Project IV (MATH4072) 2011-12

Geometry of the Heisenberg Group

John Parker

Description

The Heisenberg group H is a three dimensional group which is 2-step nilpotent (1-step nilpotent is Abelian). This means it has very interesting geometry. In this project we will investigate different ways to put a metric on H and so to investigate its geometry. There are several ways to do this and there is a possibility to develop in any of these directions:

  • The isometries and almost-Bieberbach groups.
  • Comparison of different metrics.
  • Balls, Hausdorff dimension and fractals.
  • The geometry of the visual cortex.

    Prerequisites

    Relevant pure courses, such as Differential Geometry III or Geometry III. For some of the applications, you will have to be self motivated!

    Resources

    A good reference is the book Luca Capogna, Donatella Danielli, Scott D Pauls, Jeremy T Tyson An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem Birkhäuser 2007.
    You can buy this from amazon.

    The wikipedia page on the Heisenberg group.

    • email: J R Parker

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