Geometry and Dynamics of Infinite-Dimensional Groups

• Applicant: Professor Dr. Friedrich Martin Schneider
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 561178190

Within the past three decades, topological groups of transformations of large homogeneous structures have attracted considerable attention and displayed numerous intriguing connections with operator algebra, geometry, ergodic theory, model theory, and descriptive set theory. A principal objective in the study of those infinite-dimensional groups lies in understanding relations between dynamical phenomena in a group and structural properties of the objects on which it acts. A seminal result in this direction is the celebrated Kechris-Pestov-Todorcevic correspondence unveiling an intimate link between Ramsey theory of model-theoretic structures and topological dynamics of their automorphism groups. Beyond such non-archimedean groups, there is an even wider variety of dynamical peculiarities, and the development of a useful theory requires a deep understanding of natural examples. The proposed project aims to make a strong contribution to this: our goal is the advancement of the theory of infinite-dimensional groups and their dynamics using new geometric, algebraic and functional-analytic methods. This research program is driven by the exploration of two concrete classes of topological groups, namely unit groups of von Neumann's continuous rings on the one hand, and topological groups of measurable functions along with their generalizations over submeasures on the other. We aim to shed light on intricate questions concerning these objects, interactions between them, and similarities and connections to other large groups.

• DFG Programme: Research Grants