Groups definable in tame expansions of o-minimal structures and related problems.

• Applicant: Dr. Panteleimon Eleftheriou
• Subject Area: Mathematics
• Term: from 2014 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 259198724

This project concerns work in and around o-minimality, focusing on the study of definable groups. O-minimal structures provide a rigid framework to study real algebraic geometry, which is one of the main research areas in the Department of Mathematics at the University of Konstanz. On the other hand, groups definable in o-minimal structures have been a core subject in model theory during the last decades. Examples of such groups include all compact real Lie groups. In this project we aim to extend recently developed techniques from the study of groups definable in o-minimal structures to the study of groups definable in more general, yet tame, settings. A concrete instance of such a setting is (R, 2^Q), the expansion of the real field by the multiplicative subgroup of the reals consisting of all rational powers of 2. It is known that this structure has nice model theoretic properties, but no groups definable in it have previously been studied. The ultimate goal in this setting would be a structure theorem that every such group is a quotient of a product H x K by a lattice, where H is a group locally definable in R, and K is a group which is 2^Q-internal.The proposed approach involves a uniform program for analyzing definable groups and sets in different model-theoretic settings. I expect that a successful outcome of this project will significantly advance the state-of-the-art of the proposed areas and largely influence the development of new scientific methods therein.This project also aims to answer intriguing related problems, such as lattices in locally definable groups, semilinear geometry, and the Zilber Dichotomy in the o-minimal setting.

• DFG Programme: Research Grants
• International Connection: Israel, USA
• Participating Persons: Professor Dr. Assaf Hasson; Dr. Philipp C.K. Hieronymi; Professor Dr. Yaacov Peterzil