The well known Katznelson-Tzafriri theorem states that a power-bounded operator $T$ on a Banach space $X$ satisfies $\|T^n(I-T)\| \to 0$ as $n \to \infty$ if and only if the spectrum of $T$ touches the complex unit circle nowhere except possibly at the point $\{1\}$. As it turns out, the rate at which $\|T^n(I-T)\|$ goes to zero is largely determined by estimates on the resolvent of $T$ on the unit circle minus $\{1\}$ and not only is this interesting from a purely spectral and operator theoretic perspective, the applications of such quantified decay rates are myriad, ranging from the mean ergodic theorem to so-called alternating projections, from probability theory to continuous-in-time evolution equations. In this talk, we will trace the story of these so-called quantified Katznelson-Tzafriri theorems through previously known results up to the present, ending with a new result proved just a few weeks ago that largely completes the adventure.

We present analysis and computations of a non-local thin film model developed by Kalogirou et al (2016) for a perturbed two-layer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a quasi-solution analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number R and the primary wavelength 2πν−1/2 of the disturbance, is determined analytically. We also present recent results on time periodic solutions arising from Hoof-Bifurcation of the primary solution branch.

(This work is in collaboration with D. Papageorgiou & E. Oliveira ) 
 

Deep learning has revolutionized image, text, and speech recognition. Motivated by this success, there is growing interest in developing deep learning methods for financial applications. We will present some of our recent results in this area, including deep learning models of high-frequency data. In the second part of the talk, we prove a law of large numbers for single-layer neural networks trained with stochastic gradient descent. We show that, depending upon the normalization of the parameters, the law of large numbers either satisfies a deterministic partial differential equation or a random ordinary differential equation. Using similar analysis, a law of large numbers can also be established for reinforcement learning (e.g., Q-learning) with neural networks. The limit equations in each of these cases are discussed (e.g., whether a unique stationary point and global convergence can be proven).