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27/07/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Irene Fonseca, *Carnegie Mellon University, Pittsburgh*

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Second order variational problems

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20/07/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Giovanni Leoni, *Carnegie Mellon University, Pittsburgh*

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Optimal regularity for free boundary problems and a conjecture of
De Giorgi

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08/07/2004, 11:00 — 12:00 — Room P3.10, Mathematics Building

Sérgio Oliva, *Universidade de São Paulo*

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Discretization of Unidimensional Functional Partial Equations

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06/07/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Bart Van Steirteghem, *Columbia University*

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A classification of affine smooth spherical varieties

F. Knop has translated Delzant's conjecture (1990) on compact
multiplicity free spaces (also known as noncommutative completely
integrable systems) into a conjecture on affine smooth spherical
varieties. If $G$ is a reductive algebraic group (over $C$) and $X$
is an affine $G$-variety (over $C$), then $X$ is called spherical
if its coordinate ring $C\left[X\right]$ is multiplicity free as a $G$-module.
Knop's conjecture states that if moreover $X$ is smooth, the module
structure of $C\left[X\right]$ determines the $G$-variety $X$ up to
isomorphism. As a first step towards proving the conjecture, we
have classified these varieties, "up to pesky tori and connected
components".

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29/06/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Pierre Martinetti, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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Renormalization and Hopf algebras

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22/06/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Philippe Monnier, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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A cohomology associated to a function

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15/06/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Luis Barreira, *Instituto Superior Técnico*

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Stability of nonautonomous differential equations

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01/06/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Jens Marklof, *University of Bristol*

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Ergodic theory and the distribution of ${n}^{2}x$ modulo one

I will discuss some recent results on the distribution of certain
unipotent orbits on hyperbolic surfaces and their implication on
the randomness of the fractional parts of the sequence ${n}^{2}x$.

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18/05/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

João Lopes Dias, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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Floquet solutions for linear ODEs with quasiperiodic coefficients

We investigate the local reducibility of linear analytic
multifrequency cocycles on SL(2,R) using renormalisation. In this
way we show that some linear ODEs with coefficients depending
quasiperiodically on time can be conjugated to others with constant
coefficients. Much in the same way as Floquet theory for periodic
coefficients.

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11/05/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

David Krejcirik, *Centro de Análise Matemática, Geometria e Sistemas Dinâmicos*

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A lower bound to the spectral threshold in curved tubes

Motivated by the theory of quantum waveguides, we consider the
Laplacian in curved tubes of arbitrary cross-section rotating
together with the Frenet frame along curves in Euclidean spaces of
arbitrary dimension, subject to Dirichlet boundary conditions on
the cylindrical surface and Neumann conditions at the ends of the
tube. We prove that the spectral threshold of the Laplacian is
estimated from below by the lowest eigenvalue of the Dirichlet
Laplacian in a torus determined by the geometry of the tube. This
is a joint work with Pavel Exner and Pedro Freitas.

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04/05/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Fabrice Planchon, *Université Paris 13*

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Dispersion for Schrödinger equations with variable coefficients and
applications

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27/04/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

José Mourão, *Instituto Superior Técnico*

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Generalized theta functions

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20/04/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Jorge Buescu, *Instituto Superior Técnico*

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Integral equations, positive matrices, and reproducing kernel
inequalities

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05/04/2004, 16:00 — 17:00 — Room P3.10, Mathematics Building

Margarida Mendes Lopes, *Instituto Superior Técnico*

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Godeaux surfaces with an involution

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30/03/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

Luis Barreira, *Instituto Superior Técnico*

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Ergodic theory and cohomology

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23/03/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building

José Francisco Rodrigues, *Universidade de Lisboa*

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The $N$-Membranes Problem for Nonlinear Degenerate Systems

We consider the elliptic variational inequality associated with quasilinear nonlinear systems, including those of $p$-Laplacian type, and with the ordering constraint of the $N$-membrane problem. We extend the Lewy-Stampacchia inequalities for the solution, obtaining new regularity results for the derivatives of the solution, including in the case of linear operators ( $p=2$) integrability of second derivatives for all $q>1$. Considering the $N$-membrane problem as coupled $(N-1)$-obstacle problem we obtain also the corresponding conditions for the stability of the coincidence sets for the variation of external forces.

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

Miguel Ramos, *Universidade de Lisboa*

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Asymptotic properties of the ground-state solutions of singularly perturbed elliptic Hamiltonian systems

We study the shape of the positive solutions of an elliptic system of the form

$-{\epsilon}^{2}\Delta u+u=g(v),-{\epsilon}^{2}\Delta v+v=f(u)$

as the parameter $\epsilon $ goes to zero, namely the appearance of "spike-layer patterns" for the solutions of the system having minimal energy. Here the nonlinear terms are assumed to have superlinear and subcritical growth at infinity and we consider both cases of Neumann and Dirichlet boundary conditions over a given bounded open set $\Omega \subset {R}^{n}$. The proofs combine standard estimates in elliptic equations with new ideas in the calculus of variations (variational methods and Morse theory). This is joint work with J. Yang (to appear in Trans. Amer. Math. Soc.) and with A. Pistoia (to appear in J. Differential Equations).

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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