Boundary Theory

边界的概念存在于数学的许多领域。在有关分析中，单位圆盘的经典泊松公式（以及狄利克雷问题的可解性）可以说提供了最有指导意义的例子。它建立了圆盘内部的谐波函数与边界圆盘上的函数之间的一一对应关系。由于边界圆正是单位圆盘在周围欧几里得平面上的拓扑边界，这种对应关系可能看起来不那么引人注目。然而，如果考虑到开盘上的共形不变性调和函数与双曲平面上的调和函数是相同的，而双曲平面上的调和函数最初没有任何边界，那么它就变得不那么平凡了。****这样，我们就遇到了一个问题，即为先天“无边界”的空间分配边界。从历史上看，这个问题首先是在拓扑设置中根据相关的紧化或更一般地说是边界化来研究的。紧化的构造有很多，从最小的单点（Alexandrov）紧化到Stone-Cech紧化（某种意义上是最大的），通过在原始空间（Busemann, Martin, constantinesu - cornea, Thurston, Furstenberg， Freudenthal等）上附加结构定义的各种中间结构。所有这些结构的共同特征是它们都在拓扑范畴中定义，并且产生的边界是拓扑空间。****然而，同样的问题也可以在其他类别中提出和回答。特别地，在测度范畴中，我们可以将任意合理马尔可夫链的泊松边界定义为由描述链在无穷远处的非平凡行为的次西格玛代数所决定的路径空间的商。这种结构基本上可以追溯到50年代的Feller和Blackwell，尽管它的现代表述是在很久以后，作为Dynkin， Furstenberg, Zimmer， Vershik和PI的一系列作品的结果。***本研究计划的目的是进一步研究和解决一些问题，旨在更好地理解拓扑和度量范畴中赋予附加代数或几何结构的空间的各种边界，这也应该导致更好地理解底层空间本身。这些问题也与几何、李论、几何群论、泛函分析和概率论的各个方面密切相关，我们期望基于边界考虑的统一观点能带来新的见解和方法。**

The idea of boundary is present in many areas of mathematics. In what concerns analysis, the classical Poisson formula (along with the solvability of the Dirichlet problem) for the unit disk arguably provides the most instructive example. It establishes a one-to-one correspondence between harmonic functions on the interior of the disk and functions on the boundary disk. Since the boundary circle is precisely the topological boundary of the unit disk in the ambient Euclidean plane, this correspondence may look not so striking. However, it becomes less trivial if one takes into account that by conformal invariance harmonic functions on the open disk are the same as harmonic functions on the hyperbolic plane, whereas the latter space originally does not come equipped with any boundary.****Thus, one arrives at the problem of assigning boundaries to spaces which are a priori ``borderless''. Historically, this problem was first studied in the topological setup in terms of associated compactifications or, more generally, bordifications. There are numerous constructions of compactifications ranging from the smallest one-point (Alexandrov) compactification to the Stone-Cech one (in a sense the biggest one) passing through various intermediate ones defined in terms of additional structures on the original space (Busemann, Martin, Constantinescu-Cornea, Thurston, Furstenberg, Freudenthal, etc.). The common feature of all these constructions is that all of them are defined in the topological category, and the arising boundaries are topological spaces.****However, the same questions can be asked and answered in other categories as well. In particular, in the measure category one can define the Poisson boundary of any reasonable Markov chain as the quotient of the path space determined by the sub-sigma-algebra which describes non-trivial behavior of the chain at infinity. This construction essentially goes back to Feller and Blackwell in the 50s, although its modern formulation was given much later as a result of a series of works by Dynkin, Furstenberg, Zimmer, Vershik and the PI.***The purpose of this research proposal is to further investigate and solve a number of problems aimed at a better understanding of various boundaries of spaces endowed with additional algebraic or geometric structures both in the topological and in the measure categories, which should also lead to a better understanding of the underlying spaces themselves. These problems are also closely related to various aspects of geometry, Lie theory, geometric group theory, functional analysis and probability theory, where we expect the unified point of view based on boundary considerations to bring new insights and approaches. **