ST PETER’S COLLEGE KOLENCHERY ERNAKULAM

DEPARTMENT OF MATHEMATICS

PROGRAMME - M.Sc. Mathematics

M.Sc. Mathematics- Mahatma Gandhi University, Kottayam, Kerala

Programme Outcome

PO1 Critical Thinking

PO2 Environment and Sustainability

PO3 Self-directed and Life-long learning

PO4 Computational Thinking

PO5 Problem Solving

PO6 Research Orientation

Programme Specific Outcome (PSO)

PSO 1 Provide high quality education in higher mathematics committed to

excellence in research

PSO 2 Develop the skills in objectivity, creativity, independent thinking and

analyzing

PSO 3 Abstract the core concept of modern mathematics

PSO 4 Think critically the mathematical concepts and clearly communicate

solutions to real-world problems

PROGRAMME - STRUCTURE

SEMESTER I

Course No Name of Course Credit Total

Credits/Semester

MT01C01 Linear Algebra 4

20

MT01C02 Basic Topology 4

MT01C03 Measure Theory and Integration 4

MT01C04 Graph Theory 4

MT01C05 Complex Analysis 4

SEMESTER II

Course No Name of Course Credit Total

Credits/Semester

MT02C06 Abstract Algebra 4

20

MT02C07 Advanced Topology 4

MT02C08 MT02C08 -Advanced Complex Analysis 4

MT02C09 MT02C09 -Partial Differential Equations 4

MT02C10 MT02C10 -Real Analysis 4

SEMESTER III

Course No Name of Course Credit Total

Credits/Semester

MT03C11

Multivariate Calculus and Integral

Transforms

4

20

MT03C12 Functional Analysis 4

MT03C13 Differential Geometry 4

MT03C14 Number Theory and Cryptography 4

MT03C15 Optimization Techniques 4

SEMESTER IV

Course No Name of Course Credit Total

Credits/Semester

MT04C16 Spectral Theory 3

20

MT04E01 Analytic Number Theory 3

MT04E05 Mathematical Economics 3

MT04E06 Mathematics for Computing 3

MT04E07 Operations Research 3

Project (PP) 3

Viva-voce (PV) 2

COURSE OUTCOME (CO)

Name of Course: MT01C01: LINEAR ALGEBRA

Credits given: 4

CO No. CO Statement

CO1 Understand the basic concepts of linear transformation, its algebra, null space

and range. Determinants and its properties, characteristic values,

diagonalization of linear transformations

CO2 Identify a linear transformation by a matrix

CO3 Relate matrices of linear transformations with respect to different bases

CO4 Analyze determinant function and its properties

CO5 Determine characteristic values and characteristic vectors

CO6 Understand the basic theory of simultaneous triangulations and

diagonalizations, direct sum decompositions and invariant direct sums

CO7 Realize how linear algebra uses and unifies ideas for functional analysis, the

spectral theory.

