AN INTRODUCTION TO CONTEMPORARY MATHEMATICS II

Foundations of functional analysis and noncommutative algebra from the point of view of current developments in quantum geometry. This is a self-contained follow-up of Part I.

PROGRAMME:

1. Equivalence of categories (exercises).

2. From monoids to bimodules (exercises).

3. Tensor product functor between the categories of bimodules (exercises).

4. Topological vector spaces (exercises).

5. Normed vector spaces (exercises).

6. Banach algebras (exercises).

7. Gelfand transform as a monoidal functor from spaces to C*-algebras (exercises).

The goal of this course is to develop an ability to understand at least rudimentary aspects of the state-of-the-art mathematics. A training in 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. Algebra I, Chapters 1-3, Nicolas Bourbaki, Elements of Mathematics.

3. General Topology, Chapters 5-10, Nicolas Bourbaki, Elements of Mathematics.

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

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

6. Categories for the Working Mathematician, Saunders Mac Lane, Graduate Texts in Mathematics 5.

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

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