It has been over two decades since M. Gromov initiated the study of pseudo-holomorphic curves in symplectic manifolds. In the past decade we have witnessed mathematical constructions of Gromov-Witten theory for algebraic varieties, as well as many major advances in understanding their properties. Recent works in string theory have motivated us to extend our interests to Gromov-Witten theory for Deligne-Mumford stacks. Such a theory has been constructed, but many of its properties remain to be understood. In this talk I will explain the main ingredients of Gromov-Witten theory of Deligne-Mumford stacks, and I will discuss some recent progress regarding main questions in Gromov-Witten theory of Deligne-Mumford stacks.

Using cone metrics on S2, W. Thurston proved that the triangulations of the sphere of non-negative combinatorial curvature are parameterized by the points of positive norm in a certain Eisenstein lattice. In this talk, I will discuss a different approach to this result based on the study of the degenerations of K3 surfaces. I will also discuss the connection to the compactification problem for the moduli space of polarized K3 surfaces.

Stochastic Loewner evolution (SLE) introduced by Oded Schramm is a breakthrough in studying the scaling limits of many two-dimensional lattice models from statistical physics. In this talk, I will discuss the proofs of the reversibility conjecture and duality conjecture about SLE. The proofs of these two conjectures use the same idea, which is to use a coupling technique to lift local couplings of two SLE processes that locally commute with each other to a global coupling. And from the global coupling, we can clearly see that the two conjectures hold.

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.