Regularity properties of infinite-dimensional Lie groups, and exponential laws

• Applicant: Professor Dr. Helge Glöckner
• Subject Area: Mathematics
• Term: from 2017 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 384439538

Exponential laws enable functions with values in a function space to be interpreted as ordinary functions of two variables, and thus make them easier to handle. They are central tools in infinite-dimensional differential calculus and used, for example, to establish smoothness of the group operations for prominent examples of infinite-dimensional Lie groups. Frequently, exponential laws are also the key for the proof of regularity of such groups, i.e., the existence and smooth parameter-dependence of solutions to relevant differential equations on G. One goal of the project is to provide new exponential laws. The main goal is to develop further the theory of regular infinite-dimensional Lie groups, notably the theory of measurable regularity. Recent research showed that integral curves for left-invariant vector fields with (merely) measurable dependence on time are of particular usefulness; for example, the Trotter product formula and the commutator formula for one-parameter groups (which are otherwise difficult to prove) automatically hold in measurably regular Lie groups (in which existence and smooth parameter-dependence is available for the Lie group-valued evolutions to measurable Lie algebra-valued curves). Using suitable exponential laws or alternative strategies, measurable regularity shall be established for further important classes of infinite-dimensional Lie groups.

• DFG Programme: Research Grants