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21/02/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building

Gabriela Gomes, *Instituto Gulbenkian de Ciência*

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Fundamental assumptions in models of reinfection

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14/02/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building

Pantelis Damianou, *University of Cyprus*

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The modular class and integrable systems

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07/02/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building

Maria João Oliveira, *Universidade Aberta*

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Bogoliubov functionals: from measure theory to functional analysis

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18/01/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building

Constantine Dafermos, *Brown University*

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Hyperbolic Conservation Laws with Contingent Entropies and
Involutions

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13/12/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Roman Hric, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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Topology of minimal systems

I will start with a short survey of basic facts about minimal sets
and I will briefly discuss their role in the theory of dynamical
systems. In the second part, I will present recent results obtained
with F. Balibrea, T. Downarowicz, L. Snoha and
V. Spitalsky.

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06/12/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

Michal Málek, *Silesian University, Opava*

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Basic sets and distributional chaos in dimension one

The notion of basic set was introduced by A. N. Sharkovskii in the series of his papers from late sixties. These papers focused on omega-limit sets of continuous maps of the compact interval. It turned out later that existence of basic sets for a dynamical system on the compact interval corresponds to nontrivial behavior of the dynamical system: positive topological entropy, existence of horseshoes, distributional (and other) chaos.

In this talk we present results showing that similar behavior remains true when other one-dimensional spaces are considered (the circle, trees, graphs).

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29/11/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building

João Martins, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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$2$-dimensional homotopy invariants of complements of embedded
surfaces in ${S}^{4}$

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

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

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On Hausdorff dimension in higher dimensions

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03/11/2005, 11:00 — 12:00 — Room P3.10, Mathematics Building

Arnaldo Mandel, *IME, Universidade de São Paulo*

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Faithful linear representions of the free group

Many families of matrices that generate free groups are known, notably constructions by Hausdorff (1907) and Sanov (1947). Topological arguments show that those families are ubiquitous, and Tits' Theorem of Alternatives (1972) gives deep algebraic reasons for that. On the other hand, faced with a pair of matrices, it is almost hopeless to decide whether they generate a free group. I will describe some that is known on explicit presentation of matrix generators of free groups, concentrating on joint work with Jairo Gonçalves and Mazi Shirvani, that describe explicit free groups in the multiplicative group of some finite dimensional algebras.

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28/10/2005, 15:00 — 16:00 — Room P3.31, Mathematics Building

Tomasz Downarowicz, *Wroclaw University of Technology*

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Minimal realizations of Jewett-Krieger type for nonuniquely ergodic
systems II

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).

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

Tomasz Downarowicz, *Wroclaw University of Technology*

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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).

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

Lubomír Snoha, *Matej Bel University, Banská Bystrica*

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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.

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

Rogério Martins, *Universidade Nova de Lisboa*

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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.

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

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

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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.

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

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

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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.

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

Axel Grünrock, *University of Wuppertal*

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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.

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

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

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Asymptotics of parabolic equations with possible blow-up

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

Miguel Rodríguez Olmos, *École Polytechnique Fédérale de Lausanne*

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Stability of relative equilibria in Hamiltonian mechanical systems

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

Saber Elaydi, *Trinity University, San Antonio*

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Extension of the Sharkovsky Theorem to non Autonomous Dynamical
Systems II

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

Saber Elaydi, *Trinity University, San Antonio*

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Extension of the Sharkovsky Theorem to non Autonomous Dynamical
Systems I