Georgia Institute of Technology College of Sciences

Thesis defense: Advisors: Turgay Uzer and Cristel Chandre Summary: Thirty years after the demonstration of the production of high laser harmonics through nonlinear laser-gas interaction, high harmonic generation (HHG) is being used to probe molecular dynamics in real time and is realizing its technological potential as a tabletop source of attosecond pulses in the XUV to soft X-ray range. Despite experimental progress, theoretical efforts have been stymied by the excessive computational cost of first-principles simulations and the difficulty of systematically deriving reduced models for the non-perturbative, multiscale interaction of an intense laser pulse with a macroscopic gas of atoms. In this thesis, we investigate first-principles reduced models for HHG using classical mechanics. On the microscopic level, we examine the recollision process---the laser-driven collision of an ionized electron with its parent ion---that drives HHG. Using nonlinear dynamics, we elucidate the indispensable role played by the ionic potential during recollisions in the strong-field limit. On the macroscopic level, we show that the intense laser-gas interaction can be cast as a classical field theory. Borrowing a technique from plasma physics, we systematically derive a hierarchy of reduced Hamiltonian models for the self-consistent interaction between the laser and the atoms during pulse propagation. The reduced models can accommodate either classical or quantum electron dynamics, and in both cases, simulations over experimentally-relevant propagation distances are feasible. We build a classical model based on these simulations which agrees quantitatively with the quantum model for the propagation of the dominant components of the laser field. Subsequently, we use the classical model to trace the coherent buildup of harmonic radiation to its origin in phase space. In a simplified geometry, we show that the anomalously high frequency radiation seen in simulations results from the delicate interplay between electron trapping and higher energy recollisions brought on by propagation effects.

Weinstein cobordisms give a natural relationship on the set of Weinstein domains. Flexible Weinstein domains are minimal with respect to this relationship. In this talk, I will use these minimal domains to construct maximal Weinstein domains: any two high-dimensional Weinstein domains with the same topology are Weinstein subdomains of a maximal Weinstein domain also with the same topology. Using this construction, a wide range of new Weinstein domains can be produced, for example exotic cotangent bundles of spheres containing many different closed exact Lagrangians. On the other hand, I will explain how the same line of ideas can be used to prove restrictions on which categories can arise as the Fukaya categories of certain Weinstein domains.

In joint work with J. Martinez-Garcia we study the classification problem of asymptotically log del Pezzo surfaces in algebraic geometry. This turns out to be equivalent to understanding when certain convex bodies in high-dimensions intersect the cube non-trivially. Beyond its intrinsic interest in algebraic geometry this classification is relevant to differential geometery and existence of new canonical metricsin dimension 4.

A limit-periodic function on R^d is one which lies in the L^\infty closure of the space of periodic functions. Schr\"odinger operators with limit-periodic potentials may have very exotic spectral properties, despite being very close to periodic operators. Our discussion will revolve around the transition between ``thick'' spectra and ``thin'' spectra.