The Mathematisches Forschungsinstitut Oberwolfach regularly organizes workshops in all fields of Mathematics. Their aim is to offer 45–48 experts, invited by the Institute's Director, the opportunity to present recent research results, especially new methods, and to initiate future research projects.
Mathematical research mainly aims at the study of the structure and inner correlations of mathematical objects and at the development of more comprehensive theories. Many mathematical questions are consequences of the effort to describe nature in mathematical terms, but it can also happen that the mathematical frame was created before the appearance of the applications. The process of research leads to mathematical theorems, whose proofs are typically complicated. The final write-up of a proof can best be done at the home institutes, but the development of a mathematical theory and, within such a theory, of a promising idea for a proof, is an extremely creative process depending very much on intuition and experience.
Naturally, coincidence also plays a big role. Improving the chances for progress by coincidence is one of the main purposes of the research at the Oberwolfach Institute. When getting to know the background of an important result during a talk, one can suddenly have a bright idea, perhaps leading to a considerable progress in one's own research activity. Within discussions in small groups, when presenting fresh thoughts, one sometimes can find the right direction for further work with the help of the comments of colleagues. Often it also happens that two or three colleagues, during discussions, become aware that they, though coming from different backgrounds and with different motivations, are interested in similar problems and decide to unify their potential in order to establish a common research project.
All this happens nearly daily at these workshops. A great number of important papers have been initiated at Oberwolfach in this manner. Contrary to the typically large conferences all over the world, workshops at Oberwolfach emphasize active research.
The Oberwolfach Reports are meant to capture, in an informal manner, the characteristic ideas and discussions of these workshops. As a service to the community, they are now offered by the Institute, at a nominal price, and allow the public to partake in the lively and stimulating atmosphere of these meetings. While the peer-rewiewed results will appear elsewhere, the Oberwolfach Reports will keep the reader abreast of current developments and open problems, and serve as an indispensable source of information for the active mathematician.
The following topics about subgroups of the Cremona groups are discussed: (1) maximal tori; (2) conjugacy and classification of diagonalizable subgroups of codimensions 0 and 1; (3) conjugacy of finite abelian subgroups; (4) algebraicity of normalizers of diagonalizable subgroups; (5) torsion primes.