Spectrum and Geometry

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I work at the intersection of geometric analysis, PDE and mathematical physics, studying spectra and eigenfunctions of Laplace-type and Dirac-type operators. Laplacian arises in the study of celestial mechanics, heat and wave propagation, and quantum mechanics. Eigenfunctions of Laplacian describe vibrations of a string or a drum, pure states in quantum mechanics (atoms and molecules), and many other phenomena. Related problems arise when one studies acoustics (propagation of sound) and optics; data analysis ("manifold learning"); fluid flows and plasma flows in thermonuclear reactors. I have studied the behaviour of eigenfunctions for both integrable and ergodic systems. I have worked on several main questions in Spectral Theory: (a) Study of limits of eigenfunctions for ergodic systems (Quantum Ergodicity on hyperbolic surfaces; analogous results for eigenfunctions of Dirac operator and Hodge Laplacian; QE for branching billiards, an important new class of results established in 2012); as well as for integrable systems, where some of the results were extended to solutions of Schrodinger equation; (b) Spectral function (I established lower bounds and proved an accurate trace formula for surfaces of variable negative curvature); (c) Nodal and critical sets (I answered in the negative a question raised by S.T. Yau about the number of critical points of eigenfunctions; studied topology of nodal sets of random spherical harmonics; and recently found surprising connections to conformal geometry); (d) In a recent paper, I initiated the study of random wave conjectures for eigenfunctions using probabilistic methods. In the last 2 years I have also studied spectral theory of resonances. In addition, I initiated the rigorous study of averaging over geometrically natural spaces of Riemannian metrics, with applications to the study of geometric and spectral invariants of random Riemannian metrics. My long-term objectives include continuing the study of various important questions in spectral theory, with applications to geometry and PDE. My short-term objectives include continuing the work on research programs, concentrating on the following areas: (1) Semiclassical theory of discontinuous systems; (2) Measures and averaging on manifolds of metrics, including constructing "canonical" measures on conformal classes in higher dimensions, as well as the study of Random Wave conjectures; (3) Spectral theory on asymptotically hyperbolic manifolds; (4) Spectral theory of conformally covariant operators. Systems in (1) describe wave propagation through air-water interface, semiconductors, impurities in crystals, including seismic waves and elasticity. Semiclassical and ergodic theory of systems will have important practical applications. The eigenstates studied in (2) arise in the study of Loschmidt echo effect (quantum fidelity) in physics. The novel techniques described in (2) may help to answer some fundamental conjectures in Quantum Chaos raised in 1960s and 1970s; developing those techniques may also lead to progress in related problems arising in Conformal Field Theory in Physics, as well as in Quantum Gravity. Related questions arise in the study of random maps, and have applications in astrophysics, medical imaging and other fields. Questions considered in (4) have applications in conformal geometry, relativity, and certain nonlinear PDE. Problems related to the study of resonances in (3) have important application in imaging and inverse problems. Problems in semiclassical theory arise in the theory of quantum computing ("quantum dots"); in chemical and molecular physics; and in atomic physics (atomic nuclei).