I will discuss some analogies between the second main

theorem in Nevanlinna's theory and results in holomorphic dynamics.

The two main examples will be equidistribution results for endomorphisms of $\P^k$ and

equidistribution results for singular foliations by Riemann-surfaces in $\P^2$.

This is joint work with T. C. Dinh.

Cells with the same DNA can exhibit different phenotypes, thanks in part to the nonlinear dynamics of gene regulation: feedback loops in regulatory networks give rise to multiple steady-state attractors, which correspond to alternative states of gene expression. However, noisy gene expression can enable spontaneous transitions between these states. This noise-induced switching is thought to underlie critical cellular processes, including developmental fate-decisions, phenotypic plasticity in fluctuating environments, and even carcinogenesis.

This talk will discuss computational approaches that shed light on the dynamics of spontaneous switching in multi-stable gene networks. Numerical methods will be presented that tackle two challenges: the rare-event problem (switching between gene states occurs rarely, making it difficult to achieve proper sampling), and the curse of dimensionality (switching requires coordinated changes involving many species in the network). These approaches reveal how fluctuations both in occupancies of DNA regulatory sites and protein products drive switching events in common gene network motifs. 

We study a d-dimensional branching Brownian motion, among obstacles scattered

according to a Poisson random measure with a radially decaying intensity. Obstacles

are balls with constant radius and each one works as a trap for the whole motion when

hit by a particle. Considering a general offspring distribution, we derive the decay

rate of the annealed probability that none of the particles hits a trap,

asymptotically, in time. 

This proves to be a rich problem, motivating the proof of a general result about the speed 

of branching Brownian motion conditioned on

non-extinction. We provide an appropriate `skeleton-decomposition' for the underlying

Galton-Watson process when supercritical, and show that the `doomed particles' do 

not contribute to the asymptotic decay rate.

This is joint work with Mine Caglar and Mehmet Oz.

Fluctuating hydrodynamic descriptions provide a promising approach for modelling and simulating elastic structures that interact with a fluid when subject to thermal fluctuations.  This allows for capturing simultaneously such effects as the Brownian motion of spatially extended mechanical structures as well as their hydrodynamic  coupling and responses to external flows.  A significant advantage of this approach over alternative methods is the ability to handle the hydrodynamic equations directly using spatially adaptive discretizations or using domains having complex geometries.  However, this presents the challenge of numerically approximating a set of stochastic partial differential equations whose solutions are non-classical and only defined in the generalised sense of distributions.  We introduce stochastic discretization procedures based on ideas from statistical mechanics and we show how efficient stochastic computational methods can be developed.  We demonstrate our methods in the context of applications including the simulation of particles within microfluidic devices and the rheological responses of soft materials.  We also survey the current challenges in this field and opportunities for developing new more scalable algorithms.

Particle-based stochastic reaction diffusion methods have become a popular approach for studying the behavior of cellular processes in which both spatial transport and noise in the chemical reaction process can be important. While the corresponding deterministic, mean-field models given by reaction-diffusion PDEs are well-established, there are a plethora of different stochastic models that have been used to study biological systems, along with a wide variety of proposed numerical solution methods.

In this talk I will introduce our attempt to rectify the major drawback to one of the most popular particle-based stochastic reaction-diffusion models, the lattice reaction-diffusion master equation (RDME). We propose a modified version of the RDME that converges in the continuum limit that the lattice spacing approaches zero to an appropriate spatially-continuous model. I will then discuss some application areas to which we are applying these methods, focusing on how the complicated ultrastructure within cells, as reconstructed from X-ray CT images, might influence the dynamics of cellular processes.