Name of Course: MT01C02: BASIC TOPOLOGY

Credits given: 4

CO No. CO Statement

CO1 Understand the transition from metric spaces to topological spaces

CO2 Examine whether a given family of subsets is a topology or not

CO3 Develop basic concepts in metric spaces to topological spaces

CO4 Explain smallness conditions defined in topological spaces

CO5 Distinguish between connected and disconnected spaces

CO6 Discuss the concepts of local connectedness and path connectedness

CO7 Analyze hierarchy of separation axioms

Name of Course: MT01C03: MEASURE THEORY AND INTEGRATION

Credits given: 4

CO No. CO Statement

CO1 Recall algebra of sets, open and closed sets of real numbers

CO2 Analyze Lebesgue measure, measure space and measurable functions

CO3 Define Riemann integral and Lebesgue integral

CO4 Explain differentiation of monotone functions and its applications

CO5 Apply signed measure to related theorems

CO6 Understand convergence in measure

CO7 Analyze the theorems related to measurability in product spaces

Name of Course: MT01C04 : GRAPH THEORY

Credits given: 4

CO No. CO Statement

CO1 Explain the basic concepts of graph theory

CO2 Identify induced subgraphs, cliques, vertex cuts, edge cuts, connectivity,

spanning trees, independent sets and covers in graphs

CO3 Model real world problems using graph theory

CO4 Determine whether graphs are Hamiltonian and/or Eulerian

CO5 Discuss problems involving vertex and edge colouring

CO6 Solve problems involving vertex and edge connectivity and Planarity

Name of Course: MT01C05 : COMPLEX ANALYSIS

Credits given: 4

CO No. CO Statement

CO1 Explain the fundamental concepts of complex analysis and their role in

modern mathematics

CO2 Find parametrizations of curves and compute line integrals

CO3 Utilze the residue theorem to compute several kinds of real integrals

CO4 Explain Cauchy’s theorem for a rectangle and a disk

CO5 Discuss local properties of analytic functions

CO6 Construct conformal mappings between many kinds of domain

Name of Course: MT01C06: ABSTRACT ALGEBRA

Credits given: 4

CO No. CO Statement

CO1 Apply the fundamental theorem of finitely generated abelian groups

CO2 Determine zeros of polynomials

CO3 Analyze extensions of fields

CO4 Make use of Sylow’s theorem to find the properties of subgroups of a finite

group

CO5 Understand the concepts of isomorphisms and automorphisms of fields

CO6 Understand the basics of Galois theory

C07 Develop rigorous proofs for theorems arising in the context of abstract

Algebra

Name of Course: MT02C07: ADVANCED TOPOLOGY

Credits given: 4

CO No. CO Statement

CO1 Understand the significance of the classic theorems characterising normality

CO2 Define topology on the product of an arbitrary collection of topological

spaces

CO3 Identify whether a given topological property is productive

CO4 Explain the concept of evaluation functions and embedding lemma

CO5 Apply the concept of nets to study various notions in topology

CO6 Discuss variations of compactness

C07 Construct one-point compactification of given space.

Name of Course: MT02C08: ADVANCED COMPLEX ANALYSIS

Credits given: 4

CO No. CO Statement

CO1 Illustrate the concept of power series

CO2 Explain infinite products and canonical products

CO3 Distinguish between harmonic and subharmonic functions

CO4 Analyze elliptic functions

CO5 Decide when and where a given function is analytic and be able to find it

series development

CO6 Discuss the main ideas in the proof of the Riemann mapping theorem

Name of Course: MT02C09: PARTIAL DIFFERENTIAL EQUATIONS

Credits given: 4

CO No. CO Statement

CO1 Recall basic properties of the partial differential equations

CO2 Apply the techniques to find solutions of the partial differential equations

CO3 Solve linear and nonlinear partial differential equations of both first and

second order

CO4 Classify partial differential equations and apply analytical methods to solve

