European School on Complex Analysis

9th European Intensive Course on Complex Analysis

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

Departamento de Matemática, Universidade de Coimbra, Portugal

10 -21 March 2003

Goal of the Course	Schedule of the course	Abstracts	History
Financial Support	Organizers	Coming Events	Poster

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.

There will be an Workshop on "Applications and Generalizations of Complex Analysis" on the 15th of March.

Schedule of the course

First Week

	10 March	11 March	12 March	13 March	14 March
Opening session	9h30m-10h	*	*	*	*
Van Assche	10h-12h30m	10h-12h30m		10h-12h30m	10h-12h30m
Bures and Soucek	14h30m-17h	14h30m-17h		14h30m-17h	14h30m-17h

Second Week

	17 March	18 March	19 March	20 March	21 March
Käehler	10h-12h30m	10h-12h30m	Student presentation	10h-12h30m	14h30m-17h
Chamizo	14h30m-17h	14h30m-17h		14h30m-17h	10h-12h30m

Abstracts

HERMITE-PADÉ APPROXIMATION AND MULTIPLE ORTHOGONAL POLYNOMIALS

- Walter Van Assche (Katholieke Universiteit Leuven, Belgium)

Contents: Hermite-Padé approximation is simultaneous rational approximation of a vector (f1, … ,fr) of r functions with interpolation conditions at a given point in the complex plane. In this course the following aspects will be covered:

Padé approximation of a function. Relation with orthogonal polynomials.
Hermite-Padé approximation of r functions. Type I approximation and type II approximation.
Orthogonality conditions for Hermite-Padé approximation: multiple orthogonal polynomials.
Special systems: Angelesco system, algebraic Chebyshev systems (AT systems), Nikishin systems.
Examples of multiple orthogonal polynomials.
Analytic properties of Hermite-Padé approximation for Angelesco systems and Nikishin systems.
Riemann-Hilbert approach for the asymptotic analysis of Hermite-Padé approximation.
Application: simultaneous Gauss quadrature.
Applications in number theory: transcendence of e, irrationality of ζ(3).

ELEMENTS OF QUATERNIONIC ANALYSIS AND RADON TRANSFORM

- Jarolim Bures and Vladimir Soucek (Charles University, Czech Republic)

Abstract: The presented series of lectures offers a description of basic facts of quaternionic analysis. We shall describe properties of solutions of the Fueter equation and its third power. The second topic of the series will be the Radon transform, its inversion and its applications for study of quaternionic functions in quaternionic analysis.

Refernces:

Sudbery A., Quaternionic analysis, Math. Proc. Camb. Phil. Soc. 85, 1979, 199-225.
Gurlebeck K. and Sprossig W., Quaternionic analysis and elliptic boundary value Problems, Birkhäuser, Basel, 1990.
Helgason: Radon transform, Birkhauser.
Papers by Fueter, published in Com.Math.Helv. in 30's.

FUNCTION SPACES IN CLIFFORD ANALYSIS

- Uwe Käehler (Universidade de Aveiro, Portugal)

Contents:

Orthogonal and direct decomposition of Sobolev spaces.
Weighted Bergman-spaces in the unit disk.
Hardy-Littlewood inequalities.
Dirichlet, Bloch and BMO spaces.
Q_p-spaces.
Connections with wavelets analysis and the representation theory of semi-simple Lie groups.
Open problems and generalizations to hyperbolic settings.

Prerequisits (not necessary, but recommendable):

Course of J. Bures and V. Soucek.

Refernces:

F. Brackx, R. Delanghe, and F. Sommen, Cliford analysis, Research Notes in Mathematics, n. 76, Pitman, London, 1982.
R. Delanghe, F. Sommen, nd V. Soucek, Clifford analysis and spinor-valued functions: a function theory for the Dirac operator, Math. and its Appl. 53, Kluwer Acad. Publ., Dordrecht, 1990.
K. Gurlebeck, U. Kahler, M.V. Shapiro and L.M. Tovar, On Q_p-spaces of quaternion-valued functions, Complex Variables 39 (1999), 115-135.
K. Gurlebeck and Malonek, A hypercomplex derivative of monogenic functions in Rn, and its applications, Complex Variables 39 (1999), 199-228.
K. Gurlebeck and W. Sprossig, Quaternionic and Clifford calculus for Engineers and Physicists, John Wiley and Sons, Cinchester, 1997.
J. Cnops and R. Delanghe, Moebius invariant spaces in the unit ball, Appl. Anal., 73 (1999), n. 1-2,45-64.

THE PRIME NUMBER THEOREM

- Fernando Chamizo (Universidad Autonoma de Madrid, Spain)

Abstract: This course aims to give a proof of the Prime Number Theorem with error term emphasizing the use of Complex Analysis. We shall also discuss some other results and problems in Number Theory related to Complex Analytic thechniques. In particular the original formulation to the so called "circle method"due to Hardy and Littlewood.

Probably lectures notes will be available to the participants at the beginning of the course.

References:

H. Davenport, "Multiplicative number theory", 2nd ed., revised by Hugh L. Montgomery, Graduate texts in mathematics 74, Springer Verlag, 1980.
H.M. Edwards, "Riemann's zeta function", Academic Press, 1974.
A. Ivic, "The Riemann zeta-function: the theory of the Riemann zeta-function with applications", New York, wiley, 1985.
E.C. Tichmarsh, "The theory of the Riemann zeta-function", 2nd ed. (revised by D.R. Heath-Brown), Oxford, Clarendon, 1986.
R.C. Vaughan, "The Hardy-Littlewood method", Cambridge tracts in mathematics 80, Cambridge University Press, 1981.

• • Student presentations

19 March: 10-12h

History

This intensive course follows the seven held in Coimbra and Aveiro from 1995 to 2002 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.

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)

Coming Events

6th Workshop on Applications and Generalizations of Complex Analysis

Financial Support

Living expenses can be partially covered for some students if they do not have support from their own institution and if there is enough money available.

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

Back to home page