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# Semester 4 Syllabus

Back to Int PhD course structure

### 1. Algebra ( MTH-205.4 )

Groups, subgroups, homomorphisms, normal subgroups, quotient groups,

isomorphism theorems, symmetric groups, alternating and dihedral groups.

Structure of finitely generated abelian groups, group actions and its applications, Sylow theorems, solvable groups.

Rings and homomorphisms, ideals, isomorphism theorems, prime ideals and maximal ideals, Jacobson radical and nil-radical, Chinese remainder theorem, polynomial rings and power series rings, division algorithm, roots and multiplicities,

Resultant and discriminant, elementary symmetric functions and the main theorem on symmetric functions, proof of the fundamental theorem of algebra by using symmetric functions, factorization in polynomial rings, Eisenstein criterion, unique factorization domains.

Modules, homomorphisms and exact sequences, free modules, rank of a free module (over commutative rings), hom and tensor products, chain conditions on Modules, Noetherian rings and Hilbert basis theorem, structure theorem for modules over PIDs.

Field extensions and elementary Galois theory.

### 2. Computational PDE ( MTH-206.4 )

• Review of basic numerical analysis

• Finite differences for linear equations

Linear hyperbolic equations, finite differences, theoretical concepts of stability and consistency, order of accuracy, upwind, Lax-Fredrichs and Lax-Wendroff schemes.

Linear parabolic equations-explicit and implicit schemes, Crank-Nicholson method, introduction to multi-dimensional problems.

Linear elliptic equations-finite difference schemes.

• Finite Difference schemes for nonlinear equations

One dimensional scalar conservation laws, review of basic theory, solutions of the Riemann problem and entropy conditions. First order schemes like Lax Fredrichs, Godunov, Enquist Osher and Roe's scheme. Convergence results, entropy consistency and numerical viscosity. Introduction to higher order schemes-Lax Wendroff scheme, Upwind schemes of Van Leer, ENO schemes, Central schemes, Relaxation methods. Introduction to finite volume methods. Convection-Reaction-Diffusion equations,Extension to the above methods. Splitting schemes for multi-dimensional problems.

• Finite element methods for linear equations

Review of elliptic equations, weak formulation and Lax-Milgram lemma, Galerkin approximation, basis functions, energy methods and error estimates, Cea's estimate and Babuska Brezzi theorem.

Finite elements for parabolic equations - Galerkin approximation and error estimates. A posteriori error estimates for Elliptic and Parabolic equations.

• Spectral Methods

Fast Fourier transformation, introduction to Fourier, spectral and pseudo spectral methods.

• Implementation of algorithms on computers is an integrable part of this course.

### 3. Advanced PDE ( MTH-207.4 )

Review of harmonic functions, Extension of maximum principles to 2nd order elliptic equations, existence via sub-super solutions.

A priori estimates of Schauder, existence via fixed point method.

Existence via variational methods: direct minimization and constrained minimization.

Evolution equations: existence via semigroup theory,

Well-posedness of Cauchy Problem for strictly hyperbolic systems, Initial and Initial- boundary value problems for symmetric hyperbolic systems, and linear second order systems,

Nonlinear Hyperbolic systems- Asymptotic solutions of oscillatory initial value problems (Lax Theory)

Scalar conservation laws (Hopf-Lax formula). Systems of hyperbolic conservation laws- 2 x 2 Lax theory. General Case - introduction to Glimm's theory.

### 4. Elective course ( MTH-20*.4 )

A student will choose from one of these elective courses as their fourth course in Semester 4.

4a. Harmonic Analysis