Name of the Paper
|After the completion of the course, the students will be able:
- To demonstrate the knowledge of finitely generated abelian groups and fundamental theorem.
- To apply the concept of group homomorphisms, inner automorphism and fundamental homomorphism theorems.
- To analyse group action on a set and G-sets to counting.
- To apply isomorphism theorems and Sylow theorems.
- To describe the structure and construction of field of quotients of an integral domain.
- To demonstrate knowledge of polynomial rings, the evaluation homomorphisms and factorization of polynomials over a field.
- To explain group rings, ring homomorphism, quotient rings, prime and maximal ideals.
- To illustrate the concepts of vector spaces, basis and dimension, coordinates and summary of row equivalence through examples.
- To differentiate different linear transformations, their algebra and representation of transformations by matrices.
- To apply properties of determinants while solving problems on matrices or linear transformations.
- To implement the ideas of canonical forms, characteristic values and annihilating polynomials.
- To develop concepts of triangulation, diagonalization and direct sum decomposition.
- To analyse the concept of topological spaces, base and subbase.
- To apply the concept of continuity and quotient spaces on different topology.
- To explain the concept of local connectedness and path connectedness.
- To differentiate levels of spaces based on axioms.
- To explain theorems associated with bounded variation and rectifiable curves.
- To acquire the idea about Riemann-Stieltjes integral.
- To explain uniform convergence
- To describe theorems associated with uniform convergence.
- To acquire the idea about special functions.
- To recall basic concepts of graph theory.
- To draw line graphs and graph products.
- To discuss connectivity of a graph.
- To use the application of trees in everyday problems.
- To solve problems on Eulerian and Hamiltonian graphs.
- To practice problems of graph colouring using Brook’s theorem and Vizing’s theorem.
- To discuss about planarity of graphs.
||Advanced Abstract Algebra
- To explain field extensions.
- To construct finite fields.
- To analyse the concepts of UFD and ED.
- To acquire knowledge of field automorphisms and isomorphism extension theorem.
- To describe Galois group and Galois theory.
- To explain separable extensions and cyclotomic extensions.
- To explain Urysohn characterization of normality, Tietze characterization of normality, products and co- products.
- To analyse embedding lemma, Tychonoff embedding and metrization theorem.
- To develop idea of convergence and related properties of nets and filters.
- To describe compactness and variations of compactness.
||Numerical analysis with Python 3
- To implement basics of python for programming.
- To develop basic programming involving symbolic operations.
- To apply programming techniques for finding limits and derivatives using Sympy.
- To interpret the concepts of Gaussian elimination, interpolation, curve fitting and finding roots of equations using python programme.
- To illustrate the concept of numerical integration using python.
- To explain spherical representation of complex plane and elementary properties of analytic functions.
- To analyse power series representation of analytic functions.
- To examine the concept of singularities and residues.
- To describe Cauchy’s theorem and residue theorem.
||Measure Theory andIntegration
- To explain Lebesgue measure.
- To describe theorems associated with Lebesgue measure.
- To define Lebesgue measurable functions and Lebesgue integration.
- To explain theorems associated with Lebesgue measurable functions.
- To describe general measurable space and measurable functions.
- To acquire knowledge about integration over general measurable space and product measure
||Advanced Complex Analysis
- To apply the concept of harmonic and subharmonic functions.
- To explain Weierstrass’s theorem, Gamma function and Hadamard’s theorem.
- To examine the idea of Riemann zeta function and normal families.
- To illustrate Riemann mapping theorem and Weierstrass’s theory.
||Partial Differential Equations
- To explain PDEs of first order, second and higher orders.
- To apply various analytic methods for computing solutions of various PDEs.
- To determine integral surfaces passing through a curve, characteristic curves of second order PDE and compatible systems.
- To analyse behavior of solutions of PDEs using technique of separation of variables.
||Multivariate Calculus and Integral Transforms
- To explain integral transforms and convolutions.
- To discuss multivariable differential calculus.
- To describe sufficient conditions for equality of mixed partial derivatives.
- To compute extrema of real valued functions of several variables.
- To explain integration of differential forms.
- To acquire the concepts of normed spaces and their properties.
- To discuss linear operators on finite dimensional spaces and dual space.
- To illustrate inner product spaces and properties of orthonormal sequences using examples and theorems.
- To demonstrate different forms of Hahn-Banach Theorems.
- To determine solutions to linear programming problems using simplex method, two-phase method, big-M method revised simplex method or dual simplex method.
- To apply the cutting plane method and branch and bound method for solving integer programming problems.
- To analyse the concepts of flow and potential in networks and goal programming.
- To discuss different methods for solving non-linear programming problems.
- To distinguish different forms of convergence of operators and open mapping theorem.
- To apply the concept of Banach fixed point theorem and properties of resolvent and spectrum.
- To describe compact linear operators and their properties.
- To discuss properties of bounded self adjoint linear operators, positive operators and properties of projections.
||Analytic Number Theory
- To apply the properties of arithmetical functions for solving problems.
- To acquire the knowledge about the theory of prime numbers.
- To utilize the concepts of congruences, Chines remainder theorem and Legendre symbol.
- To implement Euler’s theorem, Wilson’s theorem and Mobius inversion formula.
- To interpret the ideas of graphs and level sets, vector fields, the tangent space and vector field on surfaces and orientation.
- To summarize the fundamentals of Gauss map, geodesics and parallel transport.
- To describe the ideas of Weingarten map, curvature of plane curves and line integrals.
- To discuss curvature of surfaces and parametrized surfaces.
||Algorithmic Graph Theory
- To explain basic concepts of graphs and elementary algorithms.
- To implement the concept of trees, paths and distances using algorithms.
- To establish the max-flow min-cut algorithm and Mengers theorem for finding connectivity.
- To examine algorithm for finding maximum matching in bipartite graphs, factorizations and block designs.
- To solve permutation and combinations problems.
- To apply pigeon hole principle and Ramsey numbers.
- To use principles of inclusion and exclusion for solving problems.
- To compute generating functions and recurrence relations.
||Dissertation and Viva-voce
- To deduce their arguments in comprehensible and scholarly manner.
- To develop the spirit of research in their mind.