### Aktualne ogłoszenie

*Current announcement*

11 czerwca 2024 / June 11, 2024

** Piotr M. Hajac (IMPAN) **

*Noncommutative topology through operator algebras that one can see*

Abstract. This is an introductory talk that will start with unravelling the paradigm of noncommutative topology followed by a step-by-step construction of graph C*-algebras abundantly instantiated. Due to their tangible combinatorial nature, these C*-algebras are known as "operator algebras that one can see". The main part of the talk will be devoted to a conceptual explanation (technicalities omitted, motivating examples provided) of how we solved the mixed-functoriality problem of pullbacks of graph C*-algebras. More precisely, while some pullbacks can be understood as coming from pushouts of directed graphs via a contravariant functor assigning C*-algebras to graphs, other pullbacks require both covariant and contravariant functors assigning C*-algebras to graphs considered as objects in two different categories. In the latter case, one cannot explain the pullback structure of a graph C*-algebra in terms of a commuting diagram of underlying graphs in one category of graphs. We solve this problem by introducing a new category of directed graphs, where morphisms are relations rather than maps, and define a covariant functor from this category to the category of C*-algebras. Now, the new functor generalizes both the covariant and contravariant functors used before. Our main result is a general pullback theorem including both types of the aforementioned pullbacks as special cases. Time permitting, the talk will end by posing a tantalizing problem involving both pullbacks of noncommutative C*-algebras and the Poincaré conjecture. (Based on recent joint works with Mariusz Tobolski and Gilles Gonçalves de Castro.)