De Rham cohomology (after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.
This is a beautiful and deep part of pure mathematics which lies at the intersection of geometry, algebra and analysis and would be of interest to pure mathematicians working in various areas as well as applied mathematicians and people in other disciplines including theoretical physics.
On Friday 27th June 2008 Professor Frank Johnson (UCL) will be giving a one day course at UCL on Differential Manifolds and De Rham Cohomology. The course starts at 12.00, finishes at 16.15 and breaks for lunch between 13.00 - 14.00 (there will also be a short break in the afternoon session). It is being organised by Dr. Andrei Yafaev and Brian Tyler and is being funded by the Roberts Skills Training Fund.
The department is providing a free lunch in the college refectory for everyone who attends the course. Afterwards we might go to the union bar to continue on a more informal basis.
Professor Johnson's Abstract:
"We shall show how to define and compute cohomology groups for smooth manifolds starting from differential forms. As a first application we shall show how computing the cohomology of spheres and discs allows us to prove some of the classic results of algebraic topology, for example, the Brouwer Fixed Point Theorem and Brouwer Open mapping theorem. We shall also compute the cohomology of complex projective spaces and show how this allows us to classify complex line bundles over an arbitrary smooth manifold."
The course is open to anyone who is interested in attending; there are no specific prerequisites for the course, although some undergraduate algebra would probably be helpful. The course is aimed primarily at a postgraduate mathematician audience, but should be accessible to third year undergraduates and theoretical physicists. The course will start at 12.00 and finish at 16.15 with a break for lunch at 13.00 and another at some point during the afternoon session. It will take place in the mathematics department at UCL in room 706. See this map to find out where to go, the maths department is in square D1 and is located directly above the student union. Please email Dr. Andrei Yafaev or Brian Tyler if you are interested in attending so that we can get an idea of numbers - please note that there is no fee for this course.