Georgia Institute of Technology College of Sciences

This is the sixth (and last) of several talks discussing embeddings of manifolds. I will discuss some general results for smooth manifolds, but focus on embeddings of contact manifolds into other contact manifolds. Particular attention will be paid to embeddings of contact 3-manifolds in contact 5-manifolds. I will discuss two approaches to this last problem that are being developed jointly with Yanki Lekili.

Faltings' theorem states that curves of genus g> 1 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry and tropical geometry to show that we can also give an effective bound on the number of rational points outside of the special set of the d-th symmetric power of X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.

This talk concerns a theory of "multiparameter singularintegrals." The Calderon-Zygmund theory of singular integrals is a welldeveloped and general theory of singular integrals--in it, singularintegrals are associated to an underlying family of "balls" B(x,r) on theambient space. We talk about generalizations where these balls depend onmore than one "radius" parameter B(x,r_1,r_2,\ldots, r_k). Thesegeneralizations contain the classical "product theory" of singularintegrals as well as the well-studied "flag kernels," but also include moregeneral examples. Depending on the assumptions one places on the balls,different aspects of the Calderon-Zygmund theory generalize.

The perimeter of a convex set in R^n with respect to a given measure is the measure's density averaged against the surface measure of the set. It was proved by Ball in 1993 that the perimeter of a convex set in R^n with respect to the standard Gaussian measure is asymptotically bounded from above by n^{1/4}. Nazarov in 2003 showed the sharpness of this bound. We are going to discuss the question of maximizing the perimeter of a convex set in R^n with respect to any log-concave rotation invariant probability measure. The latter asymptotic maximum is expressed in terms of the measure's natural parameters: the expectation and the variance of the absolute value of the random vector distributed with respect to the measure. We are also going to discuss some related questions on the geometry and isoperimetric properties of log-concave measures.