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

指数定律使函数空间中有值的函数可以解释为两个变量的普通函数，从而使它们更容易处理。它们是无穷维微分学中的核心工具，例如，用于建立无穷维李群的突出例子的群运算的平滑性。通常，指数律也是证明这类群的正则性的关键，即g上相关微分方程解的存在性和光滑参数依赖性。该项目的目标之一是提供新的指数律。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，左不变向量场的积分曲线与（仅仅）可测量的时间相关是特别有用的；例如，单参数群的Trotter积公式和换易子公式（否则难以证明）在可测正则李群中自动成立（在可测正则李群中，李群值向可测李代数值曲线的演化具有存在性和光滑参数依赖性）。利用适当的指数律或替代策略，将为进一步重要的无限维李群类建立可测量的正则性。

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.