Georgia Institute of Technology College of Sciences

Unlike symplectic structures in 4-manioflds, contact structures are abundant in 3-dimension. Martinet showed that there exists a contact structure on any closed oriented 3-manifold. After that Lutz showed that there exist a contact structure in each homotopy class of plane fields. In this talk, we will review the theorems of Lutz and Martinet.

This talk will be an introduction to the homotopy principle (h-principle). We will discuss several examples. No prior knowledge about h-principle will be assumed.

The Schoenflies' conjecture proposes the following: An embedding of the n-sphere in the (n+1)-sphere bounds a standard (n+1)-ball. For n=1, this is the well known Jordan curve theorem. Depending on the type of embeddings, one has smooth and topological versions of the conjecture. The topological version was settled in 1960 by Brown. In the smooth setting, the answer is known to be yes for all dimensions other than 4, where apart from one special case, nothing is known. The talk will review the question and attempt to describe some of the techniques that have been used in low dimensions, especially in the special case, that was worked out by Scharlemann in the 1980s. There are interesting connections to the smooth 4-dimensional Poincare conjecture that will be mentioned, time permitting. The talk is aimed to be expository and not technical.

In 1957, Smale proved a striking result: we can turn a sphere inside out without any singularity. Gromov in his thesis, proved a generalized version of this theorem, which had been the starting point of the h-principle. In this talk, we will prove Gromov's theorem and see applications of it.