2001/06/11 by Jih-Hsin Cheng
Mathematics · #math.SG #math.CV #msc:57R17 #msc:32V05 #msc:32G07
published as Proceedings of the 3rd Asian Mathematical Conference 2000 (Quezon City, Philippines) ed. T. Sunada, P. W. Sy, Yang Lo (2002), 70-80. · 11 pages. Based on the author's notes for his talks at the third Asian Mathematical Conference (Quezon City, Oct. 2000) and the Contact Geometry Conference (Stanford, Dec. 2000). To appear in Proceedings of the third Asian Mathematical Conference 2000 (Quezon City, Philippines)
We study low-dimensional problems in topology and geometry via a study of contact and Cauchy-Riemann (CR) structures. A contact structure is called spherical if it admits a compatible spherical CR structure. We will talk about spherical contact structures and our analytic tool, an evolution equation of CR structures. We argue that solving such an equation for the standard contact 3-sphere is related to the Smale conjecture in 3-topology. Furthermore, we propose a contact analogue of Ray-Singer's analytic torsion. This ''contact torsion'' is expected to be able to distinguish among ''spherical space forms'' \Γ\backslash S3\ as contact manifolds. We also propose the study of a certain kind of monopole equation associated with a contact structure. In view of the recently developed theory of contact homology algebras, we will discuss its overall impact on our study.