Title. Higher dimensional picture languages

Abstract. My main goal is to develop a theory of combinatorics of higher dimensional words, or higher dimensional picture languages. These languages are closely related to the theory of C*-algebras and can be viewed as natural generalizations of free monoids, whose elements can naturally be regarded as higher-dimensional strings. We will construct concrete models in any dimension using arithmetic geometry and theory of buildings. The work we are going to present fits very well to the picture language program suggested by A.Jaffe and Z.Liu, for example, to the question L(iii)(ii). [Mathematical picture language program, PNAS 115, no. 1 (2018), pp. 81-86]