E. coli has a natural behavioral variable---the direction of rotation of its flagellar rotary motor. Monitoring this one-dimensional behavioral response in reaction to chemical perturbation has been instrumental in the understanding of how E. coli performs chemotaxis at the genetic, physiological, and computational level. We are applying this experimental strategy to the study of bacterial thermotaxis - a sensory mode that is less well understood. To investigate bacterial thermosensation we subject single cells to well defined thermal stimuli such as impulses of heat produced by an IR laser and analyze their response. Higher organisms may have more complicated behavioral responses because their motions have more degrees of freedom. Here we provide a comprehensive analysis of motor behavior of such an organism -- the nematode C. elegans. Using tracking video-microscopy we capture a worm's image and extract the skeleton of the shape as a head-to-tail ordered collection of tangent angles sampled along the curve. Applying principal components analysis we show that the space of shapes is remarkably low dimensional, with four dimensions accounting for > 95% of the shape variance. We also show that these dimensions align with behaviorally relevant states. As an application of this analysis we study the thermal response of worms stimulated by laser heating. Our quantitative description of C. elegans movement should prove useful in a wide variety of contexts, from the linking of motor output with neural circuitry to the genetic basis of adaptive behavior.

How cells determine or lose their identity is a key question in the study of development and carcinogenesis. Differentiation is regulated by a variety of factors that interact in networks. Intriguingly, some complex differentiation decisions involving many possible outcomes are not easily reduced to a defined series of binary fate decisions. I have approached the mechanism by which cells make such complex decisions in two ways.

First, in a "top-down" approach, I asked which classes of simple network designs could provide sufficiently rich behavior to account for differentiation decisions. Competitive heterodimerization networks, which are present in numerous developmental and physiological contexts, stand out as being particularly flexible. Modeling of these networks suggests unforeseen biological functions.

Second, in a "bottom-up" approach, I started to address a tractable, "real-world" experimental model of a complex differentiation decision. I chose a three-way differentiation decision made in the C. elegans germline, which provides a genetic network that has been extensively characterized. I showed experimentally that differentiation is controlled by positional and timing mechanisms. These results lay the groundwork for a systems biology analysis of differentiation in the C. elegans germline.

An important challenge for the future is to comprehensively characterize a given experimental model, by building on the understanding of simple networks that are amenable to mathematical study.

We will describe some recent applications of resolution of singularities methods to questions of interest in analysis. In particular, we will describe a recent local resolution of singularities algorithm of the speaker for real-analytic functions. This algorithm is elementary and self-contained, and makes extensive use of Newton polyhedra and local coordinate systems. Some applications will be given. These include applications to oscillatory integrals, asymptotic expansions for sublevel set volumes, and the determination of the supremum of the positive e for which |f|^{- e} is locally integrable. Here f denotes a real-analytic function.

Determining the long-term behavior of large biochemical models has proved to be a remarkably difficult problem. Yet these models exhibit several characteristics that might make them amenable to study under the right perspective. One possible approach (first suggested by Sontag and
Angeli) is their decomposition in terms of so-called monotone systems, which can be thought of as systems with exclusively positive feedback.

In this talk I discuss some general properties of monotone dynamical systems, including recent results regarding their generic convergence
towards an equilibrium. Then I will discuss the use of monotone systems to model biochemical behaviors such as switches and oscillations under
time delays.

One approach to investigate the structure of an algebraic variety X is to study the geometry of curves, especially the rational curves, that X contains. This approach relies on classical geometric ideas and strives to understand the intrinsic geometry of varieties. It is nowadays understood that if X contains many rational curves, then their geometry determines X to a large degree.

After Shigefumi Mori showed in his landmark works that many interesting varieties contain rational curves, their systematic study became a standard tool in algebraic geometry. The spectrum of application is diverse and covers long-standing problems such as deformation rigidity, stability of the tangent bundle, classification problems, and generalizations of the Shafarevich hyperbolicity conjecture.

The talk concentrates on examples and basic properties of minimal degree rational curves on projective varieties. Some of the more advanced applications will be discussed.