests include

  • the scattering and the detection of acoustic and electromagnetic waves from complex objects with applications to antenna design and to object identification or imaging (inverse scattering);
  • the inversion of static field data for the creation of magnetic or gravity profiles for geological purposes; and
  • the modelling of nonlinear effects in electronic systems leading to chaos or other dysfunctional behaviour.

Group Members:

  • Dr Stuart Hawkins: electromagnetic and acoustic wave scattering simulations, boundary integral equations, surface integral equations, boundary element methods, high order methods, preconditioning and numerical linear algebra.
  • Professor Paul Smith: Geometric mechanics and symmetry. Mathematical modelling of acoustic and electromagnetic waves, analytical and computational methods, nonlinear dynamics in electronic systems. Ordinary and partial differential equations, integral equations.
  • Dr Elena Vinogradova: Mathematical modelling of acoustic and electromagnetic wave diffraction, potential theory problems, analytical and numerical techniques, applications to resonant phenomena, antenna design, radar cross section. Direct and inverse scattering problems. Integral and series equations, ordinary and partial differential equations, special functions and integral transforms. Electronic systems models and nonlinear dynamical systems.

HDR Students:

  • Ms Audrey Markowskei (PhD):
  • Mr Leon Beale (PhD):
  • Mr Sherwin Bagheri (PhD):

Research Highlights:

5 Recent Representative Publications:

Optimisation

Content to be added

Group Members:

HDR Students:

Other research interests

The Department also has a number of staff working in other areas including

Research seminars

The Department of Statistics also runs a vibrant seminars series, with many talks of interest to the broader mathematical sciences community. Details can be found at

Back to the top of this page