Stem cells are an important component of tissue architecture. Identifying the exact regulatory circuits that can stably maintain tissue homeostasis (that is, approximately constant size) is critical for our basic understanding of multicellular organisms. It is equally critical for figuring out how tumors circumvent this regulation, thus providing targets for treatment. Despite great strides in the understanding of the molecular components of stem-cell regulation, the overall mechanisms orchestrating tissue homeostasis are still far from being understood. Typically, tissue contains the stem cells, transit amplifying cells, and terminally differentiated cells. Each of these cell types can potentially secrete regulatory factors and/or respond to factors secreted by other types. The feedback can be positive or negative in nature. This gives rise to a bewildering array of possible mechanisms that drive tissue regulation. In this talk I describe a novel stochastic method of studying stem cell lineage regulation, which is based on population dynamics and ecological approaches. The method allows to identify possible numbers, types, and directions of control loops that are compatible with stability, keep the variance low, and possess a certain degree of robustness. I will also discuss evolutionary optimization and cancer-delaying role of stem cells.

We prove that Schrodinger operators with meromorphic potentials have purely singular continuous spectrum on the set {E: \delta{(\alpha,\theta)}>L(E)\} where \alpha is the frequency, \theta is the phase, delta is an explicit function, and L is the Lyapunov exponent. This extends a result of Jitomirskaya and Liu for the Maryland model to the general class of meromorphic potentials.

Given a  Kahler manifold, the smooth Calabi flow is the parabolic version of the constant scalar curvature equation. Given that this fourth order flow has a very undeveloped regularity and existence theory, J. Streets recast it as a weak gradient flow in the abstract completion of the space of Kahler metrics. In this talk we will show how a better understanding of the abstract completion gives updated information on the large time behavior of the weak Calabi flow, and how this fits into a well known conjectural picture of Donaldson. This is joint work with Robert Berman and Chinh Lu.