Analysis I
 Measure Theory:
Sigmaalgebras, measures, outer measures, completion, construction and
properties of the Lebesgue measure, nonmeasurable sets, Measurable
functions, point wise convergence, almost uniform convergence,
convergence in measure.
 Integration:
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.
 LpSpaces : 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.
Topology I
 Topological Spaces:
Topological spaces, Bases, Continuous maps, Subspaces, Quotient
spaces, Products, Connectedness and Compactness.
 Convergence: Nets, Filters, Limits; Convergence, Countability and
Separation axioms.
 Topological groups:
Topological groups; Uniform structures, Products of Compact spaces;
Compactifications.
 Metrizability:
Metrizability and Paracompactness, Complete Metric spaces and Function
spaces.
 Monodromy:
Fundamental Group and Covering spaces.
Algebra I
 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, selfadjoint, 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
of states.
 Characteristic functions, modes of convergences, BorelCantelli
Lemma, Central Limit Theorem, Law of Large
numbers.
 Convergence Theorems in Markov Chains

Algebra II
 Topics in Galois Theory, Ring Extensions and their basic properties.
 Semisimple Rings and modules.
 Representation theory of finite groups.
Functional Analysis
 Banach spaces : Banach spaces, dual space, HahnBanach 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
spaces.
 Hilbert spaces:
Hilbert spaces, projection theorem, Riesz representation theorem,
adjoint operators.
 Compact operators: Spectral theory for compact operators.
Analysis II
 Holomorphic functions:
Holomorphic functions, power series, exponential and logarithmic
functions, Moebius transformations, CauchyRiemann 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
expansions.
 Harmonic functions:
Harmonic functionsPoisson Integral, Jensen's Inequality, idea of
analytic continuation.
 Fourier Analysis: Fourier transform and Inverse Fourier transform, Plancherel theorem in Euclidean spaces.
Topology II
 Coverings:
Covering spaces, fundamental groups.
 Homotopy:
Homotopy, homotopy lifting property, classification of covering
projections, Categories and functors.
 Homology:
Homology and cohomology, singular theory
 Simplical theory:
Simplicial theory, cell complexes.
 Excision:
Exact sequences, excision theorem, Betti numbers, Euler
characteristic.
Probability Theory II
 Martingale Theory: Radon Nikoydm Theorem, DoobMeyer decomposition.
 Weak convergence of probability meaures.
 Brownian motion.
 Markov processes and Stationary processes.
