6th European Intensive Course on Complex Analysis  

“Complex Analysis and its Generalizations (with applications to partial differential equations)”

Departamento de Matemática, Universidade de Coimbra, Portugal

  20-31 March 2000

With support from CMUC (Centro de Matemática da Universidade de Coimbra), UI&D "Matemática e Aplicações" da Universidade de Aveiro, and the Socrates programme

Table of contents

Goal of the Course

This intensive course will have a total of 40 hours of lectures and is at postgraduate level. Lecturers will have time available to discuss with the students. Successfully participating students will get a certificate. This course is organized by the Universities of Coimbra and Aveiro with the same goals as the ones organized under the programme Socrates, and is open to all young mathematicians interested in Complex Analysis and its applications.   

Schedule of the course 

Abstracts

• • TENSOR PRODUCTS AND GEOMETRY IN BANACH SPACES - Sean Dineen (Univ. College, Dublin, Ireland)

Abstract:

1. Introduction to Banach spaces, Hahn-Banach theorem, Examples.  

2. Duality theory, weak topology, reflexivity  

3. Bilinear mappings, algebraic tensor  products, linearization, projective and injective tensor products, examples.  

4. Polynomials and tensor products.  

5. Polarisation constants and geometric properties.  

6. Banach algebra, joint spectra, vector-valued spectra, polynomial spectral mapping theorems.  

Remarks: This course will not require a background in functional  analysis. The first 3 topics are standard topics that are given in  many first year graduate courses on functional analysis. The final 3 topics cover recent research.   

• • ORTHOGONAL RATIONAL FUNCTIONS - Prof. F. Marcellán (Univ. Carlos III,  Madrid, Spain)

Abstract: We will present the state of the art in the subject which constitutes a new and interesting subject of reserach with many applications in Linear Prediction, network synthesis and control theory:

1. The fundamental spaces

2. Kernel functions, recurrence and second kind functions

3. Para-orthogonality and Quadrature

4. Density of rational functions

5. Convergence

6. The boundary case

7. Applications  

• • SPACES OF ANALYTIC FUNCTIONS AND WAVELETS - Vladimir Kisil (Univ. Leeds, UK)

Abstract: Polynomials  Our purpose is to describe a general framework for generalizations of the complex analysis. As a consequence a classification scheme for different generalizations is obtained.

The framework is based on wavelets (coherent states) in Banach spaces generated by “admissible” group representations. Reduced wavelet transform allows naturally describe in abstract term main objects of an analytical function theory: the Cauchy integral formula, the Hardy and Bergman spaces, the Cauchy-Riemann equation, and the Taylor expansion.

Among considered examples are classical analytical function theories  (one complex variables, several complex variables, Clifford analysis, Segal-Bargmann space) as well as new function theories which were developed within our framework (function theory of hyperbolic type, Clifford version of Segal-Bargmann space).

We also briefly discuss applications to the operator theory (functional calculus) and quantum mechanics.  

• • MODERN TOOLS IN THE THEORY OF QUASICONFORMAL MAPPINGS - Olli T. Martio (Univ. Helsinki, Finland) 

Abstract: The theory of quasiconformal mappings was first developed in the plane and it was closely connected with the theory of analytic functions of one complex variable. The standard definitions of quasiconformality are based on the characteristic properties of conformal mappings. The most important of these are the invariance of  the 2-modulus or the 2-capacity and the angle preservation. In 1960-80 the analytic properties of quasiconformal mappings played an important role in many applications, most notably in the non-linear PDE´s and in the non-linear potential theory. In the 1990´s iterations in the complex planeand dynamical systems formed an important area where quasiconformal mappings were useful. However, recently new applications have emerged. These use quasiconformal mappings in a very general setup. Sobolev spaces on metric spaces and manifold like structures have been instrumental in this development. The purpose of these lectures is to give the basic definitions and theorems on which the modern theory is based. This will also provide a new look at the old concepts and, hopefully, open the field for new ideas and applications.  

• • Student presentations

29 March: 10-12h 

History

This intensive course follows the five held in Coimbra and Aveiro from 1995 to 1999 and there are plans for intensive courses in the following years. The lecture notes of some of the courses have been published in Coimbra and others are in print (for more information please see the URL http://www.mat.uc.pt/publicacoes/textosB.html)

Aditional Information

Informations about the Mathematics Department or the University of Coimbra can be seen in http://www.mat.uc.pt 

Organizers

• Helmuth Malonek (Departamento de Matemática da Universidade de Aveiro)

• J. Carvalho e Silva (Departamento de Matemática Universidade de Coimbra)

• Amilcar Branquinho (Departamento de Matemática Universidade de Coimbra)  

Back to home page