Phd. Program Core and Elective Courses
First Semester     Second Semester     Elective Courses
First Semester Core Courses           [Top]
Analysis I
  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.

Topology I
  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.

Algebra I
  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.

Probability Theory I

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
Second Semester Core courses           [Top]

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

Functional Analysis
  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.

Analysis II
  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.

Topology II
  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.

Probability Theory II
  1. Martingale Theory: Radon Nikoydm Theorem, Doob-Meyer decomposition.
  2. Weak convergence of probability meaures.
  3. Brownian motion.
  4. Markov processes and Stationary processes.
Elective courses           [Top]

The term elective course may refer to a, `seminar course', a `research and reading course' or a regular `non-core' course or one of the four 'core courses' not chosen by the student. The students should note that at the begining of a semester not all courses will be offered, except in semester 1 when all four core courses will be offered. The elective courses will be announced ahead of the respective semesters.
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