[ADD Description] 

Operator Algebras

Syllabus:

i) Definition and examples of Banach and C*algebras, Gelfand theory for abelian Banach and C*algebras, Spectral theorem for bounded normal operators on a Hilbert spaces.
ii) General noncommutative C*algebras: basic properties, Positive elements and states, GNS theorem, ideals and quotients of C*algebras, C*algebra of compact operators.
iii) Definition of examples of von Neumann algebras, the double commutant theorem and Kaplansky density theorem, abelian von Neumann algebras and the L -functional calculus.
iv) Time permitting: discussion on some concrete examples like AF algebras, group C*algebras for discrete countable groups, Cuntz algebras, noncommutative tori etc.

Suggested Texts :
(a) C*Algebras by Examples, K R Davidson.
(b) Functional A Course in Functional Analysis, J B Conway.
(c) C*Algebra and Operator Theory, G. J. Murphy.
(d) Fundamentals of the theory of operator algebras, volume I and II, R.V. Kadison and J. R. Ringrose.

https://www.isibang.ac.in/~adean/infsys/database/MMath/E26OA.html

Probability Theory

Syllabus:

i) Revision of Measure theory: probability spaces, distributions, random vari ables, standard random variable examples, expected value, inequalities (Holder, Cauchy-Schwarz, Jensen, Markov, Chebyshev) , convergence notions(convergence in probability and almost sure, Lp), application of DCT, MCT, Fatou with ex amples, Revision of Fubinis theorem.
ii) Independence, sum of random variables, constructing independent random vari ables, weak law of large numbers, Borel-Cantelli lemmas, First and Second Moment methods, Chernoff bounds and some applications.
iii) Strong law of large number, Kolmogorov 0 - 1 law. Convergence of random series. Kolmogorovs three series theorem.
iv) Weak convergence, tightness, characteristic functions with examples, Central limit theorem (iid sequence and triangular array).

Suggested Texts :
(a) Rick Durrett: Probability (Theory and Examples).
(b) Patrick Billingsley: Probability and measure.
(c) Robert Ash: Basic probability theory.
(d) Leo Breiman: Probability.
(e) David Williams: Probability with Martingales.

https://www.isibang.ac.in/~adean/infsys/database/MMath/C12PT.html