Every year since 1991, the IAS/Park City Mathematics Institute (PCMI) has conducted a three-week summer program on a specific theme in mathematics. The Institute’s goal is an ambitious one: to incorporate the annual theme into programs for high school teachers, undergraduate students and faculty, graduate students and university research faculty. *Mathematical Biology*, volume 14 in the series, represents the graduate lecture notes from the 2005 program.

With its home at the Institute for Advanced Study, PCMI has had no difficulty in recruiting the “heavy hitters” from each field. (One imagines a letter from Einstein Drive, written from the desk once belonging to say, Gödel, von Neumann or Borel, might read less like a polite inquiry and more like a royal summons…) The editors and authors of this volume too form an impressive group indeed.

There is an unusual and deliberate structure to *Mathematical Biology* (the text), and it says much about the authors’ take on mathematical biology (the field). John Polking, the series editor, begins with a preface detailing the PCMI mission and some of its history. Volume editors Mark Lewis and Jim Keener give a formal introduction to the field, then lay groundwork with two additional introductory chapters. In these discussions, on biochemical reactions and spatio-ecological dynamics respectively, the editors showcase what they feel to be the fundamental elements of modern mathematical biology.

Foremost in their philosophy is that a commitment to first principles — whether from chemistry, physiology, medicine or physics — is paramount. This theme resonates throughout the volume, and unifies what has evolved into a very broad field. Yet mathematical biology *is* mathematics, and it carries a responsibility for rigor. To all those involved in this project, modern mathematical biology is at once an academic discipline, and an applied science of enormous gravity.

The volume’s five main speakers focus on specialized biological topics. From the book:

The lectures of Jim Cushing show how discrete dynamical systems have been used to model the population dynamics of flour beetles; those of David Earn show how the SARS epidemic can be studied using SIR models; Leon Glass uses topological arguments to study the dynamics of oscillatory biological dynamics; Helen Byrne shows how populations of cancer cells can be modeled and studied; Paul Bressloff describes the dynamics of neural systems.

Each topic is meant as a standalone “snapshot of in-depth research,” and all chapters are mathematically serious. Yet despite its depth, the pacing of the book feels unhurried. Bressloff’s* Lectures in Mathematical Neuroscience*, for example, spans over 100 pages. Rich graphics, many in color, complement the pace, encouraging the reader to linger on the page. Those adopting the volume as a textbook will be grateful for many worked examples and exercises included in each lecture set. An exhaustive bibliography, links to online data sources and sample MATLAB code, as well as suggestions for projects, make the book ideal as a springboard for further investigation.

Attention to context is central to the volume, and most readily apparent in David Earn’s *Mathematical Epidemiology of Infectious Diseases*. His first lecture begins with a color photograph of a London “Bill of Mortality” from a one-week period in 1665:

*Live births*: *146*

*Lost to the Plague*:* 5,533*

This grim statistic sets the tone of the lectures and reminds us of the high stakes in the science. As it turns out, (largely) eradicated diseases are in fact relatively easy to model. But what of current threats? Seasonally forced diseases? Epidemics with a huge (and potentially hidden) cyclic period? In subsequent lectures, Earn focuses on measles, SARS and HIV/AIDS. He begins with time series data, and constructs the barest of models. Then, with biological principles as a touchstone, carefully adds “layers of increasing realism” to better fit observed dynamics, always weighing the cost of reliance on increasing parameterization. This deliberate, almost skeptical, approach to modeling lies at the heart of the philosophy behind mathematical biology today.

While the authors work in distinct and vastly different subfields of mathematical biology, some questions persist as common threads. First, simple assumptions can lead to differential models with startlingly complex dynamics. Yet is a continuous model too simple for essentially discrete real-world scenarios? Can a purely deterministic approach suffice, or do stochastic forces — along with the inevitable background noise they bring — represent an essential dynamic?

Graduate students and mathematics professionals will appreciate eloquent and effective discussions of these and other central questions in the field. Although undergraduates do participate in PCMI summer programs (see *Math Horizons, *Nov 2008), this book is not intended for a wide undergraduate audience. But perhaps for independent study, or small-group undergraduate research projects, *Mathematical Biology* might make an exceptional starting point.

Matthew Glomski is assistant professor of mathematics at Marist College in Poughkeepsie, New York, and a Project NExT fellow (red dot '08). He can be reached at matthew.glomski@marist.edu.