the equations

CO5 Analyze the existence and uniqueness of solutions of partial differential

equations

CO6 Create an ability to model physical phenomena using partial differential

equations

Name of Course: MT02C10: REAL ANALYSIS

Credits given: 4

CO No. CO Statement

CO1 Remember monotone functions and explore the properties

CO2 Understand bounded variation and total variation

CO3 Explain curves, paths and arc length

CO4 Develop the idea of Riemann Integral to Riemann Stieltjes Integral

CO5 Analyze the relation between uniform convergence and continuity

CO6 Construct power series of logarithmic and exponential functions

C07 Discuss Fourier series expansions of functions

Name of Course: MT02C11: MULTIVARIATE CALCULUS AND INTEGRAL

TRANSFORMS

Credits given: 4

CO No. CO Statement

CO1 Understand Weirstrass approximation theorem

CO2 Explain other forms of Fourier series

CO3 Make use of Fourier integral Theorem to find definite integrals

CO4 Develop the relation between beta and gamma functions using convolution

theorem

CO5 Evaluate extremum problems

CO6 Discuss multivariable differential calculus

Name of Course: MT03C12: FUNCTIONAL ANALYSIS

Credits given: 4

CO No. CO Statement

CO1 Make use of basic concepts of normed and inner product spaces

CO2 Understand the basic theory of bounded linear operators

CO3 Identify the role of Zorn's lemma

CO4 Apply the theory of Hilbert spaces to other areas including Fourier series

CO5 Analyze uniform boundedness principle

CO6 Discuss Hahn-Banach theorem

Name of Course: MT03C13: DIFFERENTIAL GEOMETRY

Credits given: 4

CO No. CO Statement

CO1 Define graphs, level sets, vector fields, surfaces and orientation

CO2 Explain Gauss Map

CO3 Understand geodesics and parallel transport

CO4 Determine Weingarten Map

CO5 Evaluate curvature of plane curves and higher surfaces

CO6 Utilize line integrals to find length of connected plain curves

Name of Course: MT03C14: NUMBER THEORY AND CRYPTOGRAPHY

Credits given: 4

CO No. CO Statement

CO1 Understand the basic concepts of number theory and cryptography

CO2 Explain the significance of time estimate

CO3 Make use of properties of congruence to compute solutions of problems

CO4 Solve the problems related to factoring

CO5 Evaluate discrete log using Silver Pohlig Hellman algorithm

CO6 Determine whether a given number is prime or not

Name of Course: MT03C15: OPTIMIZATION TECHNIQUES

Credits given: 4

CO No. CO Statement

CO1 Recall basic concepts of general programming and Integer programming

problem

CO2 Classify the programming problems and its importance

CO3 Discuss the importance of sensitivity analysis in programming problems

CO4 Understand the use of minimum path, maximum flow and maximum potential

difference in a network

CO5 Make use of game theory in competitive games

CO6 Understand methods in non linear programming problem and its applications

Name of Course: MT04C16: SPECTRAL THEORY

Credits given: 3

CO No. CO Statement

CO1 Extend basic concepts of convergence of elements to operators and

functionals

CO2 Understand open mapping theorem and closed graph theorem

CO3 Explain fundamentals of spectral theory of operators

CO4 Identify self adjoint operators

CO5 Analyze the theory of compact linear operators and their spectrum

CO6 Discuss basic theory of Banach algebras and unbounded operator theory

Name of Course: MT04E01: ANALYTIC NUMBER THEORY

Credits given: 3

CO No. CO Statement

CO1 Analyze arithmetical functions

CO2 Understand formal power series

CO3 Apply Euler’s summation formula

CO4 Discuss elementary theorems on the distribution of prime numbers

CO5 Apply congruences to prove Lagrange’s theorem and Euler-Fermat theorem

CO6 Understand primitive roots and partitions of a positive integer

Name of Course: MT04E05: MATHEMATICAL ECONOMICS

Credits given: 3

CO No. CO Statement

CO1 Understand the theory of consumer behavior

CO2 Analyze production function

CO3 Determine economic region of production

CO4 Apply input output analysis

CO5 Classify difference equations

CO6 Apply difference equations in economic models

Name of Course: MT04E06: COMPUTING FOR MATHEMATICS

Credits given: 3

CO No. CO Statement

CO1 Understand the basic concepts of object oriented programming

CO2 Make use of tokens to write programs in C++

CO3 Apply constructors, destructors and operator overloading in C++

programming

CO4 Analyze inheritance in C++ programming

CO5 Build formatted I/O operations

CO6 Develop a document using Latex

Name of Course: MT04E07: OPERATIONS RESEARCH

Credits given: 3

CO No. CO Statement

CO1 Understand the concepts of inventory modelling

CO2 Measure total expected cost in inventory models

CO3 Identify queuing models and their uses

CO4 Estimate measures of performance in queuing models

CO5 Discover the technique of dynamic programming to solve practical problems

CO6 Evaluate optimal sequences in sequencing problems

CO7 Develop simulation methods to solve real life situations