important roles in the life sciences. Hence, it strongly cautions against prejudging which subfields of mathematical research are most likely to contribute to biology.

Biological systems at all scales are characterized by high dimensionality, heterogeneity, robustness to perturbations, and the existence of strongly interacting, highly disparate spatial and temporal scales. While these characteristics also appear in some physical systems that have been successfully modeled mathematically—the modeling of heterogeneous and multiscale phenomena is in particular a vibrant topic of mathematical and engineering research—the modeling of biological systems will require greatly expanded capabilities in these areas. As is widely documented in the report, the characteristics enumerated above recur at all levels of biological organization, from molecules to ecosystems.

Although the committee feels strongly that mathematical research based on premature abstraction of biological problems risks irrelevance, there are more and more instances where mathematical tools have already proven their utility in a broad range of biological applications. Many examples are described in this report. In some instances, as biological applications of these tools have expanded, limitations on their effectiveness have become apparent. Nonetheless, there are opportunities here for effective and important mathematical research that is less tightly tied to particular biological applications than is typically the case. Such research will have varying degrees of innate mathematical interest but can have an important impact on biology.