27/10/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Andreas Hartmann, Université Bordeaux I, France
Extremal functions of kernels of Toeplitz operators
We will essentially discuss two points in the connection with
extremal functions of kernels of Toeplitz operators on Hardy
spaces. The first one concerns divisor properties of such extremal
functions. It turns out that in many situations such a division has
nice properties like being a contraction (case of Hedenmalm's
canonical divisors in the Bergman space), or even an isometry
(inner functions in the Hardy space). Concerning extremal functions
of kernels of a Toeplitz operator, the question has been considered
in the larger class of nearly invariant subspaces by Hitt. He
proved that in the Hilbert space situation , the division by
the extremal function of a nearly invariant subspace is isometric.
The situation changes drastically even for Toeplitz kernels when
one switches to the non Hilbert case ( ), where,
depending on the parameter , one can in general only
expect a control on the division or on the multiplication by the
extremal function. Examples show that two-sided estimates cannot be
expected in general. The second part of the talk will be devoted to
the investigation of invertibility properties of Toeplitz operators
by means of the extremal function. This understands that the
Toeplitz operator is supposed non injective in order that such an
extremal function exists. In this part we have to assume the
Hilbert situation . It turns out that certain parameters
associated with the extremal function, and that have previously
been used by Hayashi to distinguish kernels of Toeplitz operators
from general nearly invariant subspaces, enable us to characterize
the surjectivity of a (non injective) Toeplitz operator.
20/10/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Lina Oliveira, Instituto Superior Técnico, U.T. Lisboa
A non-commutative notion of topology
The Gelfand-Naimark theorem establishes that any commutative
-algebra is isometrically -isomorphic to a certain
space of continuous functions over a locally compact topological
space. In a way, the -algebra might be seen as a
translation of the topology of the space. It is the aim of this
talk to give an overview of how the generalization of these ideas
to non-commutative -algebras has lead to a non-commutative
notion of topology. A résumé of some recent results and of on going
research will be presented.
08/09/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Messoud Efendiev, GSF/TUM Munich, Germany
Nonlinear Riemann-Hilbert problem: theory and application
In my talk I will consider the recent developments in the study of global solvability of nonlinear equations. We will consider a quite large class of nonlinear mappings generated by the nonlinear elliptic problems related to the nonlinear pseudodifferential opearators. Here the geometrical properties of corresponding mappings will play a crucial role, which will allow us to define topological and other invariants. We will especially focus an application of these new ideas to the global solvability of nonlinear Riemann-Hilbert problems for general domains.
07/07/2006, 15:15 — 16:15 — Sala P3.10, Pavilhão de Matemática
Marina Dubatovskaya, Belarusian State University, Minsk, Belarus
Heat conduction in 2D-domains with symmetric inclusions: a model
and reduction to a vector-matrix problem
The problem of heat conduction of 2D bounded composite material
with symmetrically situated inclusions having different
conductivity is reduced to a vector-matrix Riemann boundary value
problem with a piecewise constant matrix. The proposed method
generalizes an approach by V.V. Mityushev presented in the book by
V.V. Mityushev and S.V. Rogosin on Constructive Methods for Linear
and Nonlinear Boundary Value Problems for Analytic Functions , CRC
Press, 1999. The talk is based upon joint work with Sergei Rogosin.
07/07/2006, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Sergei V. Rogosin, Belarusian State University, Minsk, Belarus
Hele-Shaw model for melting with several dendrits
The melting problem with several dendrits is reduced to a free
boundary Hele-Shaw problem for a multiply connected domain. Its
local in time solvability is studied on the base of a variant of
the Nirenberg-Nishida theorem concerning the Cauchy-Kovalevsky
problem and functional equations in complex domains. Other
approaches to the above problems are discussed as well. The talk is
based upon joint work with Tatsjana Vaiteakhovich.
16/06/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Cristina Diogo, Instituto Superior Técnico, U.T. Lisboa
Generalized factorization for a class of triangular symbols with a
gap around zero
It was shown in [1] that if a bounded analytic solution to the
Riemann-Hilbert problem is known which consists of
corona pairs, then admits a canonical generalized
factorization, i.e., the Toeplitz operator with symbol is
invertible. But what if the solution does not satisfy the corona
conditions? And, to begin at the beginning, how to get a particular
bounded analytic solution to the Riemann-Hilbert problem? We adress
these questions by studying a class of triangular matrix symbols
which illustrates the problems involved in those questions and for
which we can find answers.
- Bastos, M. A., Karlovich, Y. I. and dos Santos, A. F.,
Oscillatory Riemann-Hilbert problems and corona theorem,
Journal of Functional Analysis 197 (2003) 347-397.
