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Semester 1 Syllabus

Back to Int PhD course structure

 1. Real Analysis ( MTH-101.4 )

Review of Number System: Dedekinds cut; Topology of ℝ: Weierstrass theorem, Heine-Borel theorem, connectedness, series and sequences, continuous and differentiable functions, mean value theorems and their consequences, maxima, minima and curve tracing, functions of bounded variation, Riemann integration, Riemann-Stieltjes integration in ℝ.

Sequences of functions, uniform convergence, Ascoli-Arzela Theorem, functions of several variables, continuity, differentiation – directional derivatives and Frechet derivatives, mean value theorems, maxima, minima, inverse and implicit function theorems.


2. Topology ( MTH-102.4 )

Axioms of set theory, partial order, ordinality, cardinality, Schroeder-Bernstein Theorem, axiom of choice and its equivalents.

Topological spaces, induced topology, product topology, separation axioms - \(T_0\),\(T_1\),\(T_2\),\(T_3\),\(T_4\) spaces, Urysohn's lemma and Tietze extension theorem, connectedness, compactness, Tychnoff theorem, I and II axiom of countability, metric spaces, Baire category theorem, Banach fixed point theorem.


3. Linear Algebra ( MTH-103.4 )

Vector spaces, linear transformations and matrices, elementary matrices, row and column operations, multilinear algebra and determinants, rank and nullity, eigenvalues and Cayley-Hamilton theorem.

Operator norms and spectral radius formula, normal, Hermitian and unitary operators, spectral theorem, Jordan canonical form.


4. Theory of ODE ( MTH-104.4 )

Methods of solving 1st and second order, linear ODE by variation of parameters and Wronskian, solutions by series of some special 2nd order ODE.

1st order non-linear ODE, Cauchy-Picard theorem,two point boundary value problem and Sturm-Liouville theory, Weyl-Titschmarsh theorem for unbounded interval - limit cycle, limit point cases.

Linear systems with constant coefficients, Fundamental Matrix, Linear Systems with periodic coefficients. Nonlinear Autonomous system, critical points. Phase plane analysis, stability, Periodic solutions.