FROM HILBERT SPACES TO QUANTUM GEOMETRY

Wednesdays, 19:15 - 20:45, Hoża 69, room SST

The concept of a Hilbert space is the mathematical ether of quantum mechanics and the foundation of functional analysis. This lecture course is intended as a hands-on guided tour starting at this basic mathematical notion and driving towards the frontier of modern mathematics known as noncommutative or quantum geometry. Its goal is to master the rudiments of the theory of operators on Hilbert spaces to understand in this laguage both the basic quantum mechanics and the geometry and symmetry of compact Hausdorff spaces. Thus classical spaces will appear as special cases of operator algebras, and particularly useful operator algebras called C*-algebras will be main heroes acting on the stage of a Hilbert space.

PROGRAMME:

1. Why Hilbert spaces and C*-algebras?

2. Scalar products on Banach spaces (exercises).

3. Bounded linear maps (exercises).

4. GNS representation (exercises).

5. The spectrum and the spectral radius (exercises).

6. Universal C*-algebras (exercises).

7. Toeplitz algebra (exercises).

The training aim of this course is to develop an ability to understand at least basic 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 second-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. Wprowadzenie do matematyki współczesnej II, Piotr M. Hajac.

3. Od przestrzeni klasycznych do kwantowych, Piotr M. Hajac.

4. 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.

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

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

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

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

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