Robert Zimmer, a mathematical scientist who is president of the University of Chicago, speaks of the mathematical sciences as a fabric: If it is healthy—strong and connected throughout the whole—then it can be tailored and woven in many ways; if it has disconnects, then its usefulness has limitations. He also argues that, because of this interconnectivity, there is a degree of inevitability to the ultimate usefulness of mathematical sciences research. That is, important applications are the rule rather than the exception.4 Over and over, research that was internally motivated has become the foundation for applied work and underlies new technologies and start-ups. And often questions that arise because of our inability to mathematically represent important phenomena from applications prompt mathematical scientists to delve back into fundamental questions and create additional scaffolding of value both to the core and to future applications.
The fabric metaphor accurately captures the interconnectivity of the various strands of the mathematical sciences; all of the strands are woven together, each supporting the others, and collectively forming an integrated whole that is much stronger than the parts separately. The mathematical sciences function as a complex ecosystem. Ideas and techniques move back and forth—innovations at the core radiate out into applied areas; flowing back, new mathematical problems and concepts are drawn forth from problems arising in applications. The same is true of people—those who choose to make their careers in applied areas frequently got a significant part of their training from core mathematical scientists; seeing the uses and power of mathematics draws some people in to study the core. One never knows from which part of the mathematical sciences the next applications will come, and one never knows whether what is needed for a possible application is existing knowledge, a variation on what already exists, or something completely new. To maintain U.S. leadership in the mathematical sciences, the entire ecosystem must remain healthy.
In everyday life, terms that sound mathematical increasingly appear in a variety of contexts. “Doing the math” is used by politicians to mean analyzing the gains or losses of doing something, and language such as “exponential,” “algorithm,” and “in the equation” frequently appears in business and finance. A positive interpretation of this phenomenon is that more and more people appreciate the mathematical sciences, but a not-so-
4 For this reason, this report tends to avoid the terms “core mathematics” and “applied mathematics.” As can be seen in many places in the report, nearly all areas of the mathematical sciences can have applications.