### 26/10/2005, 15:00 — 16:00 — Room P3.31, Mathematics Building

Tomasz Downarowicz, *Wroclaw University of Technology*

### Minimal realizations of Jewett-Krieger type for nonuniquely ergodic systems I

In a topological dynamical system $(X,T)$ the set of invariant measures is a Choquet simplex $K(X,T)$ whose extreme points are ergodic measures. We will prove that for any zero-dimensional system $(X,T)$ having no periodic points there exists a minimal system $(Y,S)$ and an affine homeomorphism $f$ between the Choquet simplexes $K(X,T)$ and $K(Y,S)$ such that for each $m$ in $K(X,T)$ the measure-theoretic systems $(X,m,T)$ and $(Y,f(m),S)$ are isomorphic. In this sense $(Y,S)$ is a minimal model for $(X,T)$. As an application we will derive the existence of minimal models with a preset collection of invariant measures (arranged in form of a simplex).

### 25/10/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Lubomír Snoha, *Matej Bel University, Banská Bystrica*

### Small scrambled sets

In a discrete dynamical system given by a metric space $X$ (with metric $d$) and a continuous map $f:X\to X$, a subset $S$ of $X$ containing at least two points is called a *scrambled set* if for any $x,y\in S$ with $x\ne y$,

$\underset{n\to \infty}{liminf}\hspace{0.5em}d({f}^{n}(x),{f}^{n}(y))=0\hspace{0.5em}\text{and}\hspace{0.5em}\underset{n\to \infty}{limsup}\hspace{0.5em}d({f}^{n}(x),{f}^{n}(y))>0.$

In the lecture, systems with small scrambled sets will be constructed/studied. Possible cardinalities of maximal scrambled sets and the topological size of scrambled sets of minimal systems will be discussed. The results were obtained jointly with François Blanchard and Wen Huang.

### 18/10/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Rogério Martins, *Universidade Nova de Lisboa*

### When is the attractor of a dissipative system in the cylinder
homeomorphic to the circle?

We look for conditions under which the attractor of a dissipative
system in the cylinder is homeomorphic to the circle. On the other
hand, we show how the appearance of periodic inversely
unstable solutions imply that the attractor is not homeomorphic to
the circle. We apply these results to the particular case of
the damped pendulum type equation driven by a periodic force.

### 11/10/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Peter McNamara, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

### Symmetric functions and cylindric Schur functions

Although algebraic in nature, symmetric functions have long been of considerable interest in combinatorics. We will begin with a general introduction to the combinatorics of symmetric functions, where the two main highlights will be the Schur functions and the Littlewood-Richardson rule. Much of the appeal of Schur functions stems from their appearance in other areas of mathematics, and we will mention connections with representation theory, algebraic geometry (Schubert calculus, in particular) and matrix theory. This will prepare us for the second half of the talk, where we will discuss cylindric Schur functions. As well as being a natural generalization of Schur functions, they are of much relevance to a fundamental open problem in algebraic combinatorics. No knowledge of symmetric functions or combinatorics will be assumed.

### 27/09/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Michael Stessin, *State University of New York, Albany*

### Subalgebras dense in Hardy spaces

Let $A$ be a subalgebra of ${H}^{\infty}$. For each $p>0$ the Hardy space in the unit disk ${H}^{p}(D)$ has a structure of $A$-module. The problem of describing the lattice of closed submodules of this module gives an example of the lattice of closed subspaces invariant under the action of a family of commuting operators. The investigation of the module structure of the disk-algebra was initiated by Wermer in 1950s. It leads to Beurling type results and includes the classical theorem of Beurling as a special case corresponding to the subalgebra of all polynomials. In particular, if the algebra $A$ is dense in all ${H}^{p}$, $p<\infty $, every closed submodule is $z$-invariant in a weak sense, and if $A$ is weak- $*$ dense in ${H}^{\infty}$, then every closed $A$-submodule of ${H}^{p}$ is $z$-invariant. In the talk we will discuss some recent results in this area.

### 20/09/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Axel Grünrock, *University of Wuppertal*

### The Cauchy problem for nonlinear evolution equations with roughdata

The notions of local and global well-posedness for nonlinear
evolution equations, such as Schrödinger- and generalized
KdV-equations, are introduced and it is explained, how to apply the
contraction mapping principle to obtain positive results concerning
this question. The use of linear estimates, e.g.
Strichartz-estimates, within this framework is discussed in view on
lowering the regularity assumptions on the data, usually measured
in terms of the classical Sobolev spaces
${H}^{s}$. The Fourier
restriction norm method introduced by Bourgain in 1993 is sketched
at hand of several applications. Finally we indicate some recent
progress, which could be achieved by leaving the scale of the
Sobolev spaces and applying an appropriate generalization of
Bourgain's method.

### 13/09/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Radoslaw Czaja, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

### Asymptotics of parabolic equations with possible blow-up

### 06/09/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Miguel Rodríguez Olmos, *École Polytechnique Fédérale de Lausanne*

### Stability of relative equilibria in Hamiltonian mechanical systems

### 21/07/2005, 14:00 — 15:00 — Room P3.10, Mathematics Building

Saber Elaydi, *Trinity University, San Antonio*

### Extension of the Sharkovsky Theorem to non Autonomous Dynamical
Systems II

### 19/07/2005, 14:00 — 15:00 — Room P3.10, Mathematics Building

Saber Elaydi, *Trinity University, San Antonio*

### Extension of the Sharkovsky Theorem to non Autonomous Dynamical
Systems I

### 18/07/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Artur Lopes, *Universidade Federal do Rio Grande do Sul, Porto Alegre*

### Gibbs states limits and large deviations

### 08/07/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Jesus de Loera, *University of California, Davis*

### The many aspects of counting lattice points on polytopes

### 28/06/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Jack Hale, *Georgia Institute of Technology*

### Perturbation of periodic orbits of functional differential
equations

### 31/05/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Sara Fernandes, *Universidade de Évora*

### Spectral theory and discrete dynamical systems

### 17/05/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Messoud Efendiev, *Universität Stuttgart*

### Symmetry and Attractors

### 26/04/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Nuno Luzia, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

### On the variational principle for Hausdorff dimension

### 19/04/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Vitor Saraiva, *Instituto Superior Técnico*

### Average densities on invariant sets

### 12/04/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Lucian Radu, *Instituto Superior Técnico*

### Measures of maximal dimension

### 05/04/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Jorge Buescu, *Instituto Superior Técnico*

### Positivity and differentiable reproducing kernel inequalities