# Events

### September 2017

28/09/2017

Alex Novikov (UTS): Analytical and numerical approximations for maximum of weighted Brownian motion 15:00-16:00 KCL Department of Mathematics Strand Campus room C16 East Wing (ask at reception)

Motivated by the use of weighted Kolmogorov-Smirnov tests in Gene Set Enrichment Analysis (GSEA) we consider the problem of finding fast and accurate approximations for nonlinear boundary crossing probabilities. A technique based on the approximations via exact lower and upper bounds with use of piecewise linear boundaries has been developed. An alternative approach based on solving the system of two integral equations will be also discussed.
In the numerical examples, in the context of GSEA, we compare the results obtained using these two techniques to Durbin's asymptotic approximations and Monte-Carlo simulation.

### October 2017

03/10/2017

10/10/2017

Codina Cotar (UCL): Density functional theory and many-marginals optimal transport with Coulomb and Riesz costs 15:00-16:00 KCL Department of Mathematics

Multi-marginal optimal transport with Coulomb cost arises as a dilute limit of density functional theory, which is a widely used electronic structure model. The number N of marginals corresponds to the number of particles. I will discuss the question whether Kantorovich minimizers must be Monge minimizers (yes for N = 2, open for N > 2, no for N =infinity), and derive the surprising phenomenon that the extreme correlations of the minimizers turn into independence in the large N limit. I will also discuss the next order term limit and the connection of the problem to Coulomb and Riesz gases.
The talk is based on joint works with Gero Friesecke (TUM), Claudia Klueppelberg (TUM), Brendan Pass (Alberta) and Mircea Petrache (MPI/ETH).

17/10/2017

Vadim Shcherbakov (Royal Holloway): Boundary effects in the evolution of interacting birth-and-death processes 15:00-16:00 KCL Department of Mathematics

This talk concerns the long term behaviour of a continuous time Markov chain formed by two interacting birth-and-death processes. In absence of interaction the Markov chain is a pair of two independent birth-and-death processes, whose long term behaviour is well known. Presence of interaction can significantly affect the individual behaviour. We describe in detail the long term behaviour for a range of models of two interacting birth-and-death processes. The following effect is detected in some of these models. Namely, eventually one of the components tends to infinity, while the other component is confined to a finite set {0, 1,..., k}, where k is explicitly determined by the model parameters. Moreover, the bounded component of the corresponding embedded discrete time Markov chain takes any values from the above finite set infinitely many times. In particular, this effect is observed in a stochastic population model for competition between two species with Lotke-Volterra interaction.
The talk is based on joint work with M. Menshikov and S. Volkov.

### November 2017

14/11/2017

Reimer Kuehn: Heterogeneous micro-structure of percolation in sparse networks 15:00-16:00 TBC

We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using the message-passing formulation of percolation, we discover considerable variation across the network in the probability of a particular node to remain part of the giant component, and in the expected size of small clusters containing that node. In
the vicinity of the percolation threshold, weakly non-linear analysis reveals that node-to-node heterogeneity is captured by the recently introduced notion of non-backtracking centrality. We supplement these results for fixed finite networks by a population dynamics approach to analyse random graph models in the infinite system size limit, also providing closed-form approximations for the large mean degree limit of Erdos-Renyi random graphs. Interpreted in terms of the application of percolation to real-world processes, our results shed light on the heterogeneous exposure of different nodes to cascading failures, epidemic spread, and information flow.