Geometric Analysis and Spectral Theory

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I work at the intersection of geometric analysis, partial differential equations (PDE) and mathematical physics, studying spectra and eigenfunctions of Laplace type operators, arising 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, and many other phenomena. Related problems arise when one studies acoustics, optics, data analysis, fluid and plasma flows. Recently, I worked on several important questions in spectral theory: (a) Study of high energy eigenfunctions for ergodic systems, including Quantum Ergodicity for branching billiards; behaviour of Eisenstein series (scattering theory eigenfunctions on hyperbolic manifolds); study of the Loschmidt echo effect. (b) Distribution of resonances on hyperbolic surfaces, in a series of papers with F. Naud. (c) Nodal and critical sets (nodal lines of Eisenstein series; conformal invariants from nodal sets, and graph theory analogs).  I contributed to the study of gaps in geodesic length spectra on negatively curved manifolds. I continued the study of conformally covariant operators, showing in particular that 0 generically is not an eigenvalue of the conformal Laplacian. I studied the behaviour of rank and Tutte polynomials of sequences of sparse bounded degree graphs that converge Benjamini-Schramm, and showed, that natural coefficient measures converge to a delta function. I plan to focus on the following projects: study conformally covariant operators on manifolds with boundary, and related problems in geometry and PDE; study resonances and eigenfunctions on hyperbolic manifolds; extend the results about rank polynomials to Bollobas-Riordan polynomials, and to sequences of dense graphs, study applications to statistical physics, study asymptotic distribution of zeros;  study eigenfunction averages over submanifolds, including totally geodesic submanifolds;  study of spectra and eigenvectors of elements of the group ring, focussing on surface groups; study geometric and spectral invariants on "random" 3manifolds;  study extremal metrics for eigenvalue functionals, and connections to isometric embeddings; study eigenfunction localization for billiards; study length spectra;  study spectra and eigenfunctions for "towers of covers" of Riemannian manifolds, including surfaces.