On request of the some participants each of the following three meetings (28.11, 5.12, 12.12) will consist of two sessions of about 90 minutes each with a break about 15 minutes in between (from 3pm to a bit after 6pm). The second session will be focused on more examples and details of proofs. Attending only first parts should be possible without loosing continuity too much. Before Christmas we should decide how to continue next year.

In this series of talks we plan to discuss the revelation of combinatorial set-theoretic phenomena in the noncommutative mathematics. This will include introducing and motivating and checking the details of the very basics of this theory (we assume no knowledge of operator algebras or C*-algebras from the participants). We will attempt to focus on concrete combinatorial issues (and not on general algebraic theory) like the passage from ultrafilters to quantum filters, from Boolean algebras and nonmetrizable compact totally disconnected set-theoretic topology to real rank zero and AF nonseparable C*-algebras or the impact of additional set-theoretic axioms.

Like in Boolean algebras, general topology and measure theory in the classical period one could see in C*-algebras (aka operator algebras) one of the modern central notions where the set-theoretic methods can make difference. The following two texts are characteristic of such an approach and will be exploited during the talks:

- I. Farah, E. Wofsey,
*Set theory and operator algebras*. In: Cummings, James (ed.) et al., Appalachian set theory 2006--2012. London Mathematical Society Lecture Note Series 406, 63--119 (2013). - I. Farah,
*Combinatorial set theory of C*-algebras*. Springer Monographs in Mathematics, 2019.Have a look at the webpage of the seminar.