M. Sc. Mathematics

PROGRAMME OUTCOMES

Knowledge skill: Get advanced knowledge of principles, methods and clear perception of innumerous power of mathematical ideas and
Critical Thinking: Inculcate critical thinking to carry out scientific investigation objectively without being biased with preconceived
Analytical & Logical Reasoning: Equip statements with skills to analyse problems, formulate hypotheses, evaluate and validate results and draw reasonable conclusions
Scientific Communication Skills: Imbibe effective scientific or technical communication skill.
Professional & Ethical Skills: Continue to acquire relevant knowledge and skills appropriate to professional activities and demonstrate high standards when dealing with ethical issues in Mathematical
Research Skills: Prepare students for pursuing research or careers in industry in concerned subject and allied
Enlightened Citizenship: Create awareness to become an enlightened citizen with commitment to deliver one’s responsibilities within the scope of bestowed rights and privileges.
Lifelong Learning: Inculcate the habit of self-learning throughout life, through self-paced and self-directed learning aimed at personal development, and adapting to changing academic demands of work place through knowledge/ skill development/

SPECIFIC OUTCOMES

Strong Foundation in Knowledge: Have strong foundation in core areas of Mathematics, and able to communicate and apply this knowledge in all the fields of learning including higher research and
Abstract Skills: Develop abstract mathematical thinking and thereby enable them to evaluate hypothesis, theories, methods and evidence within their proper
Problem Solving: Solve complex problems by critical understanding, analysis and
Application & Research Efficiency: Provide a systematic understanding of the concepts and theories of mathematics and their application in the real world, to an advanced level, motivate and prepare students for research studies in Mathematics & related fields and enhance career prospects in a huge array of
Lifelong Practical Knowledge: Recognize the need to engage in lifelong learning through continuous education and research leading to higher

COURSE OUTCOME

Sl. No	Name of the Paper	Course Outcomes
Sl. No	Name of the Paper	After the completion of the course, the students will be able:
1	Abstract Algebra	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.
2	Linear Algebra	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.
3	Basic Topology	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.
4	Real Analysis	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.
5	Graph Theory	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.
6	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.
7	Advanced Topology	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.
8	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.
9	Complex Analysis	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.
10	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
11	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.
12	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.
13	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.
14	Functional Analysis	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.
15	Optimization Techniques	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.
16	Spectral Theory	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.
17	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.
18	Differential Geometry	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.
19	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.
20	Combinatorics	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.
21	Dissertation and Viva-voce	To deduce their arguments in comprehensible and scholarly manner. To develop the spirit of research in their mind.