We study diffusions in random environment in higher dimensions. We assume that the environment is stationary and obeys finite range dependence. Once the environment is chosen, it remains fixed in time. To restore some stationarity, it is common to average over the environment. One then obtains the so-called annealed measures, that are typically non-Markovian measures.
Our goal is to study the asymptotic behavior of the diffusion in random environment under the annealed measure, with particular emphasis on the ballistic regime
('ballistic' means that a law of large numbers with non-vanishing limiting velocity holds). In the spirit of Sznitman, who treated the discrete setting, we introduce
conditions (T) and (T'), and show that they imply, when d>1, a ballistic law of large numbers and a central limit theorem with non-degenerate covariance matrix.
As an application of our results, we highlight condition (T) as a source of new examples of ballistic diffusions in random environment.

Methods of enumeration of spanning trees in a finite graph and relations to
various areas of mathematics and physics have been investigated for more
than 150 years. We will review the history and applications. Then we will
give new formulas for the asymptotics of the number of spanning trees of a
graph. A special case answers a question of McKay (1983) for regular
graphs. The general answer involves a quantity for infinite graphs that we
call ``tree entropy", which we show is a logarithm of a normalized
[Fuglede-Kadison] determinant of the graph Laplacian for infinite graphs.
Proofs involve new traces and the theory of random walks.

A number of recent experiments have shown that several organisms
that reproduce by fissioning (e.g. E. coli bacteria)
don't share the cellular damage they have
accumulated during their lifetime equally among their offspring. Using
a stochastic PDE model, David Steinsaltz and I have shown that under quite
general conditions the optimal asymptotic growth rate for a population
of fissioning organisms is obtained when there is a non-zero but moderate
amount of preferential segregation of damage -- too much or too little
asymmetry is counter-productive. The proof uses some new results of ours
on quasi-stationary distributions of one-dimensional diffusions and
some Sturm-Liouville theory. The talk is intended for a probability
audience and I won't assume any knowledge of biology.

Classical multidimensional scaling (MDS) is a method for visualizing high-dimensional point clouds by mapping to low-dimensional Euclidean space. This mapping is defined in terms of eigenfunctions of a matrix of interpoint proximities. I'll discuss MDS applied to a specific dataset: the 2005 United States House of Representatives roll call votes. In this case, MDS outputs 'horseshoes' that are characteristic of dimensionality reduction techniques. I'll show that in general, a latent ordering of the data gives rise to these patterns when one only has local information. That is, when only the interpoint distances for nearby points are known accurately. Our results provide insight into manifold learning in the special case where the manifold is a curve. This work is joint with Persi Diaconis and Susan Holmes.