19/05/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Ana Moura Santos, Instituto Superior Técnico, U.T. Lisboa
Difracção de ondas por cunhas e estruturas periódicas no âmbito da
Feira de Conhecimento e da Inovação
Problemas de difracção de ondas têm sido estudados em cooperação
com investigadores europeus e americanos no âmbito de JNICT/BMFT
project 423/1(94-98) e project 423/2 (95-99),
FCT/FEDER/POCTI/MAT/34222 (99) e 59972 (04). Dois dos tópicos de
investigação são a difracção por cunhas e por redes periódicas. A
apresentação desta investigação no âmbito da Feira do Conhecimento
e da Inovação (Abril 2006) foi integrada na iniciativa da
divulgação de projectos dum centro de Matemática, o CEMAT, junto
dum público heterógeneo. Nesta apresentação mostraremos o
multimédia que preparámos para a ocasiâo e tentaremos analisar a
importãncia e a projecção desta iniciativa pioneira. Procuraremos
discutir a relevância dos tópicos de investigação escolhidos na
área dos Problemas de Difracção e da Teoria de Operadores.
12/05/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Anthippi Poulkou, University of Athens Panepistimioupolis, Grécia
Sampling and interpolation theories associated with boundary value
problems
This talk deals with joint work with W.N. Everritt. The link of the
sampling/interpolation theorem of Shannon-Whittaker with the
original Kramer sampling theorem is considered. Also, the
connection of these two significant results with boundary value
problems associated with linear ordinary differential equations as
defined on intervals of real line is specified. The results given
in this talk are concerned with the generation from first-order
linear, ordinary boundary value problems of Kramer analytic kernels
which introduce analytic dependence of the kernel on the sampling
parameter. These kernels are represented by unbounded self-adjoint
differential operators in Hilbert function spaces. Necessary and
sufficient conditions are given to ensure that these differential
operators have a simple, discrete spectrum which then allows the
introduction of the associated Kramer analytic kernels. Finally,
the corresponding analytic interpolation functions are defined with
the required properties, to give the Lagrange interpolation series.
21/04/2006, 14:30 — 16:00 — Sala P3.10, Pavilhão de Matemática
Cristina Câmara, Instituto Superior Técnico, U.T. Lisboa
Fredholmness and corona problems
The Fredholm properties of Toeplitz operators with \(2\times 2\)
matrix symbols \(G\), which are essentially bounded on the real
line, are studied in connection with some properties of a solution
to a homogeneous linear Riemann-Hilbert problem with coefficient
\(G\). Conditions for the Toeplitz operator to be Fredholm are
obtained and, if that is the case, formulas for the factors in a
generalized factorization of \(G\) are given in terms of the
solutions to a pair of non-standard corona problems. These results
are used to establish partial index estimates for Daniele-Khrapkov
matrix functions.
07/04/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Sergej Rjasanow, Universität Saarbrücken, Alemanha
The Boltzmann equation. Theory and numerics
In the first part of the talk we introduce the Boltzmann
equation, discuss its properties and give an overview on existence
and uniqueness of solutions. Especially our new results on mapping
properties of the Boltzmann collision operator will be presented.
Then the Direct Simulation Monte Carlo method (DSMC) which is
widely applied in numerics will be explained. In the third part of
the talk we present the Stochastic Weighted Particle Method (SWPM)
which was introduced in 90's by Rjasanow and Wagner. We apply this
method to the numerical solution of the spatially two-dimensional
Boltzmann equation. The numerical solution of the Boltzmann
equation using naive deterministic methods leads to the amount of
numerical work of the order , where denotes the number
of discrete velocities in one direction. In the next part of the
talk we give an overview on deterministic numerical methods applied
to the Boltzmann equation by a number of authors. Then, in the
final part of the talk, we present the results of our numerical
experiments obtained by a new deterministic approximation of the
Boltzmann equation using Fast Fourier Transform.
31/03/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Roland Duduchava, University of Saarland, Saarbruecken
Differential operators and boundary value problems on hypersurfaces
We explore the extent to which basic differential operators (such as Laplace-Beltrami, Lamé, Navier-Stokes, etc.) and boundary value problems on a hypersurface in can be expressed globally, in terms of the standard spatial coordinates in . The approach we develop also provides, in some important cases, useful simplifications as well as new interpretations of classical operators and equations. In particular, we obtain explicit representations of the surface deformation tensor, the Laplace-Beltrami operator, the Lamé-Beltrami operator, the operator of isotropic elasticity and the Stokes system on the hypersurface. The obtained representations are helpful in proving existence of fundamental solutions on the manifold and in writing explicit Green formulae for the classical boundary value problems on an open subsurface. Combined with the potential method, the obtained results allow to prove the unique solvability of the aforementioned boundary value problems by a standard scheme. The lecture is based upon joint research with with Dorina Mitrea and Marius Mitrea.
10/03/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich
Asymptotic formulas for traces of Toeplitz matrices with symbols in
Hölder-Zygmund spaces
We discuss new higher order asymptotic formulas for traces of
Toeplitz matrices with symbols in Hölder-Zygmund spaces. Remainders
in these formulas go to zero with the speed depending on the
smoothness parameter of the space. The Wiener-Hopf factorization of
symbols within Hölder-Zygmund spaces plays an essential role in the
proof. These results refine Widom's asymptotic trace formulas and
complement Böttcher-Silbermann's higher order asymptotic formulas
for determinants of Toeplitz matrices.
