Georgia Institute of Technology College of Sciences

In the first part of the talk(s) we are going to present a way to study numerically the complex domains of invariant Tori for the standar map. The numerical method is based on Padé approximants. For this part we are going to follow the work of C. Falcolini and R. de la LLave.In the second part we are going to present how the numerical method, developed earlier, can be used to study the complex domains of analyticity of invariant KAM Tori for the dissipative standar map. This part is work in progress jointly with R. Calleja (continuation of last talk).

The aim of this talk is to give a general overview of KAM theory, starting from its early stages untill the modern era, including infinite dimensional cases. I'll try to present the main ideas with as little technicalities as possible, and if I have time I'll also discuss some open problems in the field.

Isospectral Reduction reduces a higher dimension matrix to a lower dimension one while preserving the eigenvalues. This goal is achieved by allowing rational functions of lambda to be the entries of the reduced matrix. It has been shown that isospectral reduction also preserves the eigenvectors. Here we will discuss the conditions under which the generalized eigenvectors also get preserved. We will discuss some sufficient conditions and the reconstruction of the original network.

We present some basic results from the theory of stochastic processes and investigate the properties of some standard continuous-time stochastic processes. Firstly, we give the definition of a stochastic process. Secondly, we introduce Brownian motion and study some of its properties. Thirdly, we give some classical examples of stochastic processes in continuous time and at last prove some famous theorems.