New models describing wave propagation in transversely modulated optically induced waveguide arrays are proposed. In the weakly guided regime, a discrete nonlinear Schrodinger equation with the addition of bulk diffraction term and an external ``optical trap'' is derived. In the defocusing regime the optical trap induces a stable localized mode. In the limit of strong transverse guidance, the dynamics is governed by a model which represents the optical analogue of wave action.

We study the family of Hamiltonians which corresponds to the
adjacency operators on a percolation graph. We characterise the set of
energies which are almost surely eigenvalues with finitely supported
eigenfunctions. This set of energies is a dense subset of the algebraic
integers. The integrated density of states has discontinuities precisely
at this set of energies. We show that the convergence of the integrated
densities of states of finite box Hamiltonians to the one on the whole
space holds even at the points of discontinuity. For this we use an
equicontinuity-from-the-right argument. The same statements hold for the
restriction of the Hamiltonian to the infinite cluster. In this case we
prove that the integrated density of states can be constructed using local
data only. Finally we study some mixed Anderson-Quantum percolation models
and establish results in the spirit of Wegner, and Delyon and Souillard.

It is a classical problem to determine the span
of the theta series of a given quadraic space over
a small ring. In such a way, Jacobi proved his
famous formula of the number of ways of expressing
integers as sums of four squares.
For the norm form of a definite quaternion algebra B,
we determine the span integrally over very small ring
(for example, if B only ramifies at one prime p,
we shall determine the span over Z[1/(p-1)]).

Many widely used statistical models of evolution are algebraic varieties, that is, solutions sets of polynomial equations. We discuss this algebraic representation and its implications for the construction of maximum likelihood trees in phylogenetics. The ensuing interaction between combinatorial algebraic geometry and computational biology works as a two-way street: biologists may benefit from new mathematical tools, while mathematicians will find a rich source of open problems concerning objects reminiscent of objects familiar from classical projective geometry.