03/02/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Vladimir Mityushev, Pedagogical Academy, Krakow, Poland
Constructive solution to the -linear problem with piece-wise constant matrices
The -linear conjugation problem with a piece-wise constant matrix given on the real axis and discontinuous at the finite numbers of the points is stated as follows. To find a vector-function analytic in the upper and lower half planes continuous in the closures of the half planes except the set where is bounded. The upper and lower limit values of on the real axis are linearly related via . A polynomial growth of at infinity is admitted. This problem is solved in closed form by application of the functional equations method under some restrictions on the dimension of and .
13/01/2006, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Alexei Karlovich, Universidade do Minho, Braga
Asymptotics of block Toeplitz deteminants generated by factorable
matrix functions with equal partial indices
We prove asymptotic formulas for block Toeplitz matrices with
symbols admitting right and left Wiener-Hopf factorization such
that all partial indices are equal to some integer number. We
consider symbols and Wiener-Hopf factorizations in Wiener algebras
with weights satisfying natural submultiplicativity, monotonicity,
and regularity conditions. Our results complement known formulas
for Hoelder continuous symbols due to Boettcher and Silbermann.
16/12/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Yuri Karlovich, Universidad Autónoma del Estado de Morelos, México
A weighted analogue of the Carleson-Hunt theorem and new classes of
pseudodifferential operators
Applying a weighted analogue of the Carleson-Hunt theorem on almost
everywhere convergence, we study the boundedness and compactness of
pseudodifferential operators with symbols that are bounded
measurable functions with respect to the spatial variable and
functions of bounded variation with respect to the dual variable.
Replacement of absolutely continuous functions of bounded variation
by arbitrary functions of bounded variation allows us to study
essentially more general classes of pseudodifferential operators. A
symbol calculus and a Fredholm theory for new classes of
pseudodifferential operators with non-regular symbols are
constructed. In particular, we study pseudodifferential operators
with symbols admitting discontinuities of first kind with respect
to spatial and dual variables that generate non-commutative
algebras of Fredholm symbols.
18/11/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
David Kapanadze, A. Razmadze Mathematical Institute, Academy of Sciences, Tbilisi, Georgia
Wave diffraction by a strip with first and second kind boundary conditions: the real wave number case
We prove unique existence of solution for a class of plane wave diffraction problems by a strip with first and second kind boundary conditions. This is done in a Bessel potential spaces framework, and for a real (non-complex) wave number. At the end, results about the regularity (and data dependence) of the solution are exhibited upon the initial setting and the boundary parameters.
11/11/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Cristina Câmara, Instituto Superior Técnico, U.T. Lisboa
Riemann-Hilbert problems, factorization of functions and structure of the factors
Let $G$ be a $2\times 2$ matrix function of Daniele-Khrapkov type. An equivalence between linear Riemann-Hilbert problems with coefficient $G$ and a class of scalar boundary value problems relative to a contour in a Riemann surface $\Sigma$ is established. By studying the solutions of these problems, it can be shown that the solution of the former Riemann-Hilbert problems must satisfy certain relations. In particular, if $G$ admits a canonical bounded factorization, it follows that the factors must have a certain structure.
04/11/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Amélia Bastos, Instituto Superior Técnico, U.T. Lisboa
A generalization of the local trajectory method for C* algebras
The local trajectory method is an invertibility method for operator
algebras with generators associated to unitary representations of
groups. In 1991 Yuri Karlovich proposed a generalization of this
method. We present a new generalization of the method and discuss
some possible applications.
14/10/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Frank-Olme Speck, Instituto Superior Técnico, U.T. Lisboa
Constructive matrix factorization methods for convolution type
operators with symmetry
Since Torsten Ehrhardt succeeded in creating a factorization theory
for Toeplitz plus Hankel operators in 2003, the doors are open for
the successful investigation of various classes of singular
operators and related applications. This talk is devoted to
constructive methods based upon explicit asymmetric factorization
of matrix functions on the real line, which is quite different from
the usual Wiener-Hopf factorization. A catalogue of possible
consequences and new applications will be exposed and discussed.
07/10/2005, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
Vladimir Rabinovich, Instituto Politecnico Nacional, Mexico
Essential spectrum of the main operators of quantum mechanics.
The aim of the talk is to present a new approach to the
investigation of the essential spectra of the main operators of
quantum mechanics. We include these operators in a class of
pseudodifferential operators perturbed by non-smooth potentials.
For an operator under consideration we introduce a family of limit
operators, and prove that the essential spectrum of the original
operator is the union of spectra of limit operators. Since the
limit operators have more simple structure than the original
operator, we obtain a strong tool for the investigation of the
essential spectra of differential and pseudodifferential operators.
We apply this method to the study of the essential spectra of the
Schrödinger, Klein-Gordon, and Dirac operators and to a new simple
proof of the classical Hunziker-van Winter-Zjislin Theorem.