Imperial College London

Abstract. As the first  step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multi-plicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (Xt; Yt) of the equations dXt = Ytdt, dYt = jXtj_dBt, (X0; Y0) = (x0; y0). In particular, we prove that solutions arenonunique if 0 < _ < 1 and (x0; y0) = (0; 0) and unique if 1=2 < _ and (x0; y0) 6= (0; 0). We also show that blowup in _nite time holds if _ > 1 and (x0; y0) 6= (0; 0).

This is a joint work with A. Gomez, J.J. Lee, C. Mueller and M. Salins.

 Locality is not expected to be a fundamental aspect of a full theory of quantum gravity; it should be emergent in an appropriate semiclassical limit.  In the context of general holography, I'll define a new construct - the causal state - which provides a necessary and sufficient condition for a boundary state to have a holographic semiclassical dual causal geometry (and thus be "local").  This definition illuminates some general features of holographic quantum gravity: for instance, I'll show that the emergence of locality is "all or nothing" in the sense that it exhibits features of quantum error correction and quantum secret sharing.  In the special case of AdS/CFT, I'll also argue that the causal state is the natural boundary dual to the so-called causal wedge of a region. 

Multi-agent systems in nature oftentimes exhibit emergent behaviour, i.e. the formation of patterns in the absence of a leader or external stimuli such as light or food sources. We present a non-local two species crossinteraction model with cross-diffusion and explore its long-time behaviour. We observe a rich zoology of behaviours exhibiting phenomena such as mixing and/or segregation of both species and the formation of travelling pulses.

TBC

Lyons’ theory of rough paths allows us to solve stochastic differential equations driven by a Gaussian processes X of finite p-variation. The rough integral of the solutions against X again exists. We show that the solution also belong to the domain of the divergence operator of the Malliavin derivative, so that the 'Skorohod integral' of the solution with respect to X can also be defined. The latter operation has some properties in common with the Ito integral, and a natural question is to find a closed-form conversion formula between this rough integral and its Malliavin divergence. This is particularly useful in applications, where often one wants to compute the (conditional) expectation of the rough integral. In the case of Brownian motion our formula reduces to the classical Stratonovich-to-Ito conversion formula. There is an interesting difference between the formulae obtained in the cases 2<=p<3 and 3<=p<4, and we consider the reasons for this difference. We elaborate on the connection with previous work in which the integrand is generally assumed to be the gradient of a smooth function of X_{t}; we show that our formula can recover these results as special cases. This is joint work with Nengli Lim.

Averaging, either spatial or temporal, is a powerful technique in complex multi-scale systems.

However, in some situations it can be difficult to justify.

For example, many real-world networks in technology, engineering and biology have a function and exhibit dynamics that cannot always be adequately reproduced using network models given by the smooth dynamical systems and fixed network topology that typically result from averaging. Motivated by examples from neuroscience and engineering, we describe a model for what we call a "functional asynchronous network". The model allows for changes in network topology through decoupling of nodes and stopping and restarting of nodes, local times, adaptivity and control. Our long-term goal is to obtain an understanding of structure (why the network works) and how function is optimized (through bifurcation).

We describe a prototypical theorem that yields a functional decomposition for a large class of functional asynchronous networks. The result allows us to express the function of a dynamical network in terms of individual nodes and constituent subnetworks.

Let A be an abelian variety over a strictly henselian discretely valued field K. In his 1992 paper "Néron models and tame ramification", Edixhoven has constructed a filtration on the special fiber of the Néron model of A that measures the behaviour of the Néron model with respect to tamely ramified extensions of K. The filtration is indexed by rational numbers in [0,1], and if A is wildly ramified, it is an open problem whether the places where it jumps are always rational. I will explain how an interpretation of the filtration in terms of logarithmic geometry leads to explicit formulas for the jumps in the case where A is a Jacobian, which confirms in particular that they are rational. This is joint work with Dennis Eriksson and Lars Halvard Halle.

We propose a randomised version of the Heston model--a widely used stochastic volatility model in mathematical finance--assuming that the starting point of the variance process is a random variable. In such a system, we study the small- and large-time behaviours of the implied volatility, and show that the proposed randomisation generates a short-maturity smile much steeper (`with explosion') than in the standard Heston model, thereby palliating the deficiency of classical stochastic volatility models in short time. We precisely quantify the speed of explosion of the smile for short maturities in terms of the right tail of the initial distribution, and in particular show that an explosion rate of $t^\gamma$ (gamma in [0,1/2]) for the squared implied volatility--as observed on market data--can be obtained by a suitable choice of randomisation. The proofs are based on large deviations techniques and the theory of regular variations. Joint work with Fangwei Shi (Imperial College London)

In this talk we will give an overview of the recent research on global quantizations on spaces of different types: compact and nilpotent Lie groups, general locally compact groups, compact manifolds with boundary.