Microlocal Analysis of Random Systems and Fractal Sets

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Microlocal analysis (MA) is the study of partial differential equations through the geometry of classical mechanics and Fourier analysis. Its impact truly spans disciplines, establishing itself as the mathematical language of choice in various parts of physics and inverse problems. My research focuses on Laplace spectral problems, motivated by quantum mechanics, and their resolution via techniques from MA. On compact manifolds (or Euclidean domains), eigenfunctions and eigenvalues of Laplacians model bound states of a free quantum particle and discrete energies, respectively. On non-compact manifolds, the natural replacement for discrete eigenvalues and eigenfunctions are resonances and resonant states, respectively, providing another model for quantum phenomenon when energy is lost.  The vision of this proposal is to novelly combine randomness and fractal geometry with MA in order to approach from a new angle some enigmatic objects: eigenfunctions/resonant states on general Riemannian manifolds; MA will continue to play a crucial role. Research training is available for all levels of HQP. The components of the proposal are: (1) Growth of eigenfunctions on fractals: Norm estimates, and their relation to the ambient compact manifold's geometry, provide one manner to study eigenfunctions. These estimates are of particular interest when restricting to subsets E, shedding light on probability measures from quantum mechanics, for example. For E a fractal, we aim to build a cohesive theory for such estimates via new connections between geometric measure theory and MA. (2) Geometry of zero sets of random eigenfunctions: The emergence of randomness in mathematical structures has piqued the interest of a range of mathematicians, from number theory to dispersive equations. On compact manifolds we propose to study a classical object, the zero set (or nodal set), of random eigenfunctions.  Random eigenfunctions model deterministic eigenfunctions in negatively curved geometries, a setting where many explicit formulas are absent. (3) Resonances gaps for random systems: A natural question in the spectral theory of non-compact manifolds, a setting where energy can be lost, surrounds spectral gaps: are there quantifiable strips in the complex plane that have no resonances? Answers directly give rates of decay for waves that escape to infinity. Our objective is to develop a theory for random surfaces to gain better insight into what is a typical such rate. (4) Localization and entropy on billiards: A way to study the macroscopic behavior of eigenfunctions on compact manifolds (or domains) is to understand how they distribute: do they spread or concentrate in some fashion? Our objective is to analyze how "clusters of eigenfunctions" concentrate on dynamical subsets. Our dream setting is domains. While this question does not have explicit structures of randomness or fractals in its statement, many of its aspects display features of a close cousin: chaos.