Topics on 3- and 4-dimensional manifolds. - to develop practical algorithms for recognising surfaces, knots and 3-dimensional spaces. These will be very useful for experimentation
and to understand the computational complexity of such questions.
- to understand the properties of minimal surfaces in 3-dimensional spaces of constant negative curvature, with applications to the complexity of knots and the structure of these spaces.
- to develop a theory of 4-dimensional spaces which are covered ....Topics on 3- and 4-dimensional manifolds. - to develop practical algorithms for recognising surfaces, knots and 3-dimensional spaces. These will be very useful for experimentation
and to understand the computational complexity of such questions.
- to understand the properties of minimal surfaces in 3-dimensional spaces of constant negative curvature, with applications to the complexity of knots and the structure of these spaces.
- to develop a theory of 4-dimensional spaces which are covered by
Euclidean 4-space. New techniques will be utilised, since the powerful methods of Freedman-Quinn do not apply to most such spaces. Read moreRead less
Minimal surfaces. Recent stunning progress in topology, in particular a possible solution to one of the Clay Institute million dollar problems, using techniques from partial differential equations and minimal surfaces has made this area a hot topic. To attract researchers in this field to visit Australia and to train students in this area is a major part of this project.
Three-dimensional geometry and topology. This project will carry out important fundamental research into the geometry and topology of 3-dimensional manifolds, an area of intense activity over the last 30 years.
The work has direct applications to physics, for example recent work in cosmology aimed at determining the global structure of our universe. Our work on knotting and symmetries of molecular graphs will also be of considerable interest in chemistry and biology.
The project will also ....Three-dimensional geometry and topology. This project will carry out important fundamental research into the geometry and topology of 3-dimensional manifolds, an area of intense activity over the last 30 years.
The work has direct applications to physics, for example recent work in cosmology aimed at determining the global structure of our universe. Our work on knotting and symmetries of molecular graphs will also be of considerable interest in chemistry and biology.
The project will also provide high quality training of undergraduate and graduate students in geometry and topology, and will increase international cooperation by developing closer links with colleagues and institutions overseas.Read moreRead less
Geometric structures on 3-manifolds. Three-dimensional manifolds are of central importance in topology, algebra, and cosmology (providing models for the universe). Thurston's Geometrization Conjecture gives a beautiful conjectural picture of
3-manifolds in terms of eight uniform geometries, but the conjecture and some of its basic consequences remain unproved. This project is aimed at making advances on fundamental questions in the following areas:
* construction of geometric structures by def ....Geometric structures on 3-manifolds. Three-dimensional manifolds are of central importance in topology, algebra, and cosmology (providing models for the universe). Thurston's Geometrization Conjecture gives a beautiful conjectural picture of
3-manifolds in terms of eight uniform geometries, but the conjecture and some of its basic consequences remain unproved. This project is aimed at making advances on fundamental questions in the following areas:
* construction of geometric structures by deformation methods,
* computation of geometric structures,
* geometric and algebraic invariants.Read moreRead less
Triangulations in dimension three: algorithms and geometric structures. Perelman recently won a Fields medal for the solution of the geometrisation and Poincare conjectures on three-dimensional spaces, using a very deep heat flow method to find optimal geometries on these spaces. The project will develop a new constructive approach to building these optimal geometric structures. This will lead to effective algorithmic methods to distinguish three-dimensional spaces, with applications to the stu ....Triangulations in dimension three: algorithms and geometric structures. Perelman recently won a Fields medal for the solution of the geometrisation and Poincare conjectures on three-dimensional spaces, using a very deep heat flow method to find optimal geometries on these spaces. The project will develop a new constructive approach to building these optimal geometric structures. This will lead to effective algorithmic methods to distinguish three-dimensional spaces, with applications to the study of knots and links (for example, knotted DNA molecules) and to mathematical physics. The project will also provide new techniques to study important problems in the classification of three-dimensional spaces, such as the virtual Haken conjecture.Read moreRead less
Connections in low-dimensional topology. This project aims to resolve important open questions in low-dimensional topology, by connecting hyperbolic geometry to invariants arising from quantum topology, cluster algebras, and spinors.
The spaces studied in this project, namely 3-manifolds and knots, arise in applications across engineering and science. The project expects to generate new insights into these spaces by applying tools connecting them to hyperbolic geometry.
Expected outcomes inclu ....Connections in low-dimensional topology. This project aims to resolve important open questions in low-dimensional topology, by connecting hyperbolic geometry to invariants arising from quantum topology, cluster algebras, and spinors.
The spaces studied in this project, namely 3-manifolds and knots, arise in applications across engineering and science. The project expects to generate new insights into these spaces by applying tools connecting them to hyperbolic geometry.
Expected outcomes include efficient techniques to compute important data about 3-manifolds and knots, particularly certain polynomials encoding geometry, and exact calculations of circle packings.
This should provide significant benefits, such as progress on difficult conjectures in hyperbolic geometry.
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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
Special Research Initiatives - Grant ID: SR0354466
Funder
Australian Research Council
Funding Amount
$20,000.00
Summary
Mathematics in Contemporary Science. The Mathematics in Contemporary Science Research Network brings contemporary methods of non-linear analysis and differential equations, geometric reasoning and relevant algebraic and topological ideas to enrich six application areas in modern science: Complex Systems, Computer Vision, Optimal Transportation, Nanotechnology, Physics and Shortest Networks. MiCS will develop both the mathematics and the application areas in parallel. It will focus on postgradu ....Mathematics in Contemporary Science. The Mathematics in Contemporary Science Research Network brings contemporary methods of non-linear analysis and differential equations, geometric reasoning and relevant algebraic and topological ideas to enrich six application areas in modern science: Complex Systems, Computer Vision, Optimal Transportation, Nanotechnology, Physics and Shortest Networks. MiCS will develop both the mathematics and the application areas in parallel. It will focus on postgraduate training through workshops, summer schools and web based resources and build long-term international collaborations with EU networks and NSERC, NSF and EPSRC institutes as well as bringing together academic and industry leaders.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.
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Statistical Topology and its Application to Deriving New Geometric Invariants. This project will offer a great opportunity for talented students to engage in internationally competitive research. Statistical topology, which combines ideas in topology, geometry and statistical mechanics is becoming a rapidly increasing branch of mathematics, with many emerging applications in bio-informatics, computer science and theoretical physics.