Oxford

The convergence of Markov processes to stationary distributions is a basic topic of introductory courses in stochastic processes, and the theory has been thoroughly developed. What happens when we add killing to the process? The process as such will not converge in distribution, but the survivors may; that is, the distribution of the process, conditioned on survival up to time t, converges to a "quasistationary distribution" as t goes to infinity.

This talk presents recent work with Steve Evans, proving an analogue of the transience-recurrence dichotomy for killed one-dimensional diffusions. Under fairly general conditions, a killed one-dimensional diffusion conditioned to have survived up to time t either escapes to infinity almost surely (meaning that the probability of finding it in any bounded set goes to 0) or it converges to the quasistationary distribution, whose density is given by the top eigenfunction of the adjoint generator.

These theorems arose in solving part of a longstanding problem in biological theories of ageing, and then turned out to play a key role in a very different problem in population biology, the effect of unequal damage inheritance on population growth rates.

I will push Schanuel's conjecture in four directions: defining a dimension

theory (pregeometry), blurred exponential functions, exponential maps of

more general groups, and converses. The goal is to explain how Zilber's

conjecture on complex exponentiation is true at least in a "geometric"

sense, and how this can be proved without solving the difficult number

theoretic conjectures. If time permits, I will explain some connections

with diophantine geometry.

We survey the classification of structures interpretable in o-minimal theories in terms of thorn-minimal types. We show that a necessary and sufficient condition for such a structure to interpret a real closed field is that it has a non-locally modular unstable type. We also show that assuming Zilber's Trichotomy for strongly minimal sets interpretable in o-minimal theories, such a structure interprets a pure algebraically closed field iff it has a global stable non-locally modular type. Finally, if time allows, we will discuss reasons to believe in Zilber's Trichotomy in the present context

It is an interesting exercise to compute the iterated integrals of Brownian Motion and to calculate the expectations (of polynomial functions of these integrals).

Recent work on constructing discrete measures on path space, which give the same value as Wiener measure to certain of these expectations, has led to promising new numerical algorithms for solving 2nd order parabolic PDEs in moderate dimensions. Old work of Krylov associated finitely additive signed measures to certain constant coefficient PDEs of higher order. Recent work with Levin allows us to identify the relevant expectations of iterated integrals in this case, leaving many interesting open questions and possible numerical algorithms for solving high dimensional elliptic PDEs.