I algebraic geometry

Relation to the state-of-the-art:

Ideas from negative and nonpositive curvature were introduced into group theory first by Margulis and Gromov in late 1970 and 1980' and then developed by many researchers both in Europe and US. The developments covered very diverse topics, but were packaged as "Geometric group theory".

In the Objectives 1 and 3 the team is on the leading edge of research worldwide. Januszkiewicz was invited speaker in Section Geometry at ICM 2010. Old paper of Januszkiewicz with Davis [HoP] and recent papers of Swiatkowski provide the richest source of examples. Objective 2 is a vast and diffuse field, and the "edge" difficult to define. In some aspects, for example L2-cohomology computations for special groups Dymara and Januszkiewicz with co-authors [DDJO] provide best current results. Objective 4 has been little studied in the literature. Perhaps the best results are due to Dimitri Panov, Kings' College London (former student of M. Kontsevich, IHES), with whom the team members are in close contact. Objective 5 would require significant transfer of knowledge to catch up. The team is in contact with V. Chepoi (Marseille, a leading expert on metric graph theory) and with M. Farber (Durham, a leading expert in topological robotics). E. Gutkin and M. Wojtkowski, attached to the dynamical systems team of this project, DS, are both experts on billiards, and we hope to be able to use their expertise and their contacts with other experts.