Boundary Theory

边界的概念存在于数学的许多领域。在分析中，经典的泊松公式（沿着狄利克雷问题的可解性）为单位圆盘提供了最有启发性的例子。它建立了一个一对一的对应关系，调和函数的内部的磁盘和功能的边界磁盘。由于边界圆恰好是周围欧氏平面中单位圆盘的拓扑边界，所以这种对应关系看起来可能不那么引人注目。然而，如果考虑到通过共形不变性，开圆盘上的调和函数与双曲平面上的调和函数相同，而后者空间最初并不配备任何边界，那么它就变得不那么平凡了。这样，我们就遇到了为先验的“无边界”空间分配边界的问题。历史上，这个问题首先是在拓扑结构中研究相关的紧化，或者更一般地说，边界化。有许多紧化的构造，从最小的一点紧化（亚历山德罗夫）到斯通-切赫紧化（在某种意义上是最大的一个），通过各种中间紧化，这些中间紧化是根据原始空间上的附加结构定义的（Busemann，Martin，Constantinescu-Cornea，Thurston，Furstenberg，Freudenthal等）。所有这些构造的共同特征是它们都定义在拓扑范畴中，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，可以将任何合理的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子西格玛代数确定的路径空间的商。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由Dynkin，Furstenberg，Zimmer，Vershik和PI的一系列作品给出的。本研究提案的目的是进一步调查和解决一些问题，旨在更好地理解在拓扑和测度范畴中赋予额外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、李理论、几何群论、泛函分析和概率论的各个方面密切相关，我们期待基于边界考虑的统一观点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. **