Paradifferential Calculus and Oscillatory Motions

Abstract
Oscillatory solutions play a central role in many dynamical and physical models, but their construction is often obstructed by a loss of regularity. This loss may arise from strong nonlinearities, infinitely many near-resonant modes, or the interaction of the two. In this talk, I will explain how paradifferential calculus, a modern tool from multilinear harmonic analysis, provides a unified framework to overcome these difficulties by precisely tracking regularity in nonlinear expressions. I will illustrate the method through three applications: (1) the construction of hyperbolic invariant manifolds in free-boundary fluid models; (2) an algorithmic alternative to classical KAM theory for quasi-periodic orbits in Hamiltonian systems; and (3) a combination with bifurcation theory that produces time-periodic oscillations of capillary water droplets. The algorithmic structure revealed by the latter two applications also suggests new connections between rigorous analysis and computational approaches to more complex oscillatory models.

Speaker Biography
Dr Shao received his Ph.D. degree in mathematics from MIT under the supervision of Gigliola Staffilani. He then served as L.E. Dickson instructor at the mathematics department, University of Chicago. He is currently a postdoc at HES. His research works focus on harmonic analysis and its application to evolutionary partial differential equations arising from physical backgrounds, including Hamiltonian
dynamics and fluid mechanics.