Talk

Darmon points are a collection of conjectural generalizations of Heegner
points on modular elliptic curves. Algorithms for their effective
calculation are useful for gathering numerical evidence supporting their
conjectured properties. In addition, in some cases they can be used in
practice as very efficient methods for computing algebraic points in such
elliptic curves. In this talk I will recall two constructions of Darmon
points in different settings, and the algorithms of Darmon-Logan and
Darmon-Green-Pollack that allow for their computation in certain elliptic
curves.

Two manifolds are commensurable if they share a common finite sheered cover. Partitioning the set of hyperbolic 3-manifolds into commensurability classes organizes this set along similar geometric lines. However, however the general problem is quite difficult. Instead we will focus on a manifolds that appear rare in a commensurability class. In fact, Reid and Walsh conjecture that there are at most three hyperbolic knot complements in commensurability class. The first talk will focus on the background and motivation for this conjecture.

I will start by recalling some basic facts on the first eigenvalue of
the Laplacian on a compact Riemannian manifold. Next I will discuss a
classical theorem of Hersch, which says that on the 2-sphere the first
eigenvalue is the largest possible when the metric is the standard
one.

I will then formulate the analogous problem for Kaehler metrics and
will describe the solution in the case of Hermitian symmetric
spaces of the compact type. If time permits I will try to explain the
link with Satake-Furstenberg compactifications and coadjoint orbits.

In this lecture, I will discuss the structure of the non-singular Fourier coefficients of
the derivative at the central critical point of incoherent Eisenstein series on U(n,n). 
In certain cases, these coefficients coincide with the arithmetic degrees of
0-cycles on moduli spaces of abelian varieties.  The proof of this relation depends
on p-adic uniformization and the determination of the structure of special cycles on
Rapoport-Zink spaces. 
This is joint work with Michael Rapoport.