Colloquium

Research seminars

Clifford lectures

"Milnor invariants of string links via Chen's connections"

I will describe a monomorphism of the reduced subgroup of n-component string links and a group of homotopy classes of (n-1)-dimensional torus to  the based loop spaces of the configuration space of n points in R^3. This monomorphism allows us to express Milnor's higher linking numbers via Chen's iterated integrals. The talk is a continuation of the topic I presented in the geometry/topology seminar last Spring and should be self-contained. Results are joint with Fred Cohen from the University of Rochester.

"Milnor invariants of string links via Chen's connections- Part II"

I will describe a monomorphism of the reduced subgroup of n-component string links and a group of homotopy classes of (n-1)-dimensional torus to  the based loop spaces of the configuration space of n points in R^3. This monomorphism allows us to express Milnor's higher linking numbers via Chen's iterated integrals. The talk is a continuation of the topic I presented in the geometry/topology seminar last Spring and should be self-contained. Results are joint with Fred Cohen from the University of Rochester.

"The Quantum Layer Conjecture"

Take an embedded surface in R^3 and a tubular neighborhood about it.  This is geometrically a layer in 3-space.  When the embedded surface is compact, it is clear that the Dirichlet Laplacian acting on functions of the layer has eigenvalues.  However, when the embedded surface is non-compact, the existence of eigenvalues for the Dirichlet Laplacian is a non-trivial phenomenon.  In this talk we will survey this physically-inspired problem and talk about progress towards the Quantum Layer Conjecture, which says that a layer built over a non-flat complete, non-compact surface with integrable Gauss curvature must admit eigenvalues for the Dirichlet Laplacian.  The conjecture reinforces our understanding of surface topology.  The local geometric features of the problem is really no more sophisticated than the curves and surfaces course taught in undergraduate curriculum.  It should be interesting to a broad audience.

"TBA"

"TBA"

"TBA"

"TBA"

"TBA"

"TBA"

"TBA"

"TBA"