Abstract: We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control and random dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems -with a reasonable expectation of success - once the nonlinear system has been mapped into a high or infinite dimensional Reproducing Kernel Hilbert Space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy/Lyapunov functions for nonlinear systems, and study the ellipsoids they induce. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system. We also apply this approach to the problem of model reduction of nonlinear control systems.

In all cases the relevant quantities are estimated from simulated or observed data. These results collectively argue that there is a reasonable passage from linear dynamical systems theory to a data-based nonlinear dynamical systems theory through reproducing kernel Hilbert spaces. This is a joint work with J. Bouvrie (MIT).

There is a beautiful problem resulting from arithmetic number theory where a continuous and compactly supported function's 3-fold autoconvolution is constant. In this talk, we optimize the coefficients of a Chebyshev series multiplied by an endpoint singularity to obtain a highly accurate approximation to this constant. Convolving functions with endpoint singularities turns out to be a challenge for standard quadrature routines. However, variable transformations inducing double exponential endpoint decay are used to effectively annihilate the singularities in a way that keeps accuracy high and complexity low.

There has been a great deal of attention paid to "Big Data" over the last few years.  However, often as not, the problem with the analysis of data is not as much the size as the complexity of the data.  Even very small data sets can exhibit substantial complexity.  There is therefore a need for methods for representing complex data sets, beyond the usual linear or even polynomial models.  The mathematical notion of shape, encoded in a metric, provides a very useful way to represent complex data sets.  On the other hand, Topology is the mathematical sub discipline which concerns itself with studying shape, in all dimensions.  In recent years, methods from topology have been adapted to the study of data sets, i.e. finite metric spaces.  In this talk, we will discuss what has been
done in this direction and what the future might hold, with numerous examples.

Manifolds have been a central object of study for over a century, and the classification of them has been a core theme for the whole of this time. This talk will give an overview of the successes and failures in this effort, with some illustrative examples.

In this talk we present a new type of Soblev norm defined in the space of functions of continuous paths. Under the Wiener probability measure the corresponding norm is suitable to prove the existence and uniqueness for a large type of system of path dependent quasi-linear parabolic partial differential equations (PPDE). We have establish 1-1 correspondence between this new type of PPDE and the classical backward SDE (BSDE). For fully nonlinear PPDEs, the corresponding Sobolev norm is under a sublinear expectation called G-expectation, in the place of Wiener expectation. The canonical process becomes a new type of nonlinear Brownian motion called G-Brownian motion. A similar 1-1 correspondence has been established. We can then apply the recent results of existence, uniqueness and principle of comparison for BSDE driven by G-Brownian motion to obtain the same result for the PPDE.