__"C*-algebras and topological algebras".__Mati Abel (University of Tartu, Estonia).
Abstract:__"Weak coupling limit type generators: non-equilibrium states of quantum systems".__Luigi Accardi (Centro Vito Volterra, Università degli Studi di Roma Tor Vergata, Italy).__"Multilevel Monte Carlo methods".__Andrea Barth (Stuttgart University, Germany).

Abstract:

In this Lecture I will provide an introduction to Monte Carlo methods and their probabilistic foundations. After a primer on the Laws of Large Numbers, I will make a little excursion to random number generation. This will equip us with the foundations to define and study the Monte Carlo estimator. We will see that the key to improve the -- at first naive -- Monte Carlo approach is Variance Reduction. After a brief overview of the most common methods for the latter, I will focus on a specific one: the multilevel Monte Carlo method. I will then comment on further improvements of the multilevel Monte Carlo method and present typical applications in uncertainty quantification and stochastic (partial) differential equations.__"The inverse scattering method for solving a class of partial differential equations".__Jesús Adrián Espínola-Rocha (Universidad Autónoma Metropolitana Azcapotzalco).

Notes in PDF

Abstract:

We know that many ordinary differential equations (linear and nonlinear) as well as some partial differential equations, PDEs, (say, the classical wave, hear and Laplace equations) can be solved in closed form and there is a systematic way to extract information and their solutions

However, there is a huge range of PDEs, mainly nonlinear, which are not amenable of finding exact solutions neither a method to solve, study, or extract information out of them . In the 19th century, the Korteweg-deVries equation (a nonlinear PDE) appeared as a model to describe wave propagation in shallow water. This equation was forgotten for decades. It was not until 1967, when a group of four mathematicians in Princeton found a method to solve it: the inverse scattering method.

In this minicourse I will show how Gardner, Greene, Kruskal and Miura solved the equation. I will show some other extensions related, such as Lax pair representation, and how this allowed to solve some other nonlinear PDEs such as the nolinear Schrödinger equation. If time permits, I will talk about Hirota's method and the Bäcklund transform applied to this type of equations.

A brief introduction to topological algebras will be given. The pure algebraic background (ideals, invertibility, quasi-invertibility, Jacobson radical, involution, spectrum, unitization)will be given. main classes of algebras (q-algebras, gelfand-mazur algebras, banach algebras, C*- algebras, locally bounded algebras, locally convex algebras, locally pseudoconvex algebras) will be described and main results in there (Gelfand- Mazur theorem, Gelfand-Naimark theorem and others) will be formulated. Function algebras will be considered separately. Several open problems in topological algebras will be presented.