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Analysis, Geometry, and Dynamical Systems Seminar   RSS

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

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.

02/03/2004, 15:00 — 16:00 — Room P3.10, Mathematics Building
João Alves, Instituto Superior Técnico

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.

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

Iterates of nonpolynomial maps

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

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).

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

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.

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

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.

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

Invariant regions and self-similarity in a coagulation model

02/12/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Université Bordeaux I e Centro de Matemática e Aplicações Fundamentais

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.

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

C* -algebras arising from interval maps

18/11/2003, 16:00 — 17:00 — Room P3.10, Mathematics Building
Hui Li, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

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

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

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.

04/11/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building
Paulo Pinto, Instituto Superior Técnico

Classification of Modular Invariants and Subfactors

21/10/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building
Nara Jung, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos

Continuity properties of D2 u in BV2 with strict convergence

14/10/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building
, Université Paris 13

Homotopy equivalences between completed classifying spaces of finite groups

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

Dirac operator, instantons and holomorphic bundles

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

Symmetry groupoids, synchrony, and coupled cell networks

23/09/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building
Joana Ventura, Instituto Superior Técnico

Homological algebra for the representation Green functor for Abelian groups

09/09/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building
, The University of Strathclyde in Glasgow

Multiplicity of periodic solutions in mean-field theories of magnetisation

29/07/2003, 15:00 — 16:00 — Room P3.10, Mathematics Building
Leonardo Macarini, IMPA, Rio de Janeiro

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.

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

Measures of maximal dimension for hyperbolic diffeomorphisms

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