A complete flat Lorentz 3-manifold M is a quotient of Minkowski (2+1)- spacetime by a discrete group G of affine isometries acting freely and properly. The study of such discrete groups relates to the deformation theory of hyperbolic structures on a hyperbolic surface S corresponding to M. Properness of G's action relates to lengthening (or shortening) of geodesics on S. Determining criteria for a proper action is, in general, a difficult problem. When S is a three-holed sphere, the sign of the "Margulis invariant" on each boundary components of S determines whether G acts properly or not -- this is a result we have shown with Drumm and Goldman. We will discuss this theorem and how it applies to deformations of hyperbolic structures on S.

In the first lecture we will introduce some basic notions to study the behaviour of polynomial mappings, we will see what can happened at infinity and how the bifurcation set is related to the set of asymptotic critical values. Then, in the next two lectures we will focus on the asymptotic variety of polynomial mappings, i.e. on the set of points at which such a map fails to be proper. The geometry of this set and how we can get explicit description of it will be discussed.

The Hodge conjecture has its origins in the work of Lefschetz regarding which 2 dimensional homology classes on an algebraic surface could be represented via algebraic curves on the surface. Lefschetz's solution involved the study of a class of "Poincare normal functions" on the Riemann sphere minus a finite number of points. In this talk, I will outline Lefschetz's proof and discuss some recent work of Griffiths and Green towards studying the Hodge conjecture for higher codimension cycles using normal functions on higher dimensional parameter spaces.

There is a complete characterization of the zeros sets of the Hardy space Hp. However, at the present, we have just some partial characterizations for most of the relatives of Hp, e.g. the Bergman space and the Dirichlet space. For the latter space, we will discuss Carleson’s condition and its generalization by Shapiro-Shields. Then we give some new families of zero sets which are not covered by the preceding classical results.

The plan of the talk is to look at some explicit recipes for constructing minimal free resolutions of certain 'nice' classes of monomial and binomial ideals in a polynomial ring.

The general idea is to introduce appropriate simplicial or polytopal complexes which encode the information of a free resolution and read off the resolution from the complex.