Level: BSc, MSci, MSc
Title: Differential forms, cohomology and the Hodge operator
Supervisor:

Prof. Shahn Majid
Research Area: Geometry and Analysis
Description:

In geometry the notion of a vector field has a dual concept, that of a differential form. The project will explain this concept on any smooth manifold and the resulting exterior algebra of differential forms of different degree (the de Rham complex). The associated concept of cohomology provides some of the first invariants of manifolds. For a Riemannian manifold of dimension n, the Hodge operator turns a differential form of degree one of degree n-m and also provides a notion of codifferential, which can then be used to compute cohomology [1]. The MSci version would be expected to consider recent results and/or applications, for example a recent 6 term derivation property of the codifferential on triple products or links to other cohomology theories.

  1. I. Madsen and J. Tornhave, From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, Cambridge Univ. Press, 1997.
Further Reading:
  • R. Bott and L. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, NY, 1982.
Key Modules:

To be confirmed with the supervisor.
Other Information:
Current Availability: Yes