We consider random products of SL(2, R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense $G_\delta$ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrodinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice. This is a joint project with A.Gorodetski.

Three-dimensional (3-D) elastic wave propagation and seismic tomography is computationally challenging in large scales and high-frequency regime. In this talk, we propose the frozen Gaussian approximation (FGA) to compute the 3-D elastic wave equation and use it as t he forward modeling tool for seismic tomography with high-frequency data. Rather than standard ray-based methods (e.g. geometric optics and Gaussian beam method), the derivation requires to do asymptotic expansion in the week sense (integral form) so that one is able to perform integration by parts. In particular, we obtain the diabatic coupling terms for SH- and SV-waves, with the form closely connecting to the concept of Berry phase that is intensively studied in quantum mechanics and topology (Chern number). The accuracy and parallelizability of the FGA algorithm is illustrated by comparing to the spectral element method for 3-D elastic wave equation. With a parallel FGA solver built as an computational engine, we explore various applications in 3-D seismic tomography, including seismic traveltime tomography, full waveform inversion, and optimal transport theory-based seismic tomography (using Wasserstein distance), respectively. Global minimization for seismic tomography is investigated based on particle swarm algorithm. We also apply the FGA algorithm  to train a neural network to learn simple subsurfaces structures. 

Non-self-adjoint operators appear in many settings, from kinetic theory 
and quantum mechanics to linearizations of equations of mathematical 
physics. The spectral analysis of such operators, while often notoriously 
difficult, reveals a wealth of new phenomena, compared with their 
self-adjoint counterparts. Spectra for non-self-adjoint operators display 
fascinating features, such as lattices of eigenvalues for operators of 
Kramers-Fokker-Planck type, say, and eigenvalues for operators with 
analytic coefficients in dimension one, concentrated to unions of curves 
in the complex spectral plane. In this talk, after a general introduction, 
we shall discuss spectra for non-self-adjoint perturbations of 
self-adjoint operators in dimension two, under the assumption that the 
classical flow of the unperturbed part is completely integrable.
The role played by the flow-invariant Lagrangian tori of the completely 
integrable system, both Diophantine and rational, in the spectral analysis 
of the non-self-adjoint operators will be described. In particular, we 
shall discuss the spectral contributions of rational tori, leading to 
eigenvalues having the form of the "legs in a spectral centipede". This 
talk is based on joint work with Johannes Sj\"ostrand.

The control and prediction of interactions between high-power, nonlinear laser beams is a longstanding open problem in optics and mathematics. One of the traditional assumptions in this field has been that these interactions are deterministically modelled by the nonlinear Schrodinger equation (NLS). Lately, however, we have shown that at the presence of input noise, solutions of the NLS lose their initial phase information [1]. Thus, the interactions between beams become stochastic as well. Not all is lost, however. The statistics of many interactions are predictable by a universal model. Following experiments in elliptically-polarized laser beams, we generalized our results to a system of NLS equations and derived a “loss of polarization” result [2].

Computationally, the universal model is efficiently solved using a novel spline-based stochastic computational method. Our algorithm efficiently estimates probability density functions (PDF) of PDEs with random input [3]. This is a new and general problem in numerical uncertainty-quantification (UQ), which leads to surprising results and analysis at the intersection of probability and approximation theory.

Somewhat unexpectedly, a near consensus among theoreticians is that cryptographic theorems should be proved in the non-uniform model of complexity, rather than the standard uniform complexity model developed by Alan Turing, the “father of computer science.” In joint work with Alfred Menezes of the University of Waterloo, we have criticized the use of non-uniformity in cryptography, finding that even some of the most distinguished researchers have been led badly astray by their misplaced faith in non-uniformity