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09/03/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Pedro Resende, *Instituto Superior Técnico*

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Noncommutative topology and quantales

The classical Gelfand representation theorem tells us that any unital (complex)
$C$*-algebra is, up to isomorphism, the algebra of continuous complex valued functions on a compact Hausdorff space. The noncommutative analogue of this result is the Gelfand-Naimark theorem, which shows how any
$C$*-algebra can be concretely realized as an algebra of bounded operators on some Hilbert space. However, there is a sense in which this noncommutative "analogue" fails to provide a characterization of those operators on the Hilbert space that actually lie in the given
$C$*-algebra, and in 1971 Giles and Kummer (and also Akemann, in a related but independent way) introduced a notion of noncommutative topology in terms of which, essentially, every
$C$*-algebra becomes an algebra of "continuous functions".

But lacking in their approach is a self-contained (i.e., independent of
$C$*-algebras) characterization of what should be meant by such a noncommutative topology, and it was partly in an attempt to answer this question that quantales were proposed by Mulvey in 1983 as possible candidates for such topologies. In this talk I shall focus on the connections between quantales and
$C$*-algebras, in particular addressing as an example, if time allows, Connes' noncommutative space of Penrose tilings.

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02/03/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

João Alves, *Instituto Superior Técnico*

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Topological entropy and homological growth on graphs

We establish a precise relation between the topological entropy and
entropies arising from the homological growth and the exponential
growth rate of the number of periodic points for a piecewise
monotone graph map showing that the first one is the maximum of the
latter two. This nontrivially extends a result of Milnor and
Thurston on piecewise monotone interval maps. For this purpose we
generalize the concept of Milnor-Thurston zeta function involving
Lefschetz zeta function.

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03/02/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Henrique Oliveira, *Instituto Superior Técnico*

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Iterates of nonpolynomial maps

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27/01/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Marc Oliver Rieger, *Scuola Normale Superiore, Pisa*

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Young measures - Basic ideas and recent applications

Young measures have been successfully applied to various mathematical problems in material science and other areas. They are used as a tool to study problems which include different length scales, e.g., when microscopical structures affect the global behavior of a material.

In this talk we will give an introduction to Young measures, filling the abstract definitions with some intuitive ideas. We will apply this concept to equations modelling shape memory alloys (one-dimensional thermoelasticity with nonconvex energy density) and will present a recent existence result (in collaboration with Johannes Zimmer, MPI Leipzig).

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20/01/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Mark Holland, *University of Surrey*

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Nonuniformly hyperbolic dynamics and Lorenz attractors

We consider the statistical properties of Lorenz-like strange
attractors which arise out of two-dimensional return maps for the flow.
The questions we ask are: Does the system admit an ergodic invariant
measure which is mixing? How fast is the mixing? Is the measure physical:
ie does it describe the statistical behaviour of many points in the basin?
To answer these questions we use a "Markov extension" or "Young tower" to
reduce the dynamics to that of an expanding map over a hyperbolic base
set, with infinitely many branches having variable return times. This talk
is based on ongoing work with S. Luzzatto. I hope to make the talk
accessible to nonspecialists in ergodic theory.

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16/12/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Markus Keel, *University of Minnesota*

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Global Rough Solutions for Nonlinear Dispersive Equations

This will be a PDE talk aimed at a general audience, describing
some recent work with J. Colliander, G. Staffilani, H. Takaoka, and
T. Tao. We will focus on new results which give basic descriptions
of the long time behavior of semilinear Schroedinger equations, but
some of the techniques employed are applicable to other PDE's. In
particular, the talk will describe how so-called
almost-conservation laws and interactaction Morawetz-type
inequalities help understand the regularity properties of these
equations.

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09/12/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Fernando Pestana da Costa, *Instituto Superior Técnico*

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Invariant regions and self-similarity in a coagulation model

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02/12/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Rémi Carles, *Université Bordeaux I e Centro de Matemática e Aplicações Fundamentais*

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On the role of quadratic oscillations in nonlinear Schrödinger
equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. After a short explanation of this result, we investigate the question to know how particular such initial data are. The first step consists in finding a "linearizability condition": when is the nonlinear term relevant in the semi-classical limit? This conditions reduces the problem to the study of a linear equation. For this equation, we use a "profile decomposition" technique,
introduced by P. Gerard. This shows that according to the nonlinearity considered, quadratic oscillations are the only relevant initial data, or are not. We also present some applications to a nonlinear superposition principle, and to the study of finite time blow up.

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25/11/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Nuno Martins, *Instituto Superior Técnico*

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${C}^{*}$-algebras arising from interval maps

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18/11/2003, 16:00 — 17:00 — Room P3.10, Mathematics Building

Hui Li, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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Semi-free Hamiltonian circle actions on 6-dimensional symplectic
manifolds

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11/11/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Amílcar Sernadas, *Instituto Superior Técnico*

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Combining logic systems: Why, how, what for?

Motivated by applications in artificial intelligence and software
engineering that require the joint use of different deduction
formalisms, the interest in combination of logic systems has
recently been growing, but the topic is also of interest on purely
theoretical grounds. Several forms of combination have been
studied, like product, fusion, temporalization, parameterization,
synchronization and, more recently, fibring. In this guided tour of
the issues raised by the combination of logics, we define fibring
(the most general form of combination) in a very simple (yet
useful) context, discuss some examples and establish some
interesting transference results, namely preservation of strong
completeness and nonpreservation of congruence. We end the tour
with a brief reference to some open problems. The talk is based on
a recent overview paper (together with C. Sernadas) available
at
http://www.cs.math.ist.utl.pt/ftp/pub/SernadasA/03-SS-fiblog22.pdf
to appear in the CIM Bulletin.

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04/11/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Paulo Pinto, *Instituto Superior Técnico*

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Classification of Modular Invariants and Subfactors

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21/10/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Nara Jung, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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Continuity properties of $\text{}{D}^{2}u$ in ${\mathrm{BV}}^{2}$ with strict convergence

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14/10/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Robert Oliver, *Université Paris 13*

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Homotopy equivalences between completed classifying spaces of
finite groups

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07/10/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

João Paulo Santos, *Instituto Superior Técnico*

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Dirac operator, instantons and holomorphic bundles

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30/09/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Ana Paula Dias, *Universidade do Porto*

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Symmetry groupoids, synchrony, and coupled cell networks

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23/09/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Joana Ventura, *Instituto Superior Técnico*

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Homological algebra for the representation Green functor for
Abelian groups

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09/09/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Michael Grinfeld, *The University of Strathclyde in Glasgow*

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Multiplicity of periodic solutions in mean-field theories of
magnetisation

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29/07/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Leonardo Macarini, *IMPA, Rio de Janeiro*

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Hofer-Zehnder sensitive capacity of cotangent bundles and
symplectic submanifolds

We will consider a modified Hofer-Zehnder capacity sensitive to the
homotopy class of the periodic orbits and show that if a symplectic
manifold admits a free Hamiltonian circle action then it has
bounded Hofer-Zehnder (sensitive) capacity. We give two
applications of this result. Firstly, we prove that every closed
symplectic submanifold has a neighborhood with finite Hofer-Zehnder
capacity. Secondly, consider a closed manifold with an effective
circle action whose fixed point set has trivial normal bundle.
Then, its standard cotangent bundle has bounded Hofer-Zehnder
capacity.

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22/07/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building

Christian Wolf, *Wichita State University*

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Measures of maximal dimension for hyperbolic diffeomorphisms