Georgia Institute of Technology College of Sciences

I will describe the h-principle philosophy and explain some recent developments on the flexible side of symplectic topology, including Murphy's h-principle for loose Legendrians and Cieliebak and Eliashberg's construction of flexible symplectic manifolds in high-dimensions.

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.

Self-interacting systems of particles/agents arise in many areas of science, such as particle systems in physics, flocking and swarming models in biology, and opinion dynamics in social science. An interesting question is to learn the laws of interaction between the particles/agents from data consisting of trajectories. In the case of distance-based interaction laws, we present efficient regression algorithms to estimate the interaction kernels, and we develop a nonparametric statistic learning theory addressing learnability, consistency and optimal rate of convergence of the estimators. Especially, we show that despite the high-dimensionality of the systems, optimal learning rates can still be achieved.

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.

One variation of the Waring problem is to ask for the shortest non-trivial equations of the form f_1^d + ... + f_r^d = 0, under various conditions on r, d and where f_j is a binary form. In this talk I'll limit myself to quadratic forms, and show all solutions for r=4 and d=3,4,5. I'll also give tools for you to find such equations on your own. The talk will touch on topics from algebra, analysis, number theory, combinatorics and algebraic geometry and name-check such notables as Euler, Sylvester and Ramanujan, but be basically self-contained. To whet your appetite: (x^2 + xy - y^2)^3 + (x^2 - xy - y^2)^3 = 2x^6 - 2y^6.

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.

A set $\Omega\subset \mathbb{R}^d$ is called spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Back in 1974 B. Fuglede conjectured that spectral sets could be characterized geometrically by their ability to tile the space by translations. Although since then the subject has been extensively studied, the precise connection between spectrality and tiling is still a mystery.>In the talk I will survey the subject and discuss some recent results, joint with Nir Lev, where we focus on the conjecture for convex polytopes.

Classical Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. I will present a generalized version of this theorem for metric graphs. (Joint work with Chenxi Wu.)

Given a Hamiltonian system, normally hyperbolic invariant manifolds and their stable and unstable manifolds are important landmarks that organize the long term behaviour.

When the stable and unstable manifolds of a normally hyperbolic invarriant manifold intersect transversaly, there are homoclinic orbits that converge to the manifold both in the future and in the past. Actually, the orbits are asymptotic both in the future and in the past.

One can construct approximate orbits of the system by chainging several of these homoclinic excursions.

A recent result with M. Gidea and T. M.-Seara shows that if we consider long enough such excursions, there is a true orbit that follows it. This can be considered as an extension of the classical shadowing theorem, that allows to handle some non-hyperbolic directions

The Burau representation plays a key role in the classical theory of braid groups. When we let the complex parameter t take the value -1, we obtain a symplectic representation of the braid group known as the integral Burau representation. In this talk we will give a survey of results on braid congruence subgroups, that is, the preimages under the integral Burau representation of principal congruence subgroups of symplectic groups. Along the way, we will see the (perhaps surprising) appearance of braid congruence subgroups in a variety of other contexts, including knot theory, homotopy theory, number theory, and algebraic geometry.