Mathematical models of pattern formation have begun to play a valuable role in understanding morphogenetic processes in both normal and disease conditions. have been successfully applied to a number of phenomena. We will review several applications of Partial Differential Equation models, particularly to the formation of focal lesions in vascular calcification, which are driven by Bone Morphogenetic Proteins and their inhibitors.

However, most applications (including ours), have focused on Turing patterns, which arise as primary bifurcations of periodic patterns from a uniform equilibrium state. These linear instabilities are only the first level of the 'pattern zoo'. We will discuss further bifurcations 'far from Turing' and their associated patterns (holes, isolated spots, etc.), including some applications to physiology.

We will also discuss PDE models of branching morphogenesis in vasculature, including applications to defective branching and/or defective connections, as seen in a number of disease conditions, such as arteriovenous malformations, uneven caliber arteries, and other disease states.

Bistability plays a central role in the gene regulatory networks (GRNs) controlling many essential biological functions, including cellular differentiation and cell cycle control. However, establishing the network topologies that can exhibit bistability remains a challenge, in part due to the exceedingly large variety of GRNs that exist for even a small number of components. I will describe recent work in which the parameter-free methods of chemical reaction network theory were employed in a comprehensive in silico search for bistable network topologies among more than 40,000 simple two-gene circuits. In addition to the identification of a large number of previously unknown bistable switches, our results highlight the potential usefulness of parameter-free modeling to the study of network evolution, and are suggestive of a role in the development of novel synthetic biological switches.

We will report on the construction of energy minimizing coarse spaces built by patching solutions to appropriate saddle point problems. We first set an abstract framework for such constructions, and then we give an example of constructing coarse space and stable interpolation operator for the two level Schwarz method. We apply the theoretical results in the design of coarse spaces for discretizations of PDE with large varying coefficients. The stability and approximation bounds of the constructed interpolant are in a weighted norm and are independent of the variations in the coefficients. Such spaces can be used in two level overlapping Schwarz algorithms for elliptic PDEs with large coefficient jumps generally not resolved by a standard coarse grid. This is a joint work with Robert Scheichl (University of Bath, UK) and Panayot S. Vassilevski (Lawrence Livermore National Lab).

We review the asymptotic behavior of a class of Toeplitz (as well as related
Hankel and
Toeplitz + Hankel) determinants which arise in integrable models and other
contexts.
We discuss Szego, Fisher-Hartwig asymptotics, and a transition between them. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling
limits, to Fredholm
determinants which appear in the theory of group representations, in
random matrices, random permutations and partitions. The connection to
Toeplitz determinants
helps to evaluate the asymptotics of related Fredholm determinants in
situations of interest, and we
mention some of the corresponding results.