UCL

We will discuss two recent short-time existence results for (1) mean curvature of surface clusters, where n-dimensional surfaces in R^{n+k}, are allowed to meet at equal angles along smooth edges, and (2) for planar networks, where curves are initially allowed to meet in multiple junctions that resolve immediately into triple junctions with equal angles. The first result, which is joint work with B. White, follows from an elliptic regularisation scheme, together with a local regularity result for flows with triple junctions, which are close to a static flow of the half-planes. The second result, which is joint work with T. Ilmanen and A.Neves, relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow. 
 

We introduce sufficient conditions for the solution of a multi-dimensional, Markovian BSDE to have a density. We show that a system of BSDEs possesses a density if its corresponding semilinear PDE exhibits certain regularity properties, which we verify in the case of several examples.

Backward SDEs have proven to be a useful tool in mathematical finance. Their applications include the solution to various pricing and equilibrium problems in complete and incomplete markets, the estimation of value adjustments in the presence of funding costs, and the solution to many utility/risk optimisation type of problems.
In this work, we prove an explicit error expansion for the approximation of BSDEs. We focus our work on studying the cubature  method of solution. To profit fully from these expansions in this case, e.g. to design high order approximation methods, we need in addition to control the complexity growth of the base algorithm. In our work, this is achieved by using a sparse grid representation. We present several numerical results that confirm the efficiency of our new method. Based on joint work with J.F. Chassagneux.
 

In the first part of the talk I will review Bob Fernholz' theory of functionally generated portfolios. In the second part I will discuss questions related to the existence of short-term arbitrage opportunities.
This is joint work with Bob Fernholz and Ioannis Karatzas

Modelling catalytic kinetics is indispensable for the design of reactors and chemical processes. However, developing accurate and computationally efficient kinetic models remains challenging. Empirical kinetic models incorporate assumptions about rate-limiting steps and may thus not be applicable to operating regimes far from those where they were derived. Detailed microkinetic modelling approaches overcome this issue by accounting for all elementary steps of a reaction mechanism. However, the majority of such kinetic models employ mean-field approximations and are formulated as ordinary differential equations, which neglect spatial correlations. On the other hand, kinetic Monte Carlo (KMC) approaches provide a discrete-space continuous-time stochastic formulation that enables a detailed treatment of spatial correlations in the adlayer (resulting for instance from adsorbate-adsorbate lateral interactions), but at a significant computation expense.1,2

Motivated by these challenges, we discuss the necessity of KMC descriptions that incorporate detailed models of lateral interactions. Focusing on a titration experiment involving the oxidation of pre-adsorbed O by CO gas on Pd(111), we discuss experimental findings that show first order kinetics at low temperature (190 K) and half order kinetics at high temperature (320 K), the latter previously attributed to island formation.3 We perform KMC simulations whereby coverage effects on reaction barriers are captured by cluster expansion Hamiltonians and Brønsted-Evans-Polanyi (BEP) relations.4 By quantifying the effect of adlayer structure versus coverage effects on the observed kinetics, we rationalise the experimentally observed kinetics. We show that coverage effects lead to the half order kinetics at 320 K, rather than O-island formation as previously thought.5,6

Subsequently, we discuss our ongoing work in the development of approximations that capture such coverage effects but are much more computationally efficient than KMC, making it possible to use such models in reactor design. We focus on a model for NO oxidation incorporating first nearest neighbour lateral interactions and construct a sequence of approximations of progressively higher accuracy, starting from the mean-field treatment and continuing with a sequence of Bethe-Peierls models with increasing cluster sizes. By comparing the turnover frequencies of these models with those obtained from KMC simulation, we show that the mean-field predictions deviate by several orders of magnitude from the KMC results, whereas the Bethe-Peierls models exhibit progressively higher accuracy as the size of the explicitly treated cluster increases. While more computationally intensive than mean-field, these models still enable significant computational savings compared to a KMC simulation, thereby paving the road for employing them in multiscale modelling frameworks.

References

1    M. Stamatakis and D. G. Vlachos, ACS Catal. 2 (12), 2648 (2012).

2    M. Stamatakis, J Phys-Condens Mat 27 (1), 013001 (2015).

3    I. Nakai, H. Kondoh, T. Shimada, A. Resta, J. N. Andersen, and T. Ohta, J. Chem. Phys. 124 (22), 224712 (2006).

4    J. Nielsen, M. d’Avezac, J. Hetherington, and M. Stamatakis, J. Chem. Phys. 139 (22), 224706 (2013).

5    M. Stamatakis and S. Piccinin, ACS Catal. 6 (3), 2105 (2016).

6    S. Piccinin and M. Stamatakis, ACS Catal. 4, 2143 (2014).

Feedforward layers are integral step in processing and transmitting sensory information across different regions the brain. Yet experiments reveal the difficulty of stable propagation through layers without causing neurons to synchronize their activity. We study the limits of stable propagation in a discrete feedforward model of binary neurons. By analyzing the spectral properties of a mean-field Markov chain model, we show when such information transmission persists. Addition of inhibitory neurons and synaptic noise increases the robustness of asynchronous rate transmission. We close with an example of feedforward processing in the input layer to cerebellum. 

In numerical analysis the design and analysis of computational methods is often based on, and closely linked to, a well-posedness result for the underlying continuous problem. In particular the continuous dependence of the continuous model is inherited by the computational method when such an approach is used. In this talk our aim is to design a stabilised finite element method that can exploit continuous dependence of the underlying physical problem without making use of a standard well-posedness result such as Lax-Milgram's Lemma or The Babuska-Brezzi theorem. This is of particular interest for inverse problems or data assimilation problems which may not enter the framework of the above mentioned well-posedness results, but can nevertheless satisfy some weak continuous dependence properties. First we will discuss non-coercive elliptic and hyperbolic equations where the discrete problem can be ill-posed even for well posed continuous problems and then we will discuss the linear elliptic Cauchy problem as an example of an ill-posed problem where there are continuous dependence results available that are suitable for the framework that we propose.

Subgroup separability is a group-theoretic property that has important implications for geometry and topology, because it allows us to lift immersions to embeddings in a finite sheeted covering space. I will describe how this works in the case of graphs, and go on to motivate the construction of special cube complexes as an attempt to generalise the technique to higher dimensions.

Erdos, Faudree, Gould, Gyarfas, Rousseau and Schelp, conjectured that
whenever the edges of a complete graph are coloured using three colours
there always exists a set of at most three vertices which have at least
two-thirds of their neighbours in one of the colours.  We will describe a
proof of this conjecture. This is joint work with Rahil Baber

For Logic Seminar: Note change of time and place.