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K-State Today

December 10, 2015



Mathematics Colloquium lecture Dec.10

By Reta McDermott

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.