Abstract: In this lecture we will introduce elliptic curves
and some of the fundamental questions about them. The rank
of an elliptic curve is a measure of the number of solutions
of the equation which defines the curve. In recent years
there has been spectacular progress in the theory of elliptic
curves, but the rank remains very mysterious. Even basic
questions such as how to compute the rank, or whether the rank
can be arbitrarily large, are not settled. In this talk we
will survey what is known, as well as what is conjectured but
not known, about ranks of elliptic curves.

There is an interesting relation between Lie 2-algebras, the Kac-Moody
central extensions of loop groups, and the group String(n).
A Lie 2-algebra is a categorified version of a Lie algebra where
the Jacobi identity holds up to a natural isomorphism called the
"Jacobiator". Similarly, a Lie 2-group is a categorified
version of a Lie group. If G is a simply-connected compact simple
Lie group, there is a 1-parameter family of Lie 2-algebras
g_k each having Lie(G) as its Lie algebra of objects, but with a
Jacobiator built from the canonical 3-form on G. There appears to
be no Lie 2-group having g_k as its Lie 2-algebra, except when k = 0.
However, for integral k there is an infinite-dimensional Lie 2-group
whose Lie 2-algebra is *equivalent* to g_k. The objects of this
2-group are based paths in G, while the automorphisms of any object
form the level-k Kac-Moody central extension of the loop group of G.
This 2-group is closely related to the kth power of the canonical gerbe
over G. Its nerve gives a topological group that is an extension of G
by the Eilenberg-MacLane space K(Z,2). When k = +-1 this topological
group can also be obtained by killing the third homotopy group of G.
Thus, when G = Spin(n), it is none other than String(n).

Dissipative transport in solids can be described by a Markov
semigroup of completely positive operators on the observable algebra of the charge carriers creation and annihilation operators. A model of generators of such semigroups, called the quantum jump model, will be presented. The linear response theory will be shown to provide the expression of transport coefficients through a Green-Kubo formula. This formula will be justified rigorously through the spectral property of the
generator of the quantum jump model in various situations. The case of aperiodic solids, such as strongly disordered systems will be emphasized, in view of its relevance in the theory of the Quantum Hall effect.