In 2004 Joel E. Cohen wrote an article in PLoS Biology entitled "Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better." This short article, which has been on my "To Read" list for a long time, is a brief history and assessment of the contributions that each field has and continues to make in the other.
As the title suggests, half of Cohen's claim is that the problems in modern biology will fuel the development of new mathematics. These mathematics should address the fundamental questions posed by biologists, such as "How does it work?" and "What is it for?" There are six such questions, and they are further divided according to the many, many orders of magnitude of space and time that are spanned by biological processes: from molecular biology to global ecosystems and from ultrafast photosynthesis to evolution. Scale-dependent systems and emergent phenomena are the primary themes in modern biological problems.
To illustrate his idea of mathematics, Cohen paints a picture of a tetrahedron with the topics of data structures, theories and models, algorithms, and computers and software located at the four vertexes. Each topic affects the others and has certain strengths for addressing the problems of biology. No weaknesses in the current state of mathematics are mentioned per-se, but any real weakness is likely the notion that, for some problems, the appropriate mathematics simply don't yet exist.
For Cohen, mathematics is decidedly applied mathematics. I doubt he has much to say about topics with no direct relevance for biological applications.
The article is divided into past, present, and future. Cohen first goes into a brief review of the historical interplay between math and biology, starting with what I think is an excellent example: William Harvey's discovery of the circulation of the blood. Just enough background is given to appreciate how novel and unexpected this discovery was. Notably, empiricism, aided by calculations, was in its infancy during Harvey's time. Cohen then pays homage to several other co-developments of math and biology, some of which are nicely summarized in the article's Table 1.
For present matters, Cohen notes that issues of emergence and complexity should lead to great discoveries in mathematics. What is notable here is that emergence in biological systems at one particular level of organization is driven by events at both lower and higher levels. For example, both the genes of an organism and evolution determine many aspects of species. Cohen also provides an example of recent research that marries ideas from statistics, hierarchical clustering, and cancer cell biology. This example is a bit difficult to follow, but I think it is a good analogy of the interplay he is discussing. (To be fair, I was reading the article in an airplane flying through some turbulence, so it was difficult to give this section my full attention.)
The article finishes with a future outlook for his thesis and very briefly presents some ethical problems and opportunities for the continued correspondence between the two fields. I didn't find this section terribly insightful.
This article and those similar to it can't help but make me feel like the Age of Physics is near an end. The problems that occupy most practical people's minds today seem to be concerned with complexity. Physics, which is concerned with constructing models based on the simplest possible assumptions, is by its very nature a difficult tool for understanding phenomena that emerge from the entangled interactions of many heterogeneous parts. Biology just happens to be one field that can push forward our understanding of complex systems. Computation, information science, and neuroscience are other fields that will help further mathematics.
Physics will always be important, but the domain of natural phenomena in which it finds itself useful is lessening as the Information Age comes into full swing.