Southern California Number Theory Day, 10am - 5pm

With the proliferation of digital document collections in recent years (Web
pages, blogs, online news articles, etc) there has been an increasing
interest in automated tools that can automatically summarize, classify, and
annotate such text. In this talk I will provide an overview of statistical
topic models (also known as latent Dirichlet allocation models), which
provide a flexible framework for statistical modeling of high-dimensional
count data, in particular, for word counts in text documents.The talk will
begin by discussing the underlying statistical principles of these models,
including their interpretation within a general probabilistic
matrix-decomposition framework and contrasting them with techniques such as
principal component analysis and clustering. Estimation methods and
algorithms will be reviewed, with a focus on Gibbs sampling approaches. This
introduction will be followed by an illustration of how these models can be
to generate high-level summaries of document collections and to
automatically uncovering thematic trends in text over time. The remainder of
the talk will focus on extensions of the basic topic modeling framework,
with applications in document classification, document retrieval, as well as
discussion of parallel algorithms for scaling to very large data sets. A
number of different text data sets will be used during the talk as
illustrative examples, including archives of New York Times articles,
historical records of the Pennsylvania Gazette from the 18th century, large
databases of scientific publications such as PubMed and CiteSeer, and
publicly-available emails from the Enron corporation.

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.