Carnegie Mellon University

Educating Applied Mathematicians

An interview with Center for Nonlinear Analysis director Irene Fonseca

What is the Center for Nonlinear Analysis?

The Center for Nonlinear Analysis (CNA), founded in 1991 by the Department of Mathematical Sciences at Carnegie Mellon, offers distinctive research and scientific training in applied mathematics. Funded by the National Science Foundation (NSF), the center focuses on research in nonlinear analysis, scientific computation, mathematical finance and mechanics. The main goals of the CNA are to:

  • identify and develop new applications of mathematical sciences;
  • promote collaboration between applied mathematicians and allied scientists;
  • respond to mathematical challenges in materials science, biotechnology, etc.;
  • sustain its worldwide leadership in the areas of calculus of variations, partial differential equations and continuum mechanics; and
  • maintain its reputation for training postdoctoral fellows who are cognizant of research opportunities at the interface between mathematical and physical sciences and engineering.

What are the main applications of nonlinear analysis in the CNA's research?

The CNA focuses on the nonlinear analysis of novel man-made materials such as shape memory alloys; ferroelectric, electromagnetic and magnetostructive materials; composites; liquid crystals; thin films; and modeling and simulation in biology. Describing materials such as these through the study of mathematical models plays a central role in the research program because it leads to new frontiers in nonlinear analysis.

What is the training focus of the CNA in preparing mathematicians for this field?

The CNA supervises and mentors postdoctoral fellows. It has trained more than fifty over the past ten years. Several former postdocs now head their own institutions in the United States and abroad, while others are pursuing future leadership positions in their fields.

The CNA offers advanced nonlinear analysis courses to graduate students. In summer 2003, for example, the CNA offered five research fellowships. The department also is very proud of the consistent excellence of its graduate students' teaching. For the third year in a row, a CNA graduate student has received the Mellon College of Science's Hugh D. Young Graduate Student Teaching Award.

How does the CNA involve undergraduates?

The Undergraduate Mathematics Institute (UMI), under the direction of W. O. Williams, is a major outreach component of the CNA. The institute, supported by grants from the NSF, the National Security Agency and the CNA, helps undergraduate students, particularly those from undergraduate math programs that do not offer graduate options, make informed decisions about studying math at the graduate level by giving them the opportunity to explore graduate coursework and research interests. UMI students participate in team projects that give them the chance to discover the pleasures and frustrations of attacking open-ended research problems. These projects also give them the opportunity to discover aspects of applied mathematics that they might not otherwise see.

The UMI targets undergraduate students who are historically underrepresented in mathematics, such as African Americans, Hispanics and women. In 2002, fourteen students participated in the UMI; ten were women and four were African American.

What is the role of collaborative activity in the CNA and in applied mathematics?

To respond effectively to major technological challenges, the CAN requires collaborative and coordinated effort from a wide range of expertise that transcends institutional, disciplinary, geographical and national boundaries. Such opportunities greatly expand the horizons for training mathematicians at all levels to work effectively across these boundaries.