Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

In this pair of talks I will survey some of the machinery developed by Conant, Schneiderman, and Teichner to study Whitney towers, and their applications to the study of knot and link concordance. Whitney towers can be thought of as measuring the failure of the Whitney trick in dimension 4 and can be used, in a sense, to approximate slice disks. The talks will be based on various papers of Schneiderman, Conant-Schneiderman-Teichner, Cochran-Orr-Teichner and lecture notes by those authors.

We formulate a geometric nonlinear theory of the mechanics of accretion. In this theory the material manifold of an accreting body is represented by a time-dependent Riemannian manifold with a time-independent metric that at each point depends on the state of deformation at that point at its time of attachment to the body, and on the way the new material isadded to the body. We study the incompatibilities induced by accretion through the analysis of the material metric and its curvature in relation to the foliated structure of the accreted body. Balance laws are discussed and the initial-boundary value problem of accretion is formulated. The particular cases where the growth surface is either fixed or traction-free are studied and some analytical results are provided. We numerically solve several accretion problems and calculate the residual stresses in nonlinear elastic bodies induced from accretion.

In post-geometrization low dimensional topology, we expect to be able to relate any topological theory of 3-manifolds to the Riemannian geometry of those manifolds. On the other hand, originated from reformalization of classical mechanics, the study of contact structures has become a central topic in low dimensional topology, thanks to the works of Eliashberg, Giroux, Etnyre and Taubes, to name a few. Yet we know very little about how Riemannian geometry fits into the theory.In my oral exam, I will talk about "Ricci-Reeb realization problem" which asks which functions can be prescribed as the Ricci curvature of a "Reeb vector field" associated to a contact manifold. Finally motivated by Ricci-Reeb realization problem and using the previous study of contact dynamics by Hofer-Wysocki-Zehnder, I will prove new topological results using compatible geometry of contact manifolds. The generalization of these results in higher dimensions is the first known results achieving tightness based on curvature conditions.