After a brief review of classical function theory on $\mathbb C$, we will discuss its extension to functions with values in an algebraic variety, i.e. an entire holomorphic curve, and motivate the fact that such a curve should be constrained by birational invariants of the variety that pertains to "hyperbolicity." We will verity this conjectural fact for varieties of maximal Albanese dimension, itself a birational invariant. We will assume no prior knowledge of birational geometry. This is joint work with Joerg Winkelmann.

I will introduce the basic concepts and results in coding theory and I will explain how these can help determine the maximum number of collinear points from a finite set of points in the plane given by their coordinates.

Quantum Heisenberg Manifolds were first defined by M. Rieffel in 1989 as example of quantization of Heisenberg Manifolds along a Poisson bracket.(A typical Heisenberg Manifold is the quotient of Heisenberg group by a uniform lattice).They are interesting for several reasons, one being just because they are tractable examples of noncommutative manifolds.This means that , like the related but simpler noncommutative tori, Q-Heisenberg manifolds provide a nice setting in which to explore noncommutative geometry.

In these series of talks I will explore the different features of the noncommutative geometry on Q-Heisenberg manifolds. We introduce a class of spectral triples on Q-Heisenberg manifold, we introduce the space of L^2 -forms and then we characterize torsion less/Unitary connections. In addition, for a concrete family of unitary connections we compute Ricci curvature and scalar curvature.

Multiple zeta values (MZV) are the numbers defined by the convergent series of the form

$$\zeta(s_1,s_2,...,s_k)=\sum_{n_1>n_2>...>n_k>0}^\infty \{1/(n_1^{s_1} >... n_k^{s_k})\}$$

for $s_i$ positive integers. For these real numbers there are some beautiful relations, some of them due to Euler, like $\zeta(2,1) = \zeta(3)$ or $\zeta(2n) = q\pi^{2n}$ for $q$ a rational number. In this lecture I will present some of the most famous conjectures about MZV and its relations. I will show how we try to see the truthfulness of this conjecture by looking at them until a small degree bounded by the capacity of the actual computers.