HOLOMORPHIC CURVES, REEB FLOWS AND CONTACT TOPOLOGY. Motion of a satellite is one of many examples of a Reeb dynamical system. The aim of the project is to deepen our understanding of Reeb flows. The Reeb flows, in particular, include Hamiltonian flows on three-dimensional contact type energy surfaces. To study the behaviour of Reeb flows we construct systems of global surfaces of section and study the iterates of the Poincare map, which is obtained by following the flow until it hits a surface. ....HOLOMORPHIC CURVES, REEB FLOWS AND CONTACT TOPOLOGY. Motion of a satellite is one of many examples of a Reeb dynamical system. The aim of the project is to deepen our understanding of Reeb flows. The Reeb flows, in particular, include Hamiltonian flows on three-dimensional contact type energy surfaces. To study the behaviour of Reeb flows we construct systems of global surfaces of section and study the iterates of the Poincare map, which is obtained by following the flow until it hits a surface. The main tools in constructing systems of global surfaces of section are holomorphic curves in symplectization, which are defined on punctured Riemann surfaces and solve nonlinear Cauchy-Riemann type operator. These curves are also main ingredients of new invariants of contact and symplectic manifolds.
These new invariants are now known as Contact Homology and Symplectic Field Theory. In the second part of the project we develop analytical foundations for these theories.Read moreRead less
Eclectic problems in topology, geometry and dynamics. This project aims to resolve a number of problems across several broad areas of pure mathematics. The problems all have a geometric or topological flavour, and some deal with dynamics in the qualitative sense. The problems share two common themes: they have group theoretic aspects and homological aspects. Specifically, the problems lie in the following areas:
1. finite dimensional Lie algebras and their cohomology,
2. low dimensional combin ....Eclectic problems in topology, geometry and dynamics. This project aims to resolve a number of problems across several broad areas of pure mathematics. The problems all have a geometric or topological flavour, and some deal with dynamics in the qualitative sense. The problems share two common themes: they have group theoretic aspects and homological aspects. Specifically, the problems lie in the following areas:
1. finite dimensional Lie algebras and their cohomology,
2. low dimensional combinatorial geometry: graph drawings on surfaces,
3. topological dynamics of group actions,
4. differentiable group actions and foliation theory.
The most significant aims are to resolve two well known conjectures: Halperin's toral rank conjecture and Conway's thrackle conjecture.
Read moreRead less
Topological Optimisation of Fluid Mixing. The proposed research is aimed at improving the efficiency of fluid mixers,
which in the long term has potential to reduce vastly the economic and
environmental costs associated with large-scale mixing processes in Australian
chemical industries. The research will not only impact on practical mixer
design, but will also develop important results in the application of topology
to the the field of chaotic dynamical systems. This project will also prov ....Topological Optimisation of Fluid Mixing. The proposed research is aimed at improving the efficiency of fluid mixers,
which in the long term has potential to reduce vastly the economic and
environmental costs associated with large-scale mixing processes in Australian
chemical industries. The research will not only impact on practical mixer
design, but will also develop important results in the application of topology
to the the field of chaotic dynamical systems. This project will also provide a
graduate student and post-doctoral researcher with training to pursue a career
in fluid dynamics or general applied mathematics research.
Read moreRead less