Past

Let E be an elliptic curve over the rationals. To get an asymptotic to the number of primes p

joint work with R Gurjar

Dirichlet to Neumann maps and their generalizations are exceptionally useful tools in the study of eigenvalue problems for ODEs and PDEs. They also have real physical significance through their occurrence in electrical impedance tomography, with applications to medical imagine, landmine detection and non-destructive testing. This talk will review some of the basic properties of Dirichlet to Neumann maps, some new abstract results which make it easier to use them for a wide variety of models, and some analytical/numerical results which depend on them, including detection and elimination of spectral pollution.

We call a family of permutations A in Sn 'intersecting' if any two permutations in A agree in at least one position. Deza and Frankl observed that an intersecting family of permutations has size at most (n-1)!; Cameron and Ku proved that equality is attained only by families of the form {σ in Sn: σ(i)=j} for i, j in [n].

We will sketch a proof of the following `stability' result: an intersecting family of permutations which has size at least (1-1/e + o(1))(n-1)! must be contained in {σ in Sn: σ(i)=j} for some i,j in [n]. This proves a conjecture of Cameron and Ku.

In order to tackle this we first use some representation theory and an eigenvalue argument to prove a conjecture of Leader concerning cross-intersecting families of permutations: if n >= 4 and A,B is a pair of cross-intersecting families in Sn, then |A||B|

The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, we prove that, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss the link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on the Poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes (joint work with Zhonggen Su).

The McKean stochastic game (MSG) is a two-player version of the perpetual American put option. The MSG consists of two agents and a certain payoff function of an underlying stochastic process. One agent (the seller) is looking for a strategy (stopping time) which minimises the expected pay-off, while the other agent (the buyer) tries to maximise this quantity.

For Brownian motion one can find the value of the MSG and the optimal stopping times by solving a free boundary value problem. For a Lévy process with jumps the corresponding free boundary problem is more difficult to solve directly and instead we use fluctuation theory to find the solution of the MSG driven by a Lévy process with no positive jumps. One interesting aspect is that the optimal stopping region for the minimiser "thickens" from a point to an interval in the presence of jumps. This talk is based on joint work with Andreas Kyprianou (University of Bath).

Trading a financial asset involves a sequence of decisions to buy or sell the asset over time. A traditional trading strategy is to buy low and sell high. However, in practice, identifying these low and high levels is extremely challenging and difficult. In this talk, I will present our ongoing research on characterization of these key levels when the underlying asset price is dictated by a mean-reversion model. Our objective is to buy and sell the asset sequentially in order to maximize the overall profit. Mathematically, this amounts to determining a sequence of stopping times. We establish the associated dynamic programming equations (quasi-variational

inequalities) and show that these differential equations can be converted to algebraic-like equations under certain conditions.

The two threshold (buy and sell) levels can be found by solving these algebraic-like equations. We provide sufficient conditions that guarantee the optimality of our trading strategy.