Frontiers of operator theory – FRONT2017

Over the past decade, the interplay between operator theory (a classical subject of functional analysis) and other fields
of pure and applied mathematics has substantially increased. This interplay is interesting in both directions (from operator theory to other fields, and conversely). Let us give a couple of examples
about these interconnections, in which several members of our project have significant contributions:

1. in the study of Banach spaces of Dirichlet series, tools coming from analytic number theory are often needed to obtain fine properties of functions in these spaces; conversely, functional analytic methods
developed for the study of these spaces provide new results on classical problems in analytic number theory, like the estimation of the means of Dirichlet polynomials.
2. in ergodic theory, linear dynamics provides new, rich and sufficiently easy to handle classes of examples
which can be used to exhibit properties which were not reachable otherwise. Conversely, one may use some transfer principles to get results in the linear context knowing the existence
of their nonlinear counterparts.
3. in some chapters of geometric group theory, methods of operator theory have proved to be powerful to study famous questions.

Our main objective in this project is to undertake a general study of this interplay in the three fields of expertise (apart from operator theory)
of our members: spaces of analytic functions, ergodic theory and harmonic analysis. The questions we plan to consider are interesting for a wide community of mathematicians.
The tools we expect to construct to study these questions will be of interest to researchers working outside the field of functional analysis.

To carry out this program, we plan to organize during each of the first three years of the project a workshop and a spring school.
We will ensure the dissemination of our results by organizing an international conference at the end of the project and by giving communications in seminars and conferences throughout the project.
A particular attention will be paid to the participation of young researchers (Phd students and post-docs).
Members of the project intend to teach several graduate courses in the next four years. A website of the project will be created.