We show the existence of an N=1 supersymmetric vacuum in string theory whose observable sector has exactly the matter content of the MSSM without exotic particles and vector-like pairs of any type. This is done so by compactifying the E_8 x E_8 heterotic string on a Calabi-Yau threefold endowed with an SU(4) vector bundle which has been constructed after extensive search. In addition, we discuss the Yukawa couplings and Higgs mu-terms in this model.

Kreimer discovered a Hopf algebra structure underlying the combinatorics of renormalization in perturbative quantum field theory. Later, Connes and Kreimer explored the link to non-commutative geometry via a Hopf algebra of rooted trees and described a Hopf algebra of Feynman graphs. After reviewing these developments in some detail we show in this talk how to organize the combinatorics of renormalization in terms of unipotent triangular matrix representations. A simple decomposition of such matrices is used to characterize the process of renormalization. We thereby recover a matrix (anti-)representation of the Birkhoff decomposition of Connes and Kreimer.

Four dimensional N=4 super Yang-Mills theory contains a bigger superalgebra than AdS or superconformal algebra, su(2,2/4). It corresponds to a noncentral extension of the latter. The talk is for both physicsts and mathematicans interested in a novel way of obtaining noncentral extensions of Lie algebras.

In this talk I will present a short review on the algebraic Bethe ansatz technique and on the recently found solution of the so-called inverse problem. I will show how this solution provides a means for the explicit and exact computation of correlation functions in spin chains and summarize some of the many results obtained in this direction by members of the theory group at the ENS-Lyon in the last years. Finally I will present some work still in progress which intends the generalization of these techniques to the case of spin chains in the presence of impurities.