Analysis I
- 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.
- 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.
- 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.
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, 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
of states.
- Characteristic functions, modes of convergences, Borel-Cantelli
Lemma, Central Limit Theorem, Law of Large
numbers.
- Convergence Theorems in Markov Chains
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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, 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
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, 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
expansions.
- Harmonic functions:
Harmonic functions-Poisson 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, Doob-Meyer decomposition.
- Weak convergence of probability meaures.
- Brownian motion.
- Markov processes and Stationary processes.
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