FROM CLASSICAL TO QUANTUM SPACES

The key mathematical concept explored throughout the course is the concept of a C-algebra. This celebrated invention of Gelfand and Naimark combines algebra with functional analysis into a powerful tool to treat topological issues. The goal of these lectures is to guide one from understanding the topology of a compact Hausdorff space through its C-algebra of all complex-valued continuous functions to the brave new world of quantum geometry encoded by noncommutative C*-algebras of operators on a Hilbert space. This trading of topological point spaces as carriers of functions for Hilbert spaces as carries of operators much resembles and is much motivated by the passage from classical to quantum mechanics, where a point in a phase space is replaced by a wave function in a Hilbert space.

PROGRAMME:

1. The category of compact Hausdorff spaces (exercises).

2. Nets in compact spaces (exercises).

3. Compact Hausdorff groups (exercises).

4. Topological vector spaces and algebras (exercises).

5. Operator algebras (exercises).

6. The category of unital C*-algebras (exercises).

7. A classical vista of quantum spaces and groups.

The training aim of this course is to develop an ability to understand at least rudimentary aspects of the state-of-the-art mathematics. Exercising an effective abstract thinking in solving concrete problems is proposed as a teaching method. The only prerequisites are the working knowledge of mathematics at the first-year university level, and a clear desire to make first steps to understand current research in mathematics. Criteria to pass the course will be adapted to the level of comprehension achieved by students. To pass the course, it is necessary to attend most of the lectures. The final grade will be based on the presentation of solutions of homework assignments.

Bibliography:

1. Introduction To Commutative Algebra, Michael Atiyah, Ian G. MacDonald.

2. Topological algebras, V. K. Balachandran, North-Holland Mathematics Studies, 185. North-Holland Publishing Co., Amsterdam, 2000.

3. Wprowadzenie do topologii, Roman Duda, Biblioteka Matematyczna 61.

4. Wprowadzenie do matematyki współczesnej I, Piotr M. Hajac

5. Wprowadzenie do matematyki współczesnej II, Piotr M. Hajac

6. Fundamentals of the theory of operator algebras, Vol. I.: Elementary theory, Richard V. Kadison, John R. Ringrose, Reprint of the 1983 original, Graduate Studies in Mathematics, 15. American Mathematical Society, Providence, RI, 1997.

7. Basic Noncommutative Geometry, Masoud Khalkhali, EMS Series of Lectures in Mathematics.

8. Notes on Compact Quantum Groups, Ann Maes, Alfons Van Daele.

9. C*-algebras and Operator Theory, Gerard J. Murphy.

10. Functional Analysis, Michael Reed, Barry Simon, Methods of Modern Mathematical Physics.

11. Compact quantum groups, Stanisław L. Woronowicz, Les Houches, Session LXIV, 1995, Quantum Symmetries, Elsevier 1998.