1. 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.
   
   
2. Integration: Lebesgue integration, limit theorems, comparison with the Riemann integral, relationship with differentiation, functions of bounded variation and absolute continuity.
   
   
3. Signed Measures: Radon - Nikodym theorem, Lebesgue decomposition theorem, change of variable formula, Product Spaces, Fubini's theorem and applications.
   
   
4. Lp-Spaces : Holder and Minkowski inequalities, completeness, convolutions, approximation by smooth functions, duality.
   
   
5. Riesz representation theorem: Riesz representation theorem for positive linear functionals, Proof of the theorem, construction of the Lebesgue measure via this approach.
   
   

1. Topological Spaces: Topological spaces, Bases, Continuous maps, Subspaces, Quotient spaces, Products, Connectedness and Compactness.
2. Convergence: Nets, Filters, Limits; Convergence, Countability and Separation axioms.
3. Topological groups: Topological groups; Uniform structures, Products of Compact spaces; Compactifications.
4. Metrizability: Metrizability and Paracompactness, Complete Metric spaces and Function spaces.
5. Monodromy: Fundamental Group and Covering spaces.

1. Group theory: Group theory, permutation groups, Cayley's theorem, Sylow theorems.
2. Ring theory: Ring theory, modules, integral domains and fraction fields, polynomial rings, matrix rings.
3. Linear algebra: Vector spaces, direct sums, tensor products; Linear transformations and Matrices; Determinants; Dioganalizability and Nilpotence; Jordan form.
4. Spectral theorem: Bilinear forms; Inner product spaces; unitary, self-adjoint, normal, and isometric transformations; Spectral theorem.
5. Fields: Field theory: Algebraic and Transcendental extensions; Finite fields, Wedderburn's theorem on finite division rings.

Topics 1 and 2 will be done during the time (1 and 2) are done in Analysis.

1. Review of Basic undergraduate probability: Random variables, Standard discrete and continuous distributions, Expectation, Variance. Conditional Probability.
2. Discrete time Markov chains countable state space, classification of states.
3. Characteristic functions, modes of convergences, Borel-Cantelli Lemma, Central Limit Theorem, Law of Large numbers.
4. Convergence Theorems in Markov Chains

1. Topics in Galois Theory, Ring Extensions and their basic properties.
2. Semisimple Rings and modules.
3. Representation theory of finite groups.

1. Banach spaces : Banach spaces, dual space, Hahn-Banach theorem.
2. Baire category theorem: Baire category theorem and applications (open mapping, closed graph and uniform boundedness theorems).
3. Weak* topologies: Weak and Weak* topologies, Banach Alaoglu theorem, separable, reflexive and locally convex topological vector spaces.
4. Hilbert spaces: Hilbert spaces, projection theorem, Riesz representation theorem, adjoint operators.
5. Compact operators: Spectral theory for compact operators.

1. Holomorphic functions: Holomorphic functions, power series, exponential and logarithmic functions, Moebius transformations, Cauchy-Riemann equations, conformality, elementary conformal mappings.
2. Contour integrals: Contour integration, Cauchy's theorem.
3. Cauchy integral formula: Cauchy integral formula, Calculus of residues.
4. Open mapping theorem: Zeroes and poles, open mapping theorem, maximum modulus principle, removable singularities, poles, essential singularities. Laurent expansions.
5. Harmonic functions: Harmonic functions-Poisson Integral, Jensen's Inequality, idea of analytic continuation.
6. Fourier Analysis: Fourier transform and Inverse Fourier transform, Plancherel theorem in Euclidean spaces.

1. Coverings: Covering spaces, fundamental groups.
2. Homotopy: Homotopy, homotopy lifting property, classification of covering projections, Categories and functors.
3. Homology: Homology and cohomology, singular theory
4. Simplical theory: Simplicial theory, cell complexes.
5. Excision: Exact sequences, excision theorem, Betti numbers, Euler characteristic.

1. Martingale Theory: Radon Nikoydm Theorem, Doob-Meyer decomposition.
2. Weak convergence of probability meaures.
3. Brownian motion.
4. Markov processes and Stationary processes.