Title: Symmetric fractals and their combinatorial Loewner property
Abstract:
Many self-similar spaces have very complex geometric structure which causes challenges for developing basic tools from calculus, such as a gradient-type operator. Moreover, in general, the Euclidean metric is not an optimal way of describing the geometry of such spaces.
Graph approximations and their combinatorial modulus provide combinatorial tools for describing the geometry of a self-similar space. In case the space admits an “optimal” metric, one can study this metric through combinatorial modulus. Moreover, if the space has enough reflectional symmetries, the space admits rich structure in non-linear potential theory through Poincaré inequality.
In this talk I will introduce basic concepts and results regarding combinatorial modulus of a self-similar space and some consequences of the attainment of an optimal metric.
