4:00 pm Thursday, December 1, 2011

Kent Orr (Indiana University - Math)

Among the most active areas of research in low dimensional topology over the past 15 years takes origin in the early work of Fox and Milnor, in the 1950s. A knot is *slice* if it bounds an embedded disk in a four dimensional ball, a geometric condition which determines an abelian group of equivalence classes, the *concordance group of knots.* Understanding this relation, both as stated above, and a higher dimensional version, dominated much of the discussion on embedding theory throughout the 1960's and early 1970's.
Two modern advances in this area have once again re-energized mainstream knot theory. New approaches using Whitney towers and related geometric underpinnings, along with the introduction of L^2-Index theory, have clarified a possible path toward the classification of topological concordance. The dynamic new Heegaard-Floer invariants have revealed secrets of smoothing theory in low dimensions.
These new tools draw us to the more general problem of classifying *homology cobordism of manifolds.* We discuss the problem, some early triumphs, and how these tools potentially crack adamant problems in manifold theory, while discussing recent breakthroughs, and the hope for a future classification.