Lorentz Center - Boundary relations from 14 Dec 2009 through 18 Dec 2009
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    Boundary relations
    from 14 Dec 2009 through 18 Dec 2009


Boundary relations, extension theory, and applications

Organizers: Seppo Hassi (Vaasa), Henk de Snoo (Groningen), and Franek Szafraniec (Krakow)

Boundary value problems are ususally associated with problems involving ordinary and partial linear differential operators. In order to describe such problems in a systematic way the theory of boundary triplets has been introduced by Derkach and Malamud, building on earlier work mainly from the Russian or Ukrainian schools. An attractive feature of this approach is that connected with the boundary triplet is a so-called Weyl (operator valued holomorphic) function which carries all the spectral data of the boundary value problem.  Recently the theory of boundary triplets has been extended by Derkach, Hassi, Malamud, and de Snoo to what is now called the theory of boundary relations: this theory offers a much greater flexibility in assigning boundary values allowing unbounded multivalued Weyl functions. Almost simultaneously new concepts have  been developed in the theory  of passive and conservative state/signal systems by Arov and Staffans which appear quite close to the the concepts of boundary relations. In this connection also the notion of Dirac structures (van der Schaft and Zwart) should be mentioned.

The aim of the workshop was to make explicit what these different theories have in common and to point the direction to future work to connect the various approaches. During the workshop there were main lectures by  Arlinskii, Derkach, and Malamud on boundary triplets and their generalizations and there were double main lectures by Grubb (on boundary value problems for partial differential operators) and by Staffans (on systems theory).  Furthermore there were lectures concerning extension theory, operator theory, and applications in analysis (Szafraniec).

There were 8 posters (including 2 posters by Hassi and by de Snoo);
a special poster session was organized  in which 6 PhD students could outline their posters. The schedule of the lectures provided a substantial opportunity for discussions and joint work. The 37 participants of the workshop, including 9 PhD students, came from Austria, Denmark,  Finland, France, Germany, Hungary, Israel, Netherlands, Poland, Romania, Russia, South Africa, Sweden,  Ukraine, and the U.K.  Visitors included Haase (Delft) and Kaashoek (Haarlem).
The organizers are discussing with a publisher about the proceedings of this conference. The main speakers have agreed to provide survey articles in their respective areas.

The workshop at the Lorentz Center was sponsored by several organisations: the Johann Bernoulli Institute for Mathematics and Computer Science of the University of Groningen, the Department of Mathematics and Statistics of the University of Vaasa, the Institute of Mathematics of the Polish Academy of Sciences (European Marie Curie project TODEQ), and the Dutch Institute of Systems and Control (DISC).  We thank all these institutions for their support.

The conference at the Lorentz Center with its unique facilities was much appreciated by the participants. We thank the Lorentz Center for providing us with this opportunity and their support; and also for helping us with getting the conference in shape. We also want to thank the staff of the Lorentz Center for their impressively smooth professionalism. In particular, we would like to mention the contributions of Auke Planjer and Pauline Vincenten.

December 19, 2009

Seppo Haasi (University of Vaasa),
Henk de Snoo (University of Groningen),
Franek Szafraniec (Uniwersytet Jagiellonski, Krakow).