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Semester 3 Syllabus

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1. Introductory PDE ( MTH-201.4 )

First order PDE - solutions by characteristics.

Classification of 2nd order PDEs, fundamental solutions and Green functions for Laplace, heat and wave equations, explicit solution formulas, harmonic functions, mean value property, maximum principles, uniqueness of solutions.

Solutions by other methods: separation of variables, similarity methods, transform methods, power series method, Cauchy- Kowalewsky theorem, Holmgren's uniqueness theorem.

2. Probabilty Theory ( MTH-202.4 )

Probability spaces, probabilistic language: event, sample point, expectation, variance, moments, etc., random variables and their distributions, important examples: binomial, Poisson, hypergeometric, Gaussian, Cauchy, exponential, gamma and beta distributions.

Independence of events, of classes of events, of random variables. Kolmogorov's 0-1 Law, the Borel-Cantelli lemma, elementary conditional probability, Bayes formula.

Simple examples of Markov chains, Markov inequality, the weak and strong laws of large numbers.

Moment generating functions and characteristic functions, uniqueness theorem.

Inversion theorem, application to the dependence of random variables to the existence of moments.

Convergence of probability measures, tightness.

Convergence in distribution, Levy's continuity theorem, the central limit theorem for independent identically distributed summands.

Conditional expectation and conditional probability, basic theorems, regular conditional probability.

$$\mathbb R^2$$valued Gaussian random variables. Markov chains with countable state space: examples, transience and recurrence, stationary distributions, continuous parameter Markov chains, Poisson process, Martingales with discrete parameter, inequalities, convergence theorems, optional stopping theorem, applications in particular Markov chains.

3. Complex Analysis ( MTH-203.4 )

Complex numbers, complex differentiation, Cauchy-Riemann equations, complex integration, homotopy version of Cauchy theorem, fundamental theorem of algebra, power series expansion.

Maximum modulus and residue theorems, singularities and meromorphic functions,

Laurent series, Rouché theorem, Hurwitz theorem, Weierstrass and Mittag-Leffler theorem.

Conformal maps, Schwarz theorem, Montel's theorem, Schwarz-Christoffel formula.

Riemann mapping theorem.

4. Advanced Functional Analysis (Distribution Theory, Sobolev Spaces and Applications) ( MTH-204.4 )

Distribution theory, Sobolev Spaces, embedding theorems, Rellich's lemma, trace theorems.

Second order elliptic equations:- Formulation of Dirichlet, Neumann and oblique derivative problems, Weak formulation, Lax - Milgram Lemma, existence and regularity up to the boundary, maximum principle, elementary variational inequality.

Linear evolution equations, existence of weak solutions, energy methods.

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