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.

 The weak efficient market hypothesis can be interpreted as asserting that major market indices should lie near the Markowitz efficient frontier.  This is seen in many years, but not in years before large market downturns.  Most notably, it was not seen in most of 2007 and 2008 leading up the the crash of 2008.  Indeed, all the major indices were found on the Markowitz inefficient frontier right after the crash. More generally, we consider a frontier computed for Markowitz portfolios that hold only long positions, which lies to the right of the classical Markowitz frontier. For many years this so-called long frontier lies close to the Markowitz frontier, but not in years of market volatility. The question will be posed, but not answered, if this separation is a measure of a market exposed to large short positions that could contribute to a systemic downturn.

Let $\Gamma$ be a graph with a weight $\sigma$. Let $d$ and $\mu$ be the distance and the measure associated with $\sigma$ such that $(\Gamma, d, \mu)$ is a doubling space. Let $p$ be the natural reversible Markov kernel associated with $\sigma$ and $\mu$  and $P$ the associated
operator defined by $Pf(x) = \sum_{y} p(x, y)f(y)$. Denote by $L=I-P$ the discrete Laplacian on $\Gamma$. 
In this talk we develop the theory of Hardy spaces associated to the discrete Laplacian $H^p_L$ for $0<p\leq 1$. We then obtain boundedness of certain singular integrals on $\Gamma$ such as square functions, spectral multipliers and Riesz transforms on the Hardy spaces $H^p_L$. This is joint work with The Anh Bui.