The CAS program offered the participants an exceptional opportunity to concentrate full time on hard problems and to open up new directions of research. Collaboration with a number of foreign institutions has been strengthened, and new partnerships have been developed for ongoing and future joint work.

The progress achieved by the program can roughly be divided into three categories:

• Completion and advancement of projects already initiated and commenced before the program at CAS. A distinguished example of this kind of work is Lacey’s solution to the problem of two weight inequalities for the Hilbert transform. It has been open for three decades and was finally settled during the CAS program. Its solution opens up new perspectives in operator related function theory and is directly linked to fundamental problems about Toeplitz operators and model subspaces. Another example is Bondarenko’s remarkable solution to a question posed by David Larman in the 1970s about Borsuk’s conjecture for two-distance sets. The program provided a stimulating atmosphere to reach these remarkable results and many others documented in the list of publications.
• Solution to problems that surfaced during the program. Examples of such kind of projects are the work of Queffélec and Seip on approximation numbers of composition operators, the sharp estimates of Aistleitner, Berkes, and Seip for certain GCD sums, leading to a Carleson-Hunt inequality for systems of dilated functions of bounded variation, the work of Aleman, Lyubarskii, Malinnikova, and Perfekt, which contains a full description of Schatten classes for composition operators on model spaces, and new results of Gröchenig and Lyubarskii on multidimensional Gabor frames. The list of publications reflects a considerable interaction between the many researchers that took part in the CAS program.
• Initial steps in the development of future projects. This output of the CAS program is not documented in the list of publications, but is nevertheless an exceptional and invaluable long-term effect of the CAS program. While we cannot speak for all the participants in the program, we have recorded the following list of emerging collaborative projects for the core team of NTNU researchers: Seip is involved in projects with Berkes, Bondarenko, Hilberdink, Montes, Queffélec, Sehba, and Weber; Lyubarskii and Malinnikova are involved in projects with Aleman, Luef, Perfekt, Putinar, Dörfler, Mozolyako, and Nicolau. These new projects cross many traditional borders between mathematical subfields and intertwine subjects such as probability theory, analytic number theory, operator theory, complex analysis, potential theory, harmonic analysis, and time-frequency analysis. The broad interaction and stimulating atmosphere within the CAS program have been crucial for opening up these new research directions.

The impetus of the CAS program can be expected to be extensive for many years to come; it will be essential for the success of future projects, such as the remaining tasks of two ongoing projects supported by the Research Council of Norway: “Discrete Models in Mathematical Analysis” (2012 – 2015), a FRIPRO project headed by Yurii Lyubarskii and “Dirichlet Series and Analysis on Ploydiscs” (2013 – 2018), a FRIPRO project realizing an ERC Advanced Grant application, headed by Kristian Seip.