Heterogeneity among cells, a universal phenomenon in the noisy biological world, has stimulated much interest in systems biology. Traditional bulk measurements, however, are insufficient to understand such important phenomenon. Researchers have realized that many such complex biological systems need higher-resolution information than just ensemble average. Single cell approaches, instead, can reveal important information about how these complex systems function. We have studied two systems using a single live-cell genealogical tracking method. In one, we found that stochastic switching between different gene expression states in budding yeast is heritable. This striking behavior only became evident using genealogical information from growing colonies. Our model based on burst induced correlation can explain the bulk of our results. In the other system investigated, we have provided clear evidence for circadian gating in cyanobacteria, in which circadian rhythms regulate the timing of cell divisions. Simultaneous measurement of both circadian and cell cycle dynamics in individual cells reveals the direct relationships between these two biological clocks. We fit the data to a simple model to determine when cell cycle progression slows as a function of circadian and cell cycle phases. We infer that cell cycle progression in cyanobacteria slows during a specific circadian interval but is uniform across cell cycle phases.

Zeta functions are central topics in number theory and arithmetic algebraic geometry. They arise in many different forms. They can be viewed as generating functions for counting "points"
(or "solutions") of polynomial equations, thus contain deep arithmetic and geometric information of the equations. In this lecture, we will explain various zeta functions from this point of view, including the Riemann zeta function, the Hasse-Weil zeta function, and zeta
functions over finite fields.