13th
European Intensive Course on Complex Analysis and applications to partial differential equations 

Departamento de Matemática,
Universidade de Coimbra, Portugal
Departamento de Matemática,
Universidade de Aveiro, Portugal


Goal of the Course  
This intensive course follows the twelve held at the Universities of Coimbra and Aveiro from 1995 to 2006 (1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006) 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. 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 Socrates/Erasmus Intensive Program of Higher Education, and is opened to all young mathematicians interested in Complex Analysis and its applications. There will be a Workshop on "Applications and Generalizations of Complex Analysis" on the 23rd of June 2007. 







Title: ORTHOGONAL POLYNOMIALS OF SEVERAL VARIABLES  
Summary: While orthogonal polynomials in one variable already have numerous and varied applications in many fields of science, the theory of orthogonal polynomials in two and more variables is applied insufficiently widely. The study of orthogonal polynomials of several variables goes back to C. Hermite ( 1865). Later on, books of P.Appell (1881), P. Appell & K. de Feriet (1926) and papers of D. Jackson (1938) and T. Koornwinder (1975) introduced a considerable number of results on this theory. With the intention of making this seminar useful to a wide audience, we shall introduce standard matrix notation in order to present general properties of orthogonal polynomials of several variables and specially of two variables over a domain with arbitrary weight. Following mainly the monographs of P.K. Suetin (1988) and Ch.F. Dunkl & Y. Xu (2001), a systematic exposition and detailed discussion of many important results, examples and applications on the theory of orthogonal polynomials in two (continuous and discrete) variables is given.  




Title: BASIC CLIFFORD ALGEBRA AND CLIFFORD ANALYSIS  
Summary: Abstract and practical definitions of Clifford Algebras Cl_{p,q,r}=R_{p,q,r}. Rotation in a euclidean plane, in a 3dimensional euclidean space: C=R_{2,0,0}^+ and H=R_{3,0,0}^+ and in 4dimensional spaces. The Dirac equation and spinors. Monogenic, hypermonogenic, Clifford holomorphic and Clifford analytical functions in R_{0,n,0}. Translation of Clifford Analysis from R_{0,n,0} to R_{n+1,0,0}.  


Title: CONJUGATE HARMONICITY IN EUCLIDEAN SPACE  
Summary:
1. The real Clifford algebra R0,m+1 2. Monogenic functions versus selfconjugate differential forms 3. Conjugate harmonicity, harmonic primitives and monogenic primitives of mono genic functions. 4. Monogenic rforms versus harmonic rforms 5. Bases for the space of monogenic homogeneous vector(or para)vector valued polynomials 6. Cauchy transforms and conjugate harmonicitymore 



Title: LHOLOMORPHIC FUNCTIONS IN ELASTICITY FLUID DYNAMICS  
Summary:
1. Lecture: About the notion of holomorphy 2. Lecture: Quaternionic operator calculus  Hodge type decompositions 3. Lecture: Fluid flow problems 4. Lecture: problems in Elasticity 



Title: CLIFFORD ANALYSIS METHODS FOR TIME EVOLUTION EQUATIONS  
Summary:
1. Heat equation and related topics 2. Scattering 

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. 

REGISTRATION FORM (please copy into email)


Rafael Hernández Heredero, Universidad Politécnica de Madrid  
Odete Ribeiro, Departamento de Matemática, Instituto Politécnico de Viseu  
Márcio Sacramento, Departamento de Matemática, Instituto Politécnico de Viseu  
Ulises Fidalgo Prieto, Universidade de Aveiro  
Luis Garza, Universidad Carlos III de Madrid  
Norman Gürlebeck, Friedrich Schiller UniversityJena, Germany  
Luís Cotrim , Instituto Politécnico de Leiria  
Anabela Monteiro Paiva, Universidade da Beira Interior  
Maria da Neves Vieiro Rebocho, Univ. Beira Interior/Univ. Coimbra  
Ana Isabel Gonçalves Mendes, Instituto Politécnico de Leiria  
Herbert Dueñas Ruiz, Universidad Carlos III de Madrid  
Judit Mínguez Ceniceros, Universidad de La Rioja  
Sven Ebert, University of Freiberg  
Frank Dierich, University of Freiberg  
Matti Schneider, University of Freiberg  
André Schlichting, University of Freiberg  
Juliane Mueller , University of Freiberg  
  


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