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Geometry and Groups

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Poster

Dr Stephan Tillmann
The University of Sydney

Synopsis

According to Felix Klein's influential Erlanger program of 1872, geometry is the study of properties of a space, which are invariant under a group of transformations. In Klein's framework, the familiar Euclidean geometry consist of n-dimensional Euclidean space and its group of isometries. In general, a geometry is a pair (X, G), where X is a (sufficiently nice) space and G is a (sufficiently nice) group acting on the space. Geometric properties are precisely those that are preserved by the group. A geometry in Klein's sense may not allow the concepts of distance or angle; an example of this is affine geometry.

The study of geometry in Klein's framework motivates key ideas in different areas of mathematics, such as group theory, algebraic topology, differential geometry and representation theory. The seminal work of Bill Thurston has provided even more links between seemingly disparate fields of mathematics.

The aim of this course is to give students an introduction to geometry in the sense of Klein and Thurston, and to provide them with working knowledge of a variety of concepts and tools that are applicable in different fields of mathematics, as well as to open avenues for further study. A great emphasis will be placed on the detailed study of key examples.

Quick overview

Week 1: Model geometries in dimension two
Week 2: Notions from group theory and algebraic topology
Week 3: Hyperbolic geometry
Week 4: Geometric structures on manifolds

Detailed overview (PDF)

Textbook

Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots by Francis Bonahon
Student Mathematical Library, 49. IAS/Park City Mathematical Subseries.
American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, 2009.

Supplementary notes (PDF)

  • §1 Möbius transformations and inversions (in preparation)
  • §2 Metric spaces (in preparation)
  • §3 Groups and group actions (in preparation)
  • §4 Some algebraic topology (in preparation)
  • §5 Geometric manifolds (in preparation)

Contact hours

28 hours of hours spread over four weeks, plus consultation as required.

Prerequisites

Essential: A course in Multivariable Calculus and a course in Linear Algebra.

Helpful: A first course in group theory, metric spaces, point-set topology or algebraic topology. Neither is essential, and this course will give an introduction to key ideas in all four areas.

Bed-time reading

These books are really enjoyable to read. They give nice introductions and contain excellent illustrations.

  • "Geometry and the Imagination" by David Hilbert and S. Cohn-Vossen
  • "Indra's pearls (The vision of Felix Klein)" by David Mumford, Caroline Series and David Wright
  • "The shape of space" by Jeffrey Weeks

Games

These geometry games by Jeffrey Weeks can help to gain some intuition about tilings, curvature and group actions:

About Stephan Tillmann

My research area is geometry and topology, with a focus on surfaces and 3-dimensional manifolds but with occasional excursions into arbitrary dimensions. I was attracted to this area by its myriad connections to other fields of mathematics, as well as the intrinsic beauty--both mathematical and visual--of some of the mathematical objects involved.

After undergraduate studies in Mainz (Germany) and Melbourne, I obtained my PhD in 2002 from the University of Melbourne under supervision of Craig Hodgson and Walter Neumann. Following this, I spent three years as a postdoctoral fellow in Montreal, where I met many mathematicians from North America and Europe, some of whom became collaborators and friends. I returned to Australia to work with Hyam Rubinstein in Melbourne before taking up a lectureship at the University of Queensland. I have moved to the University of Sydney in December 2011, just in time to unpack before teaching this course!