December 10, 2015
Mathematics Colloquium lecture Dec.10
Rustam Sadykov, Dartmouth College, will present "Topological Methods of Solving Differential Relations" as part of the Mathematics Department Colloquium Lecture series at 2:30 p.m. Thursday, Dec. 10, in 122 Cardwell Hall.
The abstract for the lecture: The h-principle is a general observation that differential geometry problems can often be reduced to problems in (unstable) homotopy theory. For example, let us consider three fairly different problems in differential geometry: does a given smooth open manifold M admit a symplectic/contact structure? a foliation of a given codimension? a Riemannian metric of positive/negative scalar curvature? According to a very general theorem of Gromov, each of the mentioned differential geometry problems reduces to a purely homotopy theory problem and can be approached by standard homotopy theory methods.
Similarly, the b-principle is a general observation that differential geometry problems can often be reduced to problems in stable homotopy theory. The known instances of the b-principle type phenomena include the Barratt-Priddy-Quillen theorem, the Audin-Eliashberg theorem on Legendrian immersions, and the standard Mumford Conjecture on moduli spaces of Riemann surfaces. I will introduce a general setting for the b-principle and show that the b-principle phenomenon occurs for a fairly arbitrary differential relation imposed on smooth maps.