Sir Michael Atiyah em Portugal para Pedro Nunes Lectures

Sir Michael Atiyah em Portugal para Pedro Nunes Lectures

Um dos maiores nomes da Matemática contemporânea estará em Portugal para as Pedro Nunes Lectures de 29 de Março a 6 de Abril. Michael Francis Atiyah, britânico de origem libanesa, é detentor da Medalha Fields (1966), do Prémio Abel (2004) e da medalha Copley (1988), outorgada pela Royal Society de Londres. As suas contribuições matemáticas, bem como a sua luta pelo desarmamento nuclear, levaram-no a receber da rainha da Inglaterra o título de Sir em 1983 e a receber a Ordem do Mérito em 1992.

O matemático estará em Lisboa, Braga, Porto e Coimbra para falar dos seus trabalhos em análise, geometria e topologia, de relações entre matemática e física, bem como da profissão de matemático.

Atiyah foi professor na universidade de Oxford e no Instituto de Estudos Avançados, em Princeton, director do Trinity College e do Isaac Newton Institute for Mathematical Sciences, ambos em Cambridge, presidente da Royal Society de Londres e Chanceler da Universidade de Leicester. Actualmente é professor honorário da Universidade de Edimburgo.

As Pedro Nunes Lectures são uma iniciativa do Centro Internacional de Matemática, em colaboração com a Sociedade Portuguesa de Matemática e com o apoio da Fundação Calouste Gulbenkian.

Programa das Pedro Nunes Lectures

Geometric Models of Matter
29 de Março, 17h30 – Fundação Calouste Gulbenkian, Lisboa
Auditório 2
Web streaming: http://live.fccn.pt/fcg/
Abstract: Einstein always wanted to supplement his beautiful geometric theory of general relativity by a parallel treatment of matter. Much has happened in physics since Einstein’s time, with geometry playing an important part. In my lecture I will put forward some speculative new ideas in this area.

An unsolved problem in elementary Euclidean geometry
31 de Março, 11h30 – Universidade do Minho
Anfiteatro da Escola de Ciências, Campus de Gualtar
Abstract: I will describe a very simple problem about n distinct points in Euclidean 3 dimensional space. The problem is still unsolved after more than 10 years. The problem has a number of interesting contacts with theoretical physics, which may help in its solution.

Topology and quantum physics
4 de Abril, 15h00 – Universidade do Porto
Edifício da Matemática da FCUP
Abstract: Geometry and Physics have been closely linked, at least from the time of Galileo and Newton. Forces turn straight line motion into curved paths and the fundamental link between curvature and force persists into Maxwell’s theory of Electro-Magnetism and Einstein’s theory of General Relativity. 20th century Physics was dominated by Quantum Mechanics, and it took a long time to realize that this has close links with Topology, the branch of Geometry where measurement is unimportant and which is typified by the study of knots. I will try to explain this story in non-technical language, using only high school geometry and physics.

The index theory of Fredholm operators
6 de Abril, 14h30 – Universidade de Coimbra
Sala Pedro Nunes do Departamento de Matemática
Abstract: Fredholm operators are bounded operators in Hilbert space which, together with their adjoints, have a finite-dimensional null space. The index is the difference of the dimensions of these two null spaces, and is invariant under continuous perturbation. Fredholm operators arise naturally from elliptic differential operators, and the computation of the index is a classical problem. I will discuss this story in general terms and give simple examples. I will also introduce a number of variations on the theme.

Descarregue o cartaz do evento para divulgação.

Entrevista do boletim do CIM a Michael Atiyah

Research Track Record Overview

Currently I’m still working beyond the interplay between localization operators on the phase space and the structure of the Berezin symbol calculus in the context of polyanalytic Fock spaces.

Although localization operators has a long history of physics, they only became popular after the framework of Berezin (Berezin, 1971).  From the mathematical point of view Localization operators were used at a first time by I. Daubechies (1988) in order to localize a signal simultaneously on time and on frequency.  In the case that the symbol (window) is a Gaussian, the localization operators adopted by I. Daubechies are anti-Wick operators derived from Berezin quantization rules .

On the books:

MR1639461 (2000c:47098) Wong, M. W. Weyl transforms. Universitext. Springer-Verlag, New York, 1998. viii+158 pp. ISBN: 0-387-98414-3 (Reviewer: Khélifa Trimèche), 47G30 (42B10 43A32 44A15 81S30)

PDF Clipboard Series Book
MR1918652 (2003i:42003) Wong, M. W. Wavelet transforms and localization operators. Operator Theory: Advances and Applications, 136. Birkhäuser Verlag, Basel, 2002. viii+156 pp. ISBN: 3-7643-6789-X (Reviewer: Steen Pedersen), 42-02 (42C40 43A80 43A85 47B38 47G10)

a complete framework about Gabor-Daubechies localization operators was established using the representation theory backdrop, namely the (Weyl-)Heisenberg group.

The main interest around this subject that seems to be useful applications in time-frequency analysis, namely on the study of modulation spaces (e.g. the Feichtinger algebra $S_0$) using essentially methods from analytic function spaces (e.g. $BMO^p$ and $Q_p$ (?)) is the Coburn conjecture announced in:

Coburn, L. A. The Bargmann isometry and Gabor-Daubechies wavelet localization operators. Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), 169–17

Roughly speaking, Coburn’s conjecture states that any localization operator $L_f^{(w)}$ with symbol $f$ and window $w$ coincides with a Toeplitz operator $T_{D f}$ with polynomial differential symbol $D f$.

This conjecture were already proved in the papers

Lo, Min-Lin(1-CASSB)
The Bargmann transform and windowed Fourier localization.
Integral Equations Operator Theory 57 (2007), no. 3, 397–412.

Engliš, Miroslav(CZ-SIL-IM)
Toeplitz operators and localization operators.
Trans. Amer. Math. Soc. 361 (2009), no. 2, 1039–1052.

in the context of  Segal-Bargmann-Fock spaces for nicer class of windows under the constraint that the symbol $f$ belongs to $L^\infty(\mathbb{C})$. Recently, Joshua Isralowitz and Bo Li use the results of the papers of Lo (2007) and Englis (2009) to study regularity conditions on the symbol under certain (BMO) regularity constraints as well as the boundedness and compactness of Toeplitz operators on the Segal-Bargmann-Fock spaces. This results appeared recently in Transactions of Mathematics.

In the sequel we are still developed on an extension of this result to polyanalytic Fock spaces in the context of Gabor analysis as a continuation of the framework started in the papers below. Forthcoming results should be announced soon…

Abreu, Luís Daniel On the structure of Gabor and super Gabor spaces. Monatsh. Math. 161 (2010), no. 3, 237–253, 46Exx (33C45 42C15 81S30 94A12)

PDF Clipboard Journal Article

Abreu, Luís Daniel Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions. Appl. Comput. Harmon. Anal. 29 (2010), no. 3, 287–302,42C15 (94A20)