In this second talk, I will explain how we can probe the space of symplectic embeddings of balls in some four-manifolds through the analysis of symplectomorphism groups. After explaining the general framework, I will focus on some rational symplectic four-manifolds for which that analysis can be carried through effectively using pseudo-holomorphic curves techniques and the theory of deformations of complex structures.

I will discuss some recent results of a joint work with professor Rasul Shafikov, concerning a CR geometry problem, with a complex (yet simple) local analytic solution.

The path category P(K) of a simplicial complex K is a category which is built from vertices (objects) and 1-simplices (morphisms), subject to commutativity conditions associated to the 2-simplices of K. This construction extends to a functor from simplicial sets to categories which is left adjoint to the nerve.

Here is why one cares: path category morphisms specialize to execution paths in higher dimensional automata. These objects are geometric models for behaviour of parallel processing systems, and techniques are required to distinguish execution paths between states in such a system. This calculational problem is non-trivial, since the path category functor is not a standard homotopy invariant and produces categories with little extra structure. The known viable lines of attack arise from higher category theory and homotopy coherence theory.

35 years ago Coates and Sinnott formulated a K-theoretic analog of an even more classical result of Stickelberger's about annihilation of class groups of abelian fields. The talk will report on the current standing of this conjecture, the "correct" formulation in terms of motivic cohomology, and its relation to Iwasawa theory.