University of Oxford

Cox processes arise as a natural extension of inhomogeneous Poisson Processes, when the intensity function itself is taken to be stochastic. In multiple applications one is often concerned with characterizing the posterior distribution over the intensity process (given some observed data). Markov Chain Monte Carlo methods have historically been successful at such tasks. However, direct methods are doubly intractable, especially when the intensity process takes values in a space of continuous functions.

In this talk I'll be presenting a method to overcome this intractability that is based on the idea of "thinning" and that does not resort to approximations.

We state a conjecture about the size of the intersection between a bounded-rank progression and a sphere, and we prove the first interesting case, a result of Chang. Hopefully the full conjecture will be obvious to somebody present.

The Balog-Szemeredi-Gowers theorem states that, given any finite subset of an abelian group with large additive energy, we can find its large subset with small doubling constant. We can ask how this constant depends on the initial additive energy. In the talk, I will give an upper bound, mention the best existing lower bound and, if time permits, present an approach that gives a hope to improve the lower bound and make it asymptotically equal to the upper bound from the beginning of the talk.

Let $F \in \mathbb{Z}[x_1,\ldots,x_n]$. Suppose $F(\mathbf{x})=0$ has infinitely many integer solutions $\mathbf{x} \in \mathbb{Z}^n$. Roughly how common should be expect the solutions to be? I will tell you what your naive first guess ought to be, give a one-line reason why, and discuss the reasons why this first guess might be wrong.

I then will apply these ideas to explain the intriguing parallels between the handling of the Brauer-Manin obstruction by Heath-Brown/Skorobogotov [doi:10.1007/BF02392841] on the one hand and Wei/Xu [arXiv:1211.2286] on the other, despite the very different methods involved in each case.

Given a model dynamical system, a model of any measuring apparatus relating states to observations, and a prior assessment of uncertainty, the probability density of subsequent system states, conditioned upon the history of the observations, is of some practical interest.

When observations are made at discrete times, it is known that the evolving probability density is a solution of the Bayesian filtering equations. This talk will describe the difficulties in approximating the evolving probability density using a Gaussian mixture (i.e. a sum of Gaussian densities). In general this leads to a sequence of optimisation problems and related high-dimensional integrals. There are other problems too, related to the necessity of using a small number of densities in the mixture, the requirement to maintain sparsity of any matrices and the need to compute first and, somewhat disturbingly, second derivatives of the misfit between predictions and observations. Adjoint methods, Taylor expansions, Gaussian random fields and Newton’s method can be combined to, possibly, provide a solution. The approach is essentially a combination of filtering methods and '4-D Var’ methods and some recent progress will be described.

Everybody has heard of the Faraday cage effect, in which a wire mesh does a good job of blocking electric fields and electromagnetic waves. For example, the screen on the front of your microwave oven keeps the microwaves from getting out, while light with its smaller wavelength escapes so you can see your burrito.  Surely the mathematics of such a famous and useful phenomenon has been long ago worked out and written up in the physics books, right?

Well, maybe.   Dave Hewett and I have communicated with dozens of mathematicians, physicists, and engineers on this subject so far, and we've turned up amazingly little.   Everybody has a view of why the Faraday cage mathematics is obvious, and most of their views are different.  Feynman discusses the matter in his Lectures on Physics, but so far as we can tell, he gets it wrong. 

For the static case at least (the Laplace equation), Hewett and I have made good progress with numerical explorations based on  Mikhlin's method backed up by a theorem.   The effect seems to much weaker than we had imagined -- are we missing something?  For time-harmonic waves (the Helmholtz equation), our simulations lead to further puzzles.  We need advice!  Where in the world is the literature on this problem? 

A model of a financial market is complete if any payoff can be obtained as the terminal value of a self-financing trading strategy. It is well known that numerous models, for example stochastic volatility models, are however incomplete. We present conditions, which, in a general diffusion framework, guarantee that in such cases the market of primitive assets enlarged with an appropriate number of traded derivative contracts is complete. From a purely mathematical point of view we prove an integral representation theorem which guarantees that every local Q-martingale can be represented as a stochastic integral with respect to the vector of primitive assets and derivative contracts.

When $\partial D$ is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of $\partial D$ from the sort time behaviour of $E(s)$. Furthermore, when the Minkowski dimension exists, finer geometric fluctuations can be recovered and $E(s)$ is controlled by $s^\alpha e^{f(\log(1/s))}$ for small $s$, for some $\alpha \in (0, \infty)$ and some regularly varying function $f$. The function $f$ is not constant is general and carries some geometric information.

When $\partial D$ is statistically self-similar, the Minkowski dimension and content of $\partial D$ typically exist and can be recovered from $E(s)$. Furthermore, $E(s)$ has an almost sure expansion $E(s) = c s^{\alpha} N_\infty + o(s^\alpha)$ for small $s$, for some $c$ and $\alpha \in (0, \infty)$ and some positive random variable $N_\infty$ with unit expectation arising as the limit of some martingale. In some cases, we can show that the fluctuations around this almost sure behaviour are governed by a central limit theorem, and conjecture that this is true more generally.

This is based on joint work with David Croydon and Ben Hambly.