This school is intended for people in the second year of graduate school (M.Sc. or Ph.D.) who have had a first course in functional analysis (and know such things as Hahn-Banach Theorem, the Principle of Uniform Boundedness and the Open Mapping Theorem).

The goals of this school are manifold:

(i) To teach some rudimentary theory of operators on Hilbert space, culminating in a formulation of the Spectral Theorem for bounded self-adjoint operators (and for commuting families of normal operators, if time permits) in terms of the existence and uniqueness of appropriate continuous and measurable functional calculi;

(ii) to give a quick crash course on the basics of C*-algebras;

(iii) to give a quick crash course on the basics of von Neumann algebras;

(iv) to introduce students to the fundamentals of Hilbert C*-modules;

(v) to introduce students to the fundamentals of free probability, the combinatorics of non-crossing partitions, and the connections with random matrix theory; and

(vi) to give a brief introduction to the k-theory of C^*-algebras.For the uninitiated, already at the level of (i) above one sees the importance of the topological and probabilistic viewpoints; the `non-commutative' counterparts of these two aspects are C*-algebras and von Neumann algebras. While (ii) and (iii) may be regarded as dealing with the fundamentals of `non-commutative' general topology and measure theory respectively, (iv) and (vi) may be viewed as the beginnings of non-commutative algebraic topology, and (v) as non-commutative probability theory.

Interested people should send an email to sunder@imsc.res.in with a short description of their level of preparedness to benefit from this programme, including what chapters of what books on functional analysis/Hilbert space theory they have read.

Speakers (affiliations) - no. of lecture hours - detailed syllabus

• V.S. Sunder (IMSc)) - 6 x 1.5 : Spectral theorem for self-adjoint or normal operators: Hilbert spaces, bounded operators, self-adjoint operators, continuous and measurable functional calculi, relation to conventional formulation in terms of spectral measures, extension to normal operators (if time permits).
• Kunal Krishna Mukherjee (IIT-M) - 6 x 1.5 : $C^*$-algebras: Kunal has other commitments right now and requested some more time for sending the syllabus for his lectures.
• R. Srinivasan (CMI) - 6 x 1.5 : von Neumann algebras: 1. weak, strong topologies, double commutant theorem 2. and 3. projections and types of von Neumann algebras, (type I is B(H), finiteness is equivalent to existence of finite normal trace, semi-finite von Neumann algebras) 4, 5, 6. standard form and Tomita-Takesaki theorem and modular theory.
• Partha Sarathi Chakraborty (IMSc) - 6 x 1.5 : Hilbert $C^*$-modules:
  1) Hilbert C* modules
  2) Adjointable linear maps
  3) Multiplier Algebras
  4) Kasparov Stabilization
  5) KSGNS theorem
  6) Induced representations for C*-algebras.
  If time permits then he may discuss regular operators.Text book: Lance/ Raiburn-Williams/ Jensen-Thomsen.
• Vijay Kodiyalam (IMSc) - 6 x 1.5 : Free probability: The emphasis of this course will be on the combinatorics of free probability theory. We will begin with non-commutative probabilit
  spaces and distributions and the concepts of free independence and free product of these. Next we will discuss free cumulants and the combinatorics of non-crossing partitions. We conclude with the connection between ,free probability theory and random matrices.
  Reference: Lectures on the Combinatorics of Free Probability by Nica and Speicher
• S. Sundar (CMI) - 6 x 1.5 : K-theory of $C^*$-algebras: Brief discussion of the Serre-Swan theorem.Definition of K_{0} and K_{1}, homotopy invariance, short-exactness of K-theory, Bott periodicity, six term exact sequence.

Time table: Each day there are two lectures from 9.30 - 11.00 and 11.30 - 1.00 and one tutorial session from 3.30 - 5.00. During week 1, Sunder and Kunal will lecture, during week 2, Srinivasan and Partha and during week 3, Vijay and Sundar.. The detailed time table will be posted on the website of the workshop soon.