This award supports a coordinated set of research and educational activities with the goals of elucidating the behavior of solutions of certain nonlinear partial differential equations and stimulating the environment for research and study in analysis, differential equations, and related computation at the University of Wyoming. This activity centers on the Wyoming Applied Analysis & Computing Group, a team consisting of undergraduate and graduate students, University of Wyoming faculty, and faculty from Wyoming's network of community colleges.
Nonlinear partial differential equations arise generically in mathematical models of physical phenomena. However, a broad understanding of their properties remains a fundamental challenge in pure and applied mathematics. The differential equations that are the focus of this research are physically motivated, and their features (nonlinearity, multiple space dimensions, multiple scales) represent some of the central challenges in the analysis of such equations. The research agenda includes investigation of a recent conjecture about the formation of violent oscillations in the focusing nonlinear Schrodinger equation; this conjecture is part of an emerging belief that the formation of these oscillations may have a universal character. The validation of the appearance of such universal behavior in differential equations is certain to drive a wave of new developments in the analysis of nonlinear phenomena. On a mathematical level, the plan of attack is based on a broad array of analytical techniques, including Evans functions, singular perturbations, dynamical systems, spectral theory for linear operators, complex analysis, and numerical analysis. The anticipated results, if achieved, have the potential to make an impact on applications and other areas of mathematics. One of the research projects aims to extend the asymptotic analysis of Riemann-Hilbert problems arising in the analysis of integrable partial differential equations; such extensions are expected to be useful in the seemingly unrelated analysis of orthogonal polynomials and random matrices.
The breadth of mathematical tools required by the research agenda ensures that the Wyoming Applied Analysis & Computing Group will serve as an ideal training ground in applied analysis and computation for the next generation of researchers and students. In addition to collective work on the research agenda, the group will host an annual two-week immersion into research activities for select community-college faculty. Additionally, the group's activities will include participation in the annual "articulation conferences" that are held to connect the math and science departments at the University of Wyoming with those at Wyoming's seven community colleges. These activities will strengthen the mathematical network across the state. The group's recruitment and training strategy is partially based on the development of a novel calculus course that treats numerical methods as an intrinsic part of calculus. The course prepares students to participate in the group at an early stage of their careers.
