Coarse geometry and its application to the study of function spaces and differentiations

Coarse geometry was formed in the works of Mikhail Gromov, Gaoliang Yu, John Roe and other researchers relatively recently. The main motivation was to solve Novikov's conjecture and related questions, in particular the question of embeddability in Banach space. At the same time, coarse geometry itself, according to the speaker, has outgrown these (important) questions and is quite suitable for studying other questions as well.
In the talk I will start with basic definitions and try to give some geometric intuition. We will see which spaces are roughly equivalent (e.g. real numbers and integers) and which are not. So different equivalence classes will have: different euclidean spaces, corner (on the plane), binary tree. Banach spaces are also a natural example. So two different norms define roughly equivalent spaces only if they are equivalent.
Within this talk we will of course also discuss some coarse invariants (growth, asymptotic dimension, number of ends).
At the end of the talk we will focus on what is the main interest of the speaker. We will discuss the possibility of introducing lattices in space, considering functions on them (it is appropriate to say ‘potential functions’) and then we will see how they can be used to study derivations in group algebras. And it turns out that not only ‘ordinary’ derivations can be studied in this way, but also others: $(\sigma,\tau)$-derivations, Fox derivatives and so on.
Moreover, this coarse approach allows us to compare differentiations on different group algebras.