This informal lecture will be a bit different from others in the series, in that the primary objective will not be to discuss a very specific mathematical problem (or set of problems), but rather to introduce students to some of the main issues in the field of "systems biology" and to illustrate a few of the many possibilities for interesting mathematical research that arise from this very active field of science.

The field of (Molecular) Systems Biology mostly concerns itself with individual cells, or small collections of cells, seen as networks involving DNA, RNA, proteins, metabolites, and small molecules. An example is the study of signal transduction pathways in cells and their disruption in cancer. (Some people include in the scope of Systems Biology other fields that also deal with life processes as interactions among multiple components, but at the larger level of organs, organisms, or populations: neuronal networks in brain function, heart models, the spread of epidemics in populations, ecosystem responses to climate change, etc.)

The Life Sciences are in the midst of a major revolution in quantitative theoretical formulations, perhaps not unlike the transformation that physics underwent starting in the 17th century. It is widely recognized by leading biologists that the typical "reductionist" approach is not powerful enough to describe, analyze, and interpret the complex behaviors of such networks. Quantitative (i.e, mathematical) formalisms, concepts, tools, and models are required, and there is a major role to be played by mathematicians in applying and adapting known theory to model and understand specific systems.

On the other hand, the study of problems in systems biology also leads naturally to new mathematical questions in established areas of mathematics (for example, in probability, theoretical computer science, control theory, PDE's, or algebraic geometry). I will illustrate with some issues in dynamical systems theory.