The complex analogue of Diecke's Theorem and Brickell's Theorem in
real Finsler geometry. Complex Finsler structures naturally satisfy the complex
homogeneous Monge-Ampere equation and the analogue of Diecke's Theorem and
Brickell's Theorem can be put in the frame work of the classification of
complex manifolds admitting an exhaustion function satisfying the complex
homogeneous Monge-Ampere equation.

1st year calculus teachers use the equation Tp(cos(ϑ))=cos(pϑ),
with Tp(w) the pth Chebychev polynomial. It is a map between complex
spheres branched over three points. I will explain why we call Tp a
dihedral function. Functions similar to it form one Mobius class:
equivalent by composing with fractional transformations.

Abel used more general dihedral Mobius classes. These form what we
now call the modular curve Y0(p). In "What Gauss Told Riemann About
Abel's Theorem" a lecture at John Thompson's 70th Birthday,
I cited Otto Neuenschwanden on the 60-year-old Gauss in conversation
with the 20-year-old Rieman. Their goal was to generalize Abel using
Gauss' harmonic functions. Riemann went far, but his early death left
an incomplete program.

To see why the generalization is non-obvious, consider: What is the
alternating (group) version of taking composites of Tp to form Tpk+1,
k 0?

This talk will use (and explain) alternating versions of modular curves
to connect two famous modern problems:

1). The Strong Torsion Conjecture (on Abelian Varieties); and

2). The Regular Inverse Galois Problem.

These spaces have cusps at points on their boundary. A cusp pairing
(the shift-incidence matrix ) helps picture these spaces. They aren't
modular curves. Still, using a result with J.P. Serre, we show how their
cusps resemble those of modular curves. That gives a version of the
renown Merel-Mazur result for these alternating spaces.

The subject of enumerative geometry goes back at least to the middle
of the 19th centuary. It deals with questions of enumerating geometric
objects, e.g.
(a) how many lines pass through 2 points or
through 1 point and 2 lines in 3-space?
(b) how many conics in 2-space are tangent to k lines and
pass through 5-k points?

There has been an explosition of activity in this field over the past
twenty years, following the development of Gromov-Witten invariants in
sympletic topology and string theory. The idea of counting parameterizations
of curves in order to count curves themselves has led to solutions of
whole sets of long-standing classical problems. At the same time, string
theory has generated a multitude of predictions for the structure of
GW-invariants, as well as for the behavior of certain natural families
of Laplacians. It has in particular suggested that there is a diality
between certain symplectic and complex manifolds and that in some cases
GW-invariants see some geometric objects, that are yet to be fully
discovered mathematically.

In this talk I hope to give an indication of what enumerative geometry
is about and of the shift in the paradigm that has occured over the past
two decades.

Sending quantum information over a distance is a challenging task and the challenge increases, if the task is to be performed several times, to send a message over a large distance. In general, this can only be done with a certain probability. On several quantum networks, a recently introduced procedure, which uses special quantum measurements, enhances this probability, as follows from the properties of related classical percolation models. ALL of the above terms will be explained in the talk, which is aimed at a general mathematical audience. While mathematically rigorous, the presented results were obtained in collaboration with a physics group and are of current physical interest, leading to a number of open problems, which will be mentioned.

A non-zero Abelian differential in a compact Riemann surface of genus $g \geq 1$ endows the surface with an atlas (outside the zeroes) whose coordinate changes are translations. There is a natural ``vertical flow'' (moving up with unit speed) associated with the translation structure, generalizing the genus $1$ case of irrational flows on the torus.

The Teichm\"uller flow in the moduli space of Abelian differentials can be seen as the renormalization operator of translation flows. In this talk, we will discuss how the chaoticity of the Teichm\"uller flow dynamics reflects on the (non-chaotic) dynamics of the associated vertical flows (for typical parameters), and the closely related interval exchange transformations.