Past

Given K a separably closed field of finite ( > 1) degree of imperfection, and semiabelian variety A over K, we study the maximal divisible subgroup A^{sharp} of A(K). We show that the {\sharp} functor does not preserve exact sequences and also give an example where A^{\sharp} does not have relative Morley rank. (Joint work with F. Benoist and E. Bouscaren)

Abstract available at: http://people.maths.ox.ac.uk/~kirby/LInnocente.pdf

We consider an infinite plate being withdrawn from an infinite pool of viscous liquid. Assuming that the effects of inertia and surface tension are weak, Derjaguin (1943) conjectured that the 'load', i.e. the thickness of the liquid film clinging to the plate, is determined by a certain formula involving the liquid's density and viscosity, the plate's velocity and inclination angle, and the acceleration due to gravity.

In the present work, Deryagin's formula is derived from the Stokes equations in the limit of small slope of the plate (without this assumption, the formula is invalid). It is shown that the problem has infinitely many steady solutions, all of which are stable - but only one of these corresponds to Derjaguin’s formula. This particular steady solution can only be singled out by matching it to a self-similar solution describing the non-steady part of the film between the pool and the film’s 'tip'. Even though the near-pool region where the steady state has been established expands with time, the upper, non-steady part of the film (with its thickness decreasing towards the tip) expands faster and, thus, occupies a larger portion of the plate. As a result, the mean thickness of the film is 1.5 times smaller than the load.

The results obtained are extended to order-one inclinantion angles and the case where surface tension is present.

We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^\nu$ is constrained to satisfy an a.s.~constraint $Z^\nu(T)\in G\subset \R^{d+1}$ $\Pas$ at some final time $T>0$.  When the set is of the form $G:=\{(x,y)\in \R^d\x \R~:~g(x,y)\ge 0\}$, with $g$ non-decreasing in $y$, we provide a Hamilton-Jacobi-Bellman  characterization of the associated value function. It gives rise to a state constraint problem where the constraint can be expressed in terms of an auxiliary value function $w$ which characterizes the set $D:=\{(t,Z^\nu(t))\in [0,T]\x\R^{d+1}~:~Z^\nu(T)\in G\;a.s.$ for some $ \nu\}$. Contrary to standard state constraint problems, the domain $D$ is not given a-priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function $w$ which is itself a viscosity solution of a non-linear parabolic PDE.  Applying ideas recently developed in Bouchard, Elie and Touzi (2008), our general result also allows to consider optimal control problems with moment constraints of the form $\Esp{g(Z^\nu(T))}\ge 0$ or $\Pro{g(Z^\nu(T))\ge 0}\ge p$.

The climate is largely determined by the ocean flow, which in itself is driven by wind and by gradients in temperature and salinity. Nowadays numerical models exist that are able to describe the occurring phenomena not only qualitatively but also quantitatively. At the Institute for Marine and Atmospheric research Utrecht (IMAU) a so-called thermohaline circulation model is developed in which methods of dynamical systems theory are used to study the stability of ocean flows. Here bifurcation diagrams are constructed by varying the strength of the forcing, for instance the amount of fresh water coming in from the north due to melting. For every value of the strength we have to solve a nonlinear system, which is handled by a Newton-type method. This produces many linear systems to be solved. 

In the talk the following will be addressed: the form of the system of equations, a special purpose method which uses Trilinos and MRILU. The latter is a multilevel ILU preconditioner developed at Groningen University. Results of the approach obtained on the Dutch national supercomputer will be shown.

We solve the problem of an agent with prospect theory preferences who seeks to liquidate a portfolio of (divisible) claims.

Our methodology enables us to consider different formulations of prospect preferences in the literature (piecewise exponential or piecewise power) and various price processes. We find that these differences in specification matter - for instance, with piecewise power functions, the agent may liquidate at a loss relative to break-even, albeit the likelihood of liquidating at a gain is much higher than liquidating at a loss. This is consistent with the disposition effect documented in empirical and experimental studies. We find the agent does not choose to partially liquidate a position, but rather, if liquidation occurs, the entire position is sold. This is in contrast to partial liquidation when agents have standard concave utilities.

I will discuss the investigatation of highly nonlinear solitary waves in heterogeneous one-dimensional granular crystals using numerical computations, asymptotics, and experiments. I will focus primarily on periodic arrangements of particles in experiments in which stiffer/heavier stainless stee are alternated with softer/lighter ones.

The governing model, which is reminiscent of the Fermi-Pasta-Ulam lattice, consists of a set of coupled ordinary differential equations that incorporate Hertzian interactions between adjacent particles. My collaborators and I find good agreement between experiments and numerics and gain additional insight by constructing an exact compaction solution to a nonlinear partial differential equation derived using long-wavelength asymptotics. This research encompasses previously-studied examples as special cases and provides key insights into the influence of heterogeneous, periodic lattice on the properties of the solitary waves.

I will briefly discuss more recent work on lattices consisting of randomized arrangements of particles, optical versus acoustic modes, and the incorporation of dissipation.

Last week, I proved five theorems about fusion systems, each with a (relatively) trivial proof. All of these theorems were known, but in each case the proof was (in some cases highly) non-trivial. I will introduce fusion systems and talk a bit about why they are interesting, and then prove some, or maybe all, of the theorems I proved.

In a joint project with Christopher Voll, I have investigated the representation zeta functions of compact p-adic Lie groups. In my talk I will explain some of our results, e.g. the existence of functional equations in a suitable global setting, and discuss open problems. In particular, I will indicate how piecing together information about local zeta functions allows us to determine the precise abscissa of convergence for the representation zeta function of the arithmetic group SL3(Z).