The conjectures in the title were formulated in the late 1970's as vast generalizations of the classical theorem of Stickelberger. They make a subtle connection between the Z[G(L/k)]-module structure of the Quillen K-groups K*(OL) in an abelian extension L/k of number fields and the values at negative integers of the associated G(L/k)-equivariant L-functions.

These conjectures are known to hold true if the base field k is Q, due to work of Coates-Sinnott and Kurihara. In this talk, we will provide evidence in support of these conjectures over arbitrary totally real number fields k.

Growth estimates for orthogonal polynomials with
respect to area measure (Bergman polynomials) over the union of
finitely many Jordan regions with piecewise smooth boundary are
obtained by a careful investigation of the Green function of the
complement, and of Schwarz reflection in analytic arcs of the
boundary. As applications one derives a detailed picture of the
limiting zero distribution of Bergman's orthogonal polynomials,
and also a robust reconstruction algorithm of the
original open set, starting from incomplete data (such as obtained
by geometric tomography). A good part of the lecture will be devoted
to a non-technical discussion of the main objects and principal
techniques used in the above study, from a historical perspective.
Several numerical experiments and illustrations will be provided,
as supports for the theoretical facts. Based on recent joint work
with B. Gustafsson, E. Saff and N. Stylianopoulos.

The moduli space A_g of principally polarized abelian
g-folds may be viewed as a prime motivation for the theory of
Shimura varieties. I will explain this, along with variants of
such moduli interpretations (of Hodge-type or PEL).

I will then discuss mod p reductions and some of their moduli
interpretations which are outside the Hodge or PEL class.