Today’s scientists and engineers solve problems whose computational complexities would have amazed their predecessors only a generation ago. Statistical computations reveal that a new cancer drug is remarkably effective for patients with a certain combination of genetic markers, even though it is not particularly effective for the population as a whole. The wings of a new generation of aircraft are designed and tested in a computer rather than in a wind tunnel. Cosmological theories about the early universe are studied using computer simulations of the first billion years after the "big bang." The effectiveness of aging nuclear weapons is evaluated in computer simulations rather than in nuclear test explosions.
Such accomplishments are possible because of recent improvements in computers and in the mathematics that these new computers implement. This has led to the rise of a new field of mathematics, called ”computational mathematics.”
Computational mathematics involves finding new methods for solving classical problems and developing new mathematical theory that aims to understand what really goes on in certain computing contexts. As an example of the first type, undergraduate mathematics students have long understood how to find the largest or smallest value of a function f(x1, x2, · · · , xn) where the input variables x1, x2, · · · , xn describe the state of some system. Provided that the function f is nice enough, mathematicians of Isaac Newton’s time knew what to do. But sometimes the old methods are too slow and sometimes the function f is not nice enough, so that new mathematical techniques are required, together with airtight proofs that the new methods will work. An example of the second type comes from the attempts of computer designers to exploit the unique features of quantum systems to build a new type of computer that could outperform any digital computer available today. But quantum computing raises a host of questions that, once stated mathematically, ask new questions in traditional mathematical research areas.
Computational mathematics has already had a significant impact at the research level. But how should computational mathematics change what undergraduates study? In an attempt to find answers to that question, the National Science Foundation (NSF) launched a national program called ”Computational Science Training for Undergraduates in the Mathematical Sciences” (CSUMS). The nation’s colleges and universities were invited to submit proposals describing their ideas about how computational mathematics should find its way into the undergraduate curriculum. A team of faculty members from the College’s departments of mathematics, computer science and applied science recently received a CSUMS grant from NSF totaling more than $800,000 over a five year period. Participating faculty members include Sarah Day, Michael Lewis, Chi-Kwong Li (director), David Lutzer, David Phillips, and Junping Shi of the mathematics department, Andreas Stathopoulos and Virginia Torczon of the computer science department and Greg Smith of the applied science department.
William and Mary was one of the (fewer than 10) colleges and universities chosen for the CSUMS project. Broadly stated, the goal of the College’s CSUMS project is to enhance computational aspects of the education of undergraduates in the mathematical sciences and to better prepare these students to pursue careers and graduate study in fields that require integrated strengths in computation and mathematics. In more specific terms, the three participating departments will put more computational emphasis in their curricula by introducing new courses related to computational mathematics and including more computational issues in some of the existing courses. There will be faculty and student seminars and workshops on the topic. Moreover, students will be recruited to do research on problems involving computational mathematics with faculty members during the academic year and during the summer. For its part, the College will contribute on-campus summer housing for the program’s students. In addition to getting stipends for their research, students will have the opportunities to travel in the United States or abroad to attend professional conferences and visit other research groups.