Wednesday, February 27th, 2019
04:00 pm - 04:50 pm, in McCormack 01-0421

Daniel Álvarez-Gavela

IAS, Princeton

The flexibility of singularities

Abstract: The problem of understanding the generic singularities of smooth maps is intractable, in a way that can be made precise. Nevertheless, the problem of simplifying these singularities is not only tractable, but in most cases reduces completely to the underlying homotopy theoretic problem (which can be tackled using the tools of algebraic topology). The first instance of this phenomenon was discovered by M. Hirsch and S. Smale, who proved that the space of immersions of a manifold M into a manifold N is (weakly) homotopy equivalent to the space of bundle monomorphisms from TM to TN whenever dim(M) < dim(N). For example, that one can realize the eversion of the sphere through immersions follows immediately from this theorem together with the fact that \pi_2(O_3)=0. These results were vastly generalized by M. Gromov and others to a discipline of mathematics known as flexible topology, or the study of h-principles. In this talk we will review this story, starting with immersion theory and ending with a recent h-principle for the simplification of singularities of wavefronts which I established in my PhD thesis.