L4

The talk is concerned with adapted solution of a multi-dimensional BSDE with a "diagonally" quadratic generator, the quadratic part of whose iith component only depends on the iith row of the second unknown variable. Local and global solutions are given. In our proofs, it is natural and crucial to apply both John-Nirenberg and reverse Holder inequalities for BMO martingales. 

Recently, a series of unprecedented leaks by Edward Snowden had made it possible for the first time to get a glimpse into the actual capabilities and limitations of the techniques used by the NSA and GCHQ to eavesdrop to computers and other communication devices. In this talk, I will survey some of the things we have learned, and discuss possible countermeasures against these capabilities.

I will discuss a recent joint work with A. Malchiodi (Pisa) and M. Micallef (Warwick) in which we show that not every harmonic map can be approximated by a sequence of $\alpha$-harmonic maps.

The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $H^1$ - Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This a consequence of a self-improving property that so-called reverse Hölder inequality implies. In the case of nonlocal equations a self-improving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire. 

We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients.
The algorithm is based on the construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regularity or growth assumption; in particular it applies to SDEs with coefficients that are nowhere continuous and that grow superlinearly. We show that if the diffusion coefficient is bounded and bounded away from 0, then our algorithm has a weak convergence rate of order 1/4. Finally, we illustrate the algorithm's performance with several examples.

 If $R = F_q[t]$ is the polynomial ring over a finite field
then the group $SL_2(R)$ is not finitely generated. The group $SL_3(R)$ is
finitely generated but not finitely presented, while $SL_4(R)$ is
finitely presented. These examples are facets of a larger picture that
I will talk about.

Lein’s confocal systems make accurate and precise measurements in many different applications. In applications where the object under test introduces variability and/or optical aberrations to the optical signal, the accuracy and precision may deteriorate. This technical challenge looks for mathematical solutions to improve the accuracy and precision of measurements made in such circumstances.

The presentation will outline the confocal principle, show “perfect” signals, give details of how we analyse such signals, then move on to less perfect signals and the effects on measurement accuracy and precision.