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Grantor: European Commission

**Programme type**: FP7 Marie Curie Action "International Research Staff Exchange Scheme"

**Objectives**:

The theory of asymptotic behaviour of operator semigroups is a comparatively new field serving as a common denominator for many other areas of mathematics, such as for instance the theory of partial differential equations, complex analysis, harmonic analysis and topology.

The primary interest in the study of asymptotic properties of strongly continuous operator semi-groups comes from the fact that such semigroups solve abstract Cauchy problems which are often models for various phenomena arising in natural sciences, engineering and economics.

Knowledge of the asymptotics of semigroups allows one to determine the character of long-time evolution of these phenomena.

Despite an obvious importance, the asymptotic theory of one-parameter strongly continuous operator semigroups was for a very long time a collection of scattered facts rather than an organized area of research. The interest increased in the 1980s and the theory has witnessed a dramatic development over the past thirty years. Still there is a number of notorious open problems that have been left open. These missing blocks prevent the theory from being complete, slow down the development of the theory and discourage specialists from related fields to engage into the theory.

The goal of the project is to give new impetus to the theory of asymptotic behaviour of operator semigroups. To this aim we plan to extend and unify various aspects of the asymptotic theory of operator semigroups: stability, hyperbolicity, rigidity, boundedness, relations to Fredholm property, to work out new methods and to solve several long-standing open problems thus giving the theory its final shape.

We intend to create an international forum that enables and promotes a multi- and cross- disciplinary exchange of ideas, methods and tools under the common umbrella of asymptotic theory of operator semigroups. Thus we expect that, moreover, a wide range of modern analysis will benefit from the project.

IM leader :

Main investigator:

**Tomilov** Yuri

Participating institutions:

Institute of Mathematics of the Polish Academy of Sciences, Poland (**Coordinator**)

The Chancellor, Masters and Scholars of the University of Oxford, Great Britain

University of Ulm, Germany

Karlsruhe Institute of Technology, Germany

Universite Lille, France

Technical University of Dresden, Germany

Nicolaus Copernicus University, Poland

Institute of Mathematics, AS CR

Aix-Marseille University, France

Tel-Aviv University, Israel

University of Missouri, Columbia, USA

Northwestern University, USA

William and Mary College, USA

University of North Carolina, USA

Kent State University, USA

University of New South Wales, Australia

University of Auckland, New Zealand