The simple harmonic urn is a discrete-time stochastic process on $\mathbb Z^2$ approximating the phase portrait of the harmonic oscillator using very basic transitional probabilities on the lattice, incidentally related to the Eulerian numbers.
This urn which we consider can be viewed as a two-colour generalized Polya urn with negative-positive reinforcements, and in a sense it can be viewed as a “marriage” between the Friedman urn and the OK Corral model, where we restart the process each time it hits the horizontal axes by switching the colours of the balls. We show the transience of the process using various couplings with birth and death processes and renewal processes. It turns out that the simple harmonic urn is just barely transient, as a minor modification of the model makes it recurrent.
We also show links between this model and oriented percolation, as well as some other interesting processes.
This is joint work with E. Crane, N. Georgiou, R. Waters and A. Wade.
Lattices in semisimple Lie groups have been studied from the point of view of number theory, algebraic groups, topology and geometry, and geometric group theory. The Fragestellung of one line of investigation is to what extent the properties of the lattice determine, and are determined by, the properties of the group. This talk reviews a number of results about lattices, and in particular looks at Mostow--Margulis rigidity.
Given any two diagrams of the same knot or link, we
provide an explicit upper bound on the number of Reidemeister moves required to
pass between them in terms of the number of crossings in each diagram. This
provides a new and conceptually simple solution to the equivalence problem for
knot and links. This is joint work with Marc Lackenby.