We will describe an explicit reciprocity law for generalised Heegner cycles, relating the images of certain twists of these classes under the Bloch-Kato dual exponential map to certain Rankin-Selberg L-values, and explain the applications of this formula to the proof of certain rank 0 cases of the Bloch-Kato conjecture. This is a joint work with M.-L. Hsieh.

In 1980, A. P. Calderon published a short seminal paper entitled "On an inverse boundary value problem", which has become the starting point in the mathematical analysis of the following inverse problem: Can one determine the electrical conductivity of a medium by making current and voltage measurements at the boundary of the medium? To this day, this problem serves as a fundamental source of motivation and inspiration for many developments in the field of inverse boundary problems. In this talk we shall give an introduction to the field of inverse boundary problems, survey some of the most important developments, and state some open problems. 

Given a Lie group G and a lattice \Gamma in G, we consider a flow on G/\Gamma induced by the action of a one-parameter subgroup of G. If this flow is mixing then a generic orbit is dense, but nevertheless one can discuss the dimension of the set of exceptions. We discuss work of S. G. Dani, in which such estimates are made in certain cases and a connection to diophantine approximation is established, and also generalizations due to D. Kleinbock and G. Margulis. In particular, we outline a dynamical proof, due to Kleinbock, that the set of badly approximable systems of affine forms has full Hausdorff dimension.

The Webster curvature of a strongly pseudoconvex real hypersurface in C^2 (with respect to the standard pseudo-hermitian structure) is a scalar quantity, invariant under volume-preserving biholomorphic mapping.