8 th European Intensive Course on Complex Analysis

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

Departamento de Matemática , Universidade de Coimbra , Portugal

11 -22 March 2002

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

 Photoalbum

#### 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 16th of March.

#### Schedule of the course

 11 March 12 March 13 March 14 March 15 March Opening session 9h30m-10h * * * * 15h -17h30m 10h-12h30m 14h30m -17h * 10h-12h30m * López Lagomasino 10h-12h30m * 10h-12h30m 14h30m - 17h 10h -12h30m
 18 March 19 March 20 March 21 March 22 March 10h-12h30m 10h-12h30m Student Presentation 10h-12h30m 10h-12h30m 14h30m -17h 14h30m -17h * 14h30m -17h 14h30m -17h

#### Abstracts

APPROXIMATION THEORY AND WEIGHTED SOBOLEV SPACES
- Francisco Marcellán (Univ. Carlos III, Madrid, Spain)

Contents:

1.- Basic results concerning weights.
2.- Weighted Sobolev spaces. Isoperimetric inequalities. Sobolev type inequalities.
3.- Potentials and capacities. Meyer's Theory for Lp capacities. Some examples: Bessel,  Riesz and Haussdorf capacities.
4.- Applications of Potential Theory to Sobolev spaces. Poincaré type inequalities.
5.- Polynomial Approximation in Sobolev spaces.

Basic references that will be distributed to the participants in the course:
[1].- B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics 1736. Springer Verlag. 2000.
[2].-  C. Bernardi, Y. Maday, Approximations spectrales de problèmes aux limites elliptiques. Springer Verlag. 1992.

GEOMETRIC THEORY OF FUNCTIONS OF A COMPLEX VARIABLE
- Guillermo López Lagomasino (Univ. Carlos III, Madrid, Spain)

Abstract: The course aims to cover basic questions related with the theory of univalent conformal mappings of simply connected and multiply connected domains, conformal mapping of multiply connected domains onto a disk (generalization of Riemann's Mapping Theorem), applications of conformal mappings  to the study of interior and boundary properties of analytic functions, and general questions of geometric nature dealing with analytic functions.

Basic references that will be distributed to the participants in the course:
[1].- G.M. Goluzin, Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc. Transl. Monographs, vol. XXVI.
[2].- J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Plane, Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., Providence R. I., 1935; rev. ed., 1965.

SPECIAL TOPICS IN COMPLEX ANALYSIS: UNIVERSAL FUNCTIONS AND BANACH ALGEBRAS OF ANALYTIC FUNCTIONS
- Richard Aron (Kent State University, USA)

Abstract: We plan to give five introductory, expository lectures on each of the topics: (1) Universal Entire Functions, (2) Banach algebras of analytic functions.

(1). Universal entire functions ~ $f$ is an analytic function on $\Cl$ such that some iterated operation on it produces a dense set. For instance, we will show the following:
(a). There is a function $f \in {\mathcal H}(\Cl)$ such that given any function $g \in {\mathcal H}(\Cl),$ any $R > 0,$ and any $\epsilon > 0,$ there is some $n \in \N$ such that $\max_{|z| \leq R} |f^{(n)}(z) - g(z)| < \epsilon.$  (Here, the iterated operation is differentiation.)
(b). There is a function $f \in {\mathcal H}(\Cl)$ such that given any function $g \in {\mathcal H}(\Cl),$ any $R > 0,$ and any $\epsilon > 0,$ there is some $n \in \N$ such that $\max_{|z| \leq R} |f(z+n) - g(z)| < \epsilon.$ (Here, the iterated operation is translation.)
Both of these results seem, to me at least, to be quite surprising. Even more surprising, then, are the following:
(c). There is a function $f \in {\mathcal H}(\Cl)$ which works' in both (a) and (b) above!
(d). There is an infinite dimensional subspace $X \subset {\mathcal H}(\Cl)$ such that for any $f \in X, f \not\equiv 0, f$ works' in both (a) and (b) above!
Such functions are part of a broader area known as  hypercyclicity , which is related to the famous Invariant Subspace Problem.
(2). Banach algebras of analytic functions ~ We will introduce the concept of Banach algebra
${\mathcal A}$, specializing to the case when ${\mathcal A}$ is an algebra of analytic functions. Associated to any Banach algebra ${\mathcal A}$ is a compact set known as the {\em maximal ideal space} ${\mathcal M(A)},$ which consists of the homomorphisms $\phi:{\mathcal M(A)} \to \Cl.$ Knowledge of ${\mathcal M(A)}$ enables us to learn a lot about ${\mathcal A}$ itself.
The two most useful examples of Banach algebras for us will be ${\mathcal A} = {\mathcal H}^\infty(D),$
the algebra of bounded analytic functions on the open unit disc $D,$ and ${\mathcal A} = A(D),$  the algebra of analytic functions on $D$ which can be continuously extended to $\overline{D}.$ We will study ${\mathcal M(A)}$ for both these algebras. We will also look at homomorphisms between such algebras.

INTRODUCTION TO INFINITE DIMENSIONAL HOLOMORPHY
- Chris Boyd (Univ. College, Dublin, Ireland)

Abstract: Given Banach spaces $E$ and $F$ we let ${\cal P}(^nE;F)$ denote the vector space of all $n$--homogeneous polynomials from $E$ into $F$. This vector space will become a Banach space when endowed with the norm $\|P\|:=\sup_{x\in B_E}\|P(x)\|$. A classical (linear) Banach space result is that $B(\ell_2)$, the space of all linear operators from $\ell_2$ into $\ell_2$, is the bidual of $K(\ell_2)$, the space of all compact operator from $\ell_2$ into itself. The polynomial analogue of $K(\ell_2)$ is ${\cal P}_w(^nE;F)$, the space of all polynomials from $E$ into $F$ which are weakly continuous on bounded sets. In the course we will investigate necessary and sufficient conditions for the bidual of ${\cal P}_w(^nE;F)$ to be isomorphic to ${\cal P}(^nE'',F'')$. Among the concepts we will meet will be the approximation property, Asplund Banach spaces, the Radon-Nikod\'ym property and symmetric tensor products of Banach spaces, integral polynomials and nuclear polynomials. We will also examine conditions on when the above isomorphism becomes an isometry. This leads us to investigate extreme and exposed points on the unit ball of
Banach spaces. As a result of our investigation we will be able to give necessary and sufficient conditions for the space ${\cal P}(^nE;F)$ to be reflexive. We will also relate the topic discussed to the concept of Q-reflexive Banach space a concept which has been the centre of much research over the last ten years.

• • Student presentations

20 March: 10-12h

#### History

This intensive course follows the seven held in Coimbra and Aveiro from 1995 to 2001 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.

#### 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.