During Fall and Winter 1999-2000, while visiting the Max Planck Institute in Leipzig, I was enticed by Matthias Schwarz of the University of Leipzig to think about contact geometry, and specifically how to apply parabolic methods to build a contact form on an odd-dimensional compact manifold with a nontrivial fundamental group or other homotopy groups below the middle dimension. This has led to an onging collaboration involving also Hansjorg Geiges of the University of Leiden. We have developed a method which has promise for finding contact structures on a class of manifolds which includes the product of any contact manifold with an oriented surface. In particular, this would include contact structures on all odd-dimensional tori. A brief report [51] (see the PostScript version or the PDF version), with an introduction to contact geometry, has appeared in the Proceedings of the Symposium on Geometric Analysis and Applications, CMA, Canberra, Australia. A complete proof, by very different methods, has now been published by Frederic Bourgeois in Int. Math. Research Notices 30, 1571-4 (2002).

[51]. The Heat-Flow Method in Contact Geometry. Pp. 106--117 of Proceedings of Symposium on Geometric Analysis and Applications (2000), Proceedings of the Centre for Mathematics and its Applications, vol. 39, Australian National University, 2001. Postscript version or PDF version