The processes of adaptation, learning, discovery, and invention are expected to play an increasingly significant role in emerging large-scale computationally intelligent (CI) systems. . The meanings of adaptation and learning are well-known. By discovery we mean the process of creating a new hypothesis based on sufficient new data that does not fit existing hypotheses; while by invention, the process of creating (synthesizing) new prototypes by interpolation or extrapolation of existing ones.
In this lecture we will present a kernel-based mathematical approach to the modeling and design of the processes of adaptation, learning, discovery and invention in nonlinear dynamical systems. The approach is based on our previous work in which uncertainty is handled by mathematical approximation theory methods, specifically, by best approximation of nonlinear functionals in an appropriate Reproducing Kernel Hilbert Space (RKHS). This formulation leads to optimal nonlinear system models in the form of artificial neural networks (ANNs) in a natural way, that is, without imposing, a-priori, a neural structure on the system being modeled. For this reason, while connecting with the mathematical foundations on which they are based, we will use ANNs as generic representations of the nonlinear dynamical systems under discussion.
Computer simulation results, some using real data, will be presented to establish and illustrate the theoretical developments.

The values of the partial zeta functions for an abelian extension of number fields at non-positive integers are rational numbers with known bounds on their denominators. David Hayes conjectured that when the associated fields satisfy certain algebraic conditions, the bound at s=0 can be sharpened. I will present a counterexample to Hayes's conjecture. I will then propose a new conjecture sharpening the bounds at arbitrary non-positive integers that implies a weaker version of Hayes conjecture at s=0. I will conclude by proving that the new conjecture is a consequence of the Coates-Sinnott conjecture.

We give a direct proof of a theorem of Jech
and Shelah addressing the existence of some
set system on $\omega_1$.

We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u+V_n when it visits a particular state 0 at time n, with V_n representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. Typically the probability of an excursion of length n for the underlying Markov chain is taken to decay as a power of n (called the loop exponent), perhaps with a slowly varying correction. A particular case not covered in a number of previous studies is that of loop exponent one, which includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of u in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ at least at low temperatures. The work is joint with N. Zygouras.