Name of the Speaker with affiliation

No. of Lectures

Detailed Syllabus

Sundar Sobers

IMSc, Chennai

6

Basics of C^*-algebras, Gelfand-Naimark theorem, continuous functional calculus, spectral theorem for normal operators, positive linear functionals, representations and the GNS construction, Universal C^*-algebras, examples (including universal C^*-algebra generated by an isometry, rotational algebra and Cuntz algebra), and if time permits: inductive limits of C^*-algebras and AF algebras.

Ved Prakash Gupta

Jawaharlal Nehru University

6

Different topologies on the algebra of bounded linear operators on a Hilbert space (strong, weak and σ-weak), von Neumann’s double commutant theorem and Kaplansky’s density theorem, normal linear functionals, geometry of projections in a von Neumann algebra and type decomposition of von Neumann algebras, abelian von Neumann algebras, the GNS construction, the Tomita-Takesaki theorem (for states), examples of von Neumann algebras (for instance, group von Neumann algebras and group measure space construction), and if time permits: discrete crossed products of von Neumann algebras.

Prahlad Vaidyanathan

IISER Bhopal

6

Projections and Unitary Elements: Homotopy classes of Unitary elements, Equivalence of Projections; K_0 group of a unital C*-algebra: Definition, Functoriality, Examples; The functor K_0: Standard Picture of K_0, Half and Split Exactness, Stability; The K_1 group: Definition, Functoriality, Half and Split Exactness, Stability; Examples: Cuntz algebra, Irrational Rotation Algebra, etc.

Bata Krishna Das

IIT Bombay

6

Jaydeb Sarkar

IISc Bangalore

6

Lecture 1. Basic introduction to operator and function theory. Introduction to H*p spaces. Radial limits. H*2 space and invariant subspaces, Beurling theorem, inner outer factorizations.

Lecture 2. Introduction to model spaces, characteristic functions, and commutant lifting theorem.

Lecture 3. Toeplitz operators. Schur functions and representations of Schur functions.

Lecture 4. Nevanlinna-Pick interpolations. The Carathéodory-Fejér interpolation.

Lecture 5. Schur functions and interpolation on polydisc and ball.

Lecture 6. Hankel operators. Applications to system theory/electrical engineering (if time permits).

Manjunath Krishnapur

IISc Bangalore

6

Models of random matrices such as GUE (Gaussian unitary ensemble), CUE (circular unitary ensemble) and $\beta$-tridiagonal random matrices. Fundamental questions in the subject and methods to tackle them. Free probability and its use in first-order calculations in random matrix theory. Orthogonal polynomials, determinantal processes and finer analysis of eigenvalues. If time permits, we shall also touch upon non-hermitian matrices and Brown measure. (Any probability facts beyond very basics will be covered.)