Dr Daniel Mathews & Associate Professor Jessica Purcell, Monash University
The study of low-dimensional topology is the study of spaces of dimensions 2, 3, and 4, including the study of surfaces and their symmetries, knots and links, and structures on 3 and 4-manifolds. It is an area of active research with deep connections to mathematical fields such as geometry and dynamics; it also has modern applications to microbiology, chemistry, and quantum physics. It requires a different set of tools from higher dimensional topology. Perhaps this was first realised when Smale proved the Poincare conjecture in dimensions five and higher in 1961. The proofs in dimensions four (by Freedman in 1982) and three (by Perelman in 2003) required completely new ideas.
In this course we will cover some foundational results of low-dimensional topology. In two dimensions, we will study surfaces, their symmetries, and the mapping class group, proving a beautiful theorem of Lickorish that the mapping class group is generated by Dehn twists (which we will define). In three dimensions, we will study knots – knotted loops in three-dimensional space – and 3-manifolds. We will investigate different ways of describing 3-manifolds, including Heegaard splittings and Dehn fillings, and knot invariants including the Jones and Alexander polynomials. Along the way we will mention some 4-dimensional applications.
- Surfaces and their homeomorphisms
- The mapping class group and Dehn twists
- 3-manifolds: Heegaard splittings and Dehn filling
- Decompositions of 3-manifolds
- Knots and knot invariants
A strong foundation in basic analysis and/or algebra, gained through at least one abstract course focusing on proofs (e.g. a course on groups or on metric spaces). A first course in topology would be helpful, but not required for the motivated student.
- Assignments and presentations: 40%
- Final examination: 60%
- “Knots and links” by D. Rolfsen.
- B. R. Lickorish, “A representation of orientable combinatorial 3-manifolds”
Dr Daniel Mathews, Monash University
Daniel is a Lecturer in the School of Mathematical Sciences at Monash. He previously worked at Boston College in the USA and at Nantes in France. His research includes low-dimensional topology and hyperbolic, contact and symplectic geometry.
Associate Professor Jessica Purcell, Monash University
Jessica is an Australian Research Council Future Fellow at Monash University. Prior to that, she worked at Brigham Young University in the USA, at Oxford in the UK, and at the University of Texas at Austin in the USA. Jessica’s research focuses on 3-manifolds and hyperbolic geometry, as well as applications to knot theory.