One of the oldest fields of mathematics, algebra is the study of the abstract structures that underlie symmetry and the solution of polynomial equations. It has applications throughout mathematics as well as to mathematical physics and theoretical computer science.
Specific Research Areas: Algebraic geometry, algebraic combinatorics, group theory, number theory, quantum groups, representation theory, semigroup theory, totally disconnected groups, Yangians.
Researchers: D. Badziahin, N.Brownlowe, K.Coulembier, Z. Dancso, D. Easdown, M.Ehrig, A. Fish, T. Gobet, A. Henderson, G. Lehrer, M. Liu, A. Mathas, A. Molev, V. Nandakumar, J. Parkinson, J. Ramagge, B. Roberts, U. Thiel, A. Thomas, G. Williamson, O. Yacobi, R. Zhang
Contact person: Prof Ruibin Zhang
Many questions about particular algebraic structures can be answered by designing efficient algorithms with a clever use of machinery from different branches of mathematics. Such algorithms have applications in cryptography, coding theory, digital signal processing and quantum computation. The Magma Computational Algebra System, developed at The University of Sydney, is used by mathematicians, corporations and governments worldwide.
Specific research areas: Algorithmic group theory, algorithmic number theory, commutative and noncommutative algebra, computational algebraic geometry, finite geometry, programming languages, and applications.
Researchers: G. Bailey, J. Cannon, A. Steel, N. Sutherland, B. Unger, M. Watkins
Contact Person: Prof. J. Cannon
Geometry and topology are the study of shapes and spaces of all dimensions, their properties and measures. Many aspects of modern geometry make heavy use of analysis. On the other hand, geometric methods are common in the modern study of groups.
Specific research areas: amenability, buildings, differential geometry, geometric evolution equations, geometric group theory, harmonic analysis, integrable systems, knot theory, low-dimensional topology, manifolds, pluripotential theory, real and complex singularities, Ricci flow, spectral curves, stratifications and subanalytic sets, surface theory, topology of algebraic varieties.
Researchers: E. Carberry, A. Fish, A. Henderson, J. Hillman, T.-C. Kuo, K.-K. Kwong, B. Lishak, J. Parkinson, L. Paunescu, M. Radnovic, J. Ramagge, J. Spreer, A. Thomas, S. Tillmann, G. Williamson, H. Wu, Z. Zhang
Contact person: Professor L. Paunescu
Nonlinear ordinary and partial differential equations have a rich mathematical structure and arise naturally in biology, physics and other scientific applications. Even simple nonlinear systems can behave in a complicated way, for example by displaying chaotic behaviour.
Specific research areas: complete integrability, elliptic PDE, mathematical biosciences
Researchers: F.-C. Cirstea, N. Dancer, D. Daners, D. Hauer, M. Myerscough, L. Tzou
Contact Person: A/Prof D. Daners
Operator Algebras and Ergodic Theory study dynamical systems from algebra-analytic point of view. By producing certain invariants they enable answering questions of non-isomophicity of various natural systems. Ergodic Theory has various applications in Number Theory, especially in Diophantine approximation.
Specific research areas: C^*-algebras, Diophantine approximation, ergodic theory, homogeneous dynamics, operator algebras.
Contact Person: A/Prof. D. Badziahin