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

指数定律使函数在函数空间中的值可以被解释为普通的两个变量的函数，从而使它们更容易处理。它们是无穷维微分学的核心工具，例如，用于建立无穷维李群的著名例子的群运算的光滑性。通常，指数定律也是证明此类群体规律性的关键，即，G上相关微分方程解的存在性和光滑参数依赖性。该项目的一个目标是提供新的指数定律。主要目标是进一步发展正则无限维李群理论，特别是可测正则性理论。最近的研究表明，积分曲线的左不变向量场与（仅）可测量的依赖于时间是特别有用的;例如，在一个实施例中，单参数群的Trotter乘积公式和换位子公式（否则很难证明）在可测正则李群中自动成立（其中李群值演化到可测李代数值曲线的存在性和光滑参数依赖性是可用的）。使用合适的指数律或替代策略，可以为更重要的无穷维李群类建立可测正则性。

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.