- Measure Theory:
Sigma-algebras, measures, outer measures, completion, construction and
properties of the Lebesgue measure, non-measurable sets, Measurable
functions, point wise convergence, almost uniform convergence,
convergence in measure.
Lebesgue integration, limit theorems, comparison with the Riemann
integral, relationship with differentiation, functions of bounded
variation and absolute continuity.
- Signed Measures:
Radon - Nikodym theorem, Lebesgue decomposition theorem, change of
variable formula, Product Spaces, Fubini's theorem and applications.
- Lp-Spaces : Holder and Minkowski inequalities, completeness,
convolutions, approximation by smooth functions, duality.
- Riesz representation theorem:
Riesz representation theorem for positive linear functionals, Proof of
the theorem, construction of the Lebesgue measure via this approach.
- Topological Spaces:
Topological spaces, Bases, Continuous maps, Subspaces, Quotient
spaces, Products, Connectedness and Compactness.
- Convergence: Nets, Filters, Limits; Convergence, Countability and
- Topological groups:
Topological groups; Uniform structures, Products of Compact spaces;
Metrizability and Paracompactness, Complete Metric spaces and Function
Fundamental Group and Covering spaces.
- Group theory:
Group theory, permutation groups, Cayley's theorem, Sylow theorems.
- Ring theory:
Ring theory, modules, integral domains and fraction fields, polynomial
rings, matrix rings.
- Linear algebra: Vector spaces, direct sums, tensor products;
Linear transformations and Matrices; Determinants; Dioganalizability
and Nilpotence; Jordan form.
- Spectral theorem:
Bilinear forms; Inner product spaces; unitary, self-adjoint, normal,
and isometric transformations; Spectral theorem.
- Fields: Field theory: Algebraic and Transcendental extensions;
Finite fields, Wedderburn's theorem on finite division rings.
Probability Theory I
Topics 1 and 2 will be done during the time (1 and 2) are done in Analysis.
- Review of Basic undergraduate probability: Random variables,
Standard discrete and continuous distributions, Expectation,
Variance. Conditional Probability.
- Discrete time Markov chains countable state space, classification
- Characteristic functions, modes of convergences, Borel-Cantelli
Lemma, Central Limit Theorem, Law of Large
- Convergence Theorems in Markov Chains
- Topics in Galois Theory, Ring Extensions and their basic properties.
- Semisimple Rings and modules.
- Representation theory of finite groups.
- Banach spaces : Banach spaces, dual space, Hahn-Banach theorem.
- Baire category theorem:
Baire category theorem and applications (open mapping, closed graph
and uniform boundedness theorems).
- Weak* topologies: Weak and Weak* topologies, Banach Alaoglu
theorem, separable, reflexive and locally convex topological vector
- Hilbert spaces:
Hilbert spaces, projection theorem, Riesz representation theorem,
- Compact operators: Spectral theory for compact operators.
- Holomorphic functions:
Holomorphic functions, power series, exponential and logarithmic
functions, Moebius transformations, Cauchy-Riemann equations,
conformality, elementary conformal mappings.
- Contour integrals: Contour integration, Cauchy's theorem.
- Cauchy integral formula:
Cauchy integral formula, Calculus of residues.
- Open mapping theorem:
Zeroes and poles, open mapping theorem, maximum modulus principle,
removable singularities, poles, essential singularities. Laurent
- Harmonic functions:
Harmonic functions-Poisson Integral, Jensen's Inequality, idea of
- Fourier Analysis: Fourier transform and Inverse Fourier transform, Plancherel theorem in Euclidean spaces.
Covering spaces, fundamental groups.
Homotopy, homotopy lifting property, classification of covering
projections, Categories and functors.
Homology and cohomology, singular theory
- Simplical theory:
Simplicial theory, cell complexes.
Exact sequences, excision theorem, Betti numbers, Euler
Probability Theory II
- Martingale Theory: Radon Nikoydm Theorem, Doob-Meyer decomposition.
- Weak convergence of probability meaures.
- Brownian motion.
- Markov processes and Stationary processes.