The purpose of this talk is to introduce the concept of quantum ergodicity of the eigenfunctions of the Laplacian and show its relation to the classical ergodicity of the geodesic flow on Riemannian manifolds. The talk is somewhat elementary and no background on Riemannian geometry or dynamical systems is necessary.

More and more, traveling waves are observed inside individual cells. These waves can be pulses of biochemical factors (diffusing proteins or metabolites) but also mechanical factors (such as the cell cortex). One example of mechanical traveling wave is offered by cellular blebs, pressure-driven “bubbles” on the cell surface implicated in cell division, apoptosis and cell motility. Blebs exhibit a range of behaviors including contracting in place, travel around the cell’s periphery, or repeated blebbing, making them biophysically interesting. Mechanical traveling waves are naturally modeled using “non-local” integro-PDEs, which lack the theoretical tools available for reaction-diffusion waves. This lack obfuscates simple questions such as what determines if a bleb will travel or not, and, if it travels, what determines its velocity? We present results in two parts: First, we develop a simple model of the cell surface describing the membrane, cortex, and adhesions, including the slow timescale cortical healing (treating implicitly the fast timescale of fluid motion). We find traveling and stationary blebs, which we characterize through numerical simulation. In the second part, we review the so-called Maxwell condition for reaction-diffusion systems that determines whether an excitation will travel or recover in place. We present our progress in deriving an analogue of the Maxwell condition for non-local integro-PDEs suitable for our cell surface model. This condition allows the theoretical (simulation-free) elucidation of blebbing including bleb travel. 

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.