Rigorous Computations for Infinite Dimensional Nonlinear Problems

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As we develop evermore powerful computational tools, so it is incumbent upon scientists to exploit newly-developed computational techniques to maintain the pace of research progress, effectively leading us to the point where high-performance computing must be mastered to progress. As a society we continuously undertake more and more ambitious tasks, from whole genome sequencing, to whole-ocean numerical simulation and weather prediction to molecular-level simulation of nanofluids and onwards. However, while the pace of progress increases, sometimes we need to take a step back and pose the question, just how reliable are our computations? The goal of this proposal is to develop rigorous computational tools to prove, in a constructive mathematical way, the existence of qualitative phenomena in infinite dimensional nonlinear problems such as partial differential equations, variational equations and delay differential equations, which are successful widely used mathematical models that describe, explain and predict phenomena in areas as broad as physics, chemistry, biology and economics. More precisely, we are interested in answering each of the following questions. Can we mathematically demonstrate the reliability of the solutions computed using the Navier-Stokes equations or a closely related fluid dynamical model? Can we undertake the same program of research for the difficult, nonlinear molecular dynamics models thrown up by cell biological problems? More generally, can we develop algorithms to construct rigorously and automatically invariant sets of a given infinite dimensional nonlinear dynamical system? Finally, can we use computational methods to prove long-standing open problems in mathematical fields as broad as the calculus of variations and delay differential equations?