Boundary Theory

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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.