**Conferencias**

#### A Robuster Scott Rank

**Antonio Montalbán - University of California, Berkeley**

Abstract: The Scott rank was introduced in the 60''s as a measure of complexity for algebraic structures. There are various other ways to measure the complexity of structures that give ordinals that are close to each other, but are not necessarily equal. We will introduce a new definition of Scott rank where all these different ways of measuring complexity always match, obtaining what the author believes it the correct definition of Scott Rank. We won''t assume any background in logic, and the talk will consist mostly of an introduction to these topics.

#### Weyl groupoids

**Nicolás Andruskiewitsch - Universidad Nacional de Córdoba**

The structure of the finite-dimensional semisimple complex Lie algebras and groups is governed by the rich combinatorial notion of root systems. Root systems had then a large number of applications in very dissimilar areas like finite groups and singularities that through them show unexpected connections. Some decades ago root systems appeared again in the study of a large class of Hopf algebras named quantum groups. In the search of understanding the role of these Hopf algebras in the general picture, it was proposed to study some objects called Nichols algebras of diagonal type. The classification of those with finite dimension was obtained by Heckenberger with a generalization of the notion of root system as the primary tool. This notion, axiomatized later by Heckenberger, Yamane and Cuntz, turned to be ubiquitous, appearing naturally in the contexts of Lie superalgebras and modular Lie algebras. The richness and beauty of these generalized root systems with their corresponding Weyl groupoids is just starting to be unveiled. The purpose of the talk is to introduce generalized root systems and Weyl groupoids explaining the relations with Lie algebras and superalgebras, and quantum groups.

#### Breathers solutions and the generalized Korteweg-de Vries equation

**Gustavo Ponce - University of California**

This talk is centered in the generalized Korteweg-de Vries (KdV) equation (1) \(∂_tu + ∂_x^3u + ∂_xf(u)=0, x, t ∈ R\). The case \(f(u) = u^2\) corresponds to the famous KdV eq., and \(f(u) = u^3\) to the modified KdV eq. We shall review some results concerning the initial value problem associated to the equation (1). These include local and global well-posedness, existence and stability of traveling waves, and the existence and non-existence of “breathers” solutions. The aim is to understand how the non-linearity \(f(u)\) induces these results.

#### Transfer operators and atomic decomposition

**Daniel Smania - Universidade de São Paulo**

Since the groundbreaking contributions of Ruelle, the study of transfer operators has been one of the main tools to understand the ergodic theory of expanding maps, that is, discrete dynamical systems that locally expand distances. Questions on the existence of interesting invariant measures, as well the statistical properties of such dynamics system, as exponential decay of correlations and Central Limit Theorem, can be answered studying the spectral properties of the action of these operators on suitable spaces of functions. Using the method of atomic decomposition, we consider new Banach spaces of functions (that in some cases coincides with Besov spaces) that have a remarkably simple definition and allows us to obtain very general results on the quasi-compactness of the transfer operator acting in these spaces, even when the underlying phase space and expanding map are very irregular. Joint work with Alexander Arbieto (UFRJ-Brazil).

#### Algorithmic graph theory

**Flavia Bonomo - Universidad de Buenos Aires**

Intersection graphs: interval graphs, chordal graphs, circular-arc graphs, circle graphs, permutation graphs. Characterizations and algorithms for combinatorial optimization problems on these classes. Graph decompositions: modular decomposition, cliquewidth, treewidth, pathwidth, thinness, clique-cutsets, claw-free graphs decompositions. Algorithms based on decompositions. Graph classes defined by forbidden induced subgraphs: structural properties and algorithmical applications. Bibliography: - Brandstadt A., Bang Le V. and Spinrad J., Graph classes: A survey, SIAM, 1999. - Golumbic M.C., Algorithmic graph theory and perfect graphs, Annals of Discrete Mathematics, Vol 57, 2004. - McKee T. and McMorris F., Topics in intersection graph theory, SIAM, 1999. Recent and classical papers.