Applied Analysis and Partial Differential Equations
Mathematical models of natural phenomena often present themselves in the form of nonlinear partial differential equations (PDEs) and/or minimization problems. Their rigorous treatment is the historical root for the entire field of mathematical analysis. However, applied analysis has the distinctive feature that it develops not simply for its own sake, but with an eye toward finding effective solutions to concrete problems. Moreover, this relationship is symbiotic: mathematical models motivate general analytic techniques, while the analysis itself informs the modeling and computational experiments. For example, a sharp existence theorem can reveal either the adequacy or the inadequacy of a model to predict an observed phenomenon. Or by knowing the specific type of discontinuities present in the solution of a differential equation, one can often build a hybrid (analytic/numerical) approximation that is more accurate and efficient than what would result from a naïve/standard discretization.
Employing various techniques of nonlinear functional analysis and PDEs, the calculus of variations, bifurcation theory, probability theory, stochastic processes, geometric group theory, numerical analysis and computational science, current research at Cornell in Applied Analysis and PDEs includes problems from nonlinear elasticity and thin structures, mechanics of materials, mathematical aspects of materials science, homogenization theory, optimal control and differential games, seismic imaging and inverse problems, heat diffusion on manifolds, condensed matter physics, and nano-scale electronic systems.