Course outcomes
M.A/M.Sc. (Mathematics)
Teaching Pattern for Semester I, II, III and IV

Four lectures per week per course. Each lecture is of 60 minutes duration.

In addition, there shall be tutorials, seminars as necessary for each of the five courses.
SemesterIII
PSMT301 / PAMT301: ALGEBRA III
Course Outcomes:

Students will learn about classical groups like Simple groups, Solvable groups and
Nilpotent groups and applications of these classical groups.

Motive of an introduction of the Zariski topology is to take a glance at algebraic
geometry which is like ocean in its own right. This topic provides geometric point of
view of algebra. It gives students a wider perspective to rethink the nature of a prime
ideal, maximal ideal and very precise the geometric position of prime and maximal
ideals. It also helps to visualize radical of an ideal in geometric setting and local nature
of a prime ideal.

Students will learn Finitely generated modules, Free modules, Free module of rank n.

Students will understand the Structure theorem for finitely generated modules over a
PID and Applications to the Structure theorem for finitely generated Abelian groups
and linear operators.
PSMT302 / PAMT302: Functional Analysis
Course Outcomes:

Students will learn Hilbert spaces and Banach spaces.

Students will be able to understand the concept of dimension of a Hilbert space,
bounded linear transformations, norms, inner products, dual spaces and their differ
ence from the finite dimensional cases.

Students should know about ℓ^{p}, L^{p} spaces, dual spaces and their properties.

Students should understand the fundamental theorems as mentioned in the syllabus.
PSMT 303/PAMT 303: Differential Geometry
Course Outcomes:

Students will be able to grasp parametrization of curves and surfaces.

Students will be able to understand the various geometrical aspects like tangent, arc
length, curvature, torsion etc of plane and space curves.

Students will be able to understand the role of first fundamental theorem and second
fundamental theorem in the computation of Gaussian curvature, mean curvature and
principal curvature.

Students will aware about properties of various special types of curves and surfaces.
ELECTIVE COURSES
1. Algebraic Topology
Course Outcomes:

Students will learn about homotopy of maps, homotopy of paths and the fundamental
group and its applications.

Students will learn about universal covering spaces.

Students will understand the Singular homology and Excision theorem.
2. Advanced Complex Analysis
Course Outcomes:

Students will learn about monodromy and the monodromy theorem.

Students will gain knowledge of Elliptic Functions and Zeta functions.

Students will understand Uniform convergence, Ascoli’s theorem, Riemann mapping
theorem.
3. Commutative Algebra
Course Outcomes:

Students will learn about basics of rings and modules, primary decomposition and
associated primes.

Students will gain knowledge of integral extensions, valuation rings, discrete valuation
rings, Dedekind domains.

Students will understand the Going up theorem and Going down theorem.
4. Algebraic Number Theory
Course Outcomes:

Students will learn about algebraic numbers, algebraic integers and further properties
of rings of integers.

Students will understand the class group.

Students will gain knowledge of Ramification theory and Diophantine equations.
5. Advanced Partial Differential Equations
Course Outcomes:

Students will be able to grasp nature of the differential operator viz parabolic, hyper
bolic and elliptic.

Students will be able to understand the solution and various properties of the Laplacian
operator, heat operator and wave operator.

Students will aware about applications of the Laplacian operator, heat operator and
wave operator.
6. Integral Transforms
Course Outcomes:

Students will be able to grasp the concept of integral transforms and development of
corresponding kernels.

Students will be able to understand various properties of the Laplace transform,
Fourier transform, Mellin transform and Z−transform.

Students will aware about applications of the integral transform in the solution of
initial and boundary value problems.
7. Numerical Analysis
Course Outcomes:

Students will be able to grasp the concept of numerical solution of various mathemat
ical problems and corresponding erorrs.

Students will be able to understand the approximation of functions by least square
method.

Students will aware about applications of various numerical techniques in the solution
of difference equations, ordinary and partial differential equations.
8. Graph Theory
Course Outcomes:

Overview of Graph theoryDefinition of basic concepts such as Graph, Subgraphs,
Adjacency and incidence matrix, Eigen values of graph, Friendship Theorem. Degree, Connected graph, Components, Isomorphism, Bipartite graphs etc., Shortest
path problemDijkstra’s algorithm, Vertex and Edge connectivityResult κ ≤ κ′ ≤δ, Blocks, Blockcut point theorem, Construction of reliable communication network, Menger’s theorem.

Cut vertices, Cut edges, Bond.

Trees, Characterizations of Trees, Spanning trees, Fundamental cycles.

Vector space associated with graph, Cayley’s formula, Connector problem Kruskal’s
algorithm, Proof of correctness, Binary and rooted trees, Huffman coding, Searching
algorithmsBFS and DFS algorithms.

Eulerian Graphs Characterization of Eulerian Graph, Randomly Eulerian graphs,
Chinese postman problem Fleury’s algorithm with proof of correctness.

Hamiltonian graphs Necessary condition, Dirac’s theorem, Hamiltonian closure
of a graph, Chvatal theorem, Degree majorisation, Maximum edges in a non
Hamiltonian graph, Traveling salesman problem.

Matchingsaugmenting path, Berge theorem, Matching in bipartite graph, Hall’s
theorem (Necessary and sufficient condition for complete Matching), Konig’s theorem (Maximum matching is same as minimum vettex cover), Tutte’s theorem,
Personal assignment problem,Matchingsaugmenting path, Berge theorem, Matching in bipartite graph, Hall’s
theorem (Necessary and sufficient condition for complete Matching), Konig’s the
orem (Maximum matching is same as minimum vettex cover), Tutte’s theorem,
Personal assignment problem,

Independent sets and covering α + β
= p, Gallai’s theorem.

Ramsey theoremExistence of r(k, l), Upper bounds of r(k, l), Lower bound for
r(k, l) ≥ 2m/2 where m = min{k, l}, Generalize Ramsey numbersr(k1, k2, . . . , kn),
Graph Ramsey Theorem, Evaluation of r(G, H) when for simple graphs G =
P_{3}, H = C_{4}.
9. Coding Theory
Course Outcomes:

Error detection, Correction and Decoding. Communication channels, Maximum
likelihood decoding, Hamming distance, Nearest neighbor / minimum distance decoding, Distance of a code.

Vector spaces over finite fields, Linear codes, Hamming weight, Bases of linear codes,
Generator matrix and parity check matrix, Equivalence of linear codes, Encoding
with a linear code, Decoding of linear codes, Cossets, Nearest neighbor decoding
for linear codes, Syndrome decoding.

Definition of cyclic codes, Generator polynomials, Generator and parity check ma
trices, Decoding of cyclic codes, Bursterrorcorrecting codes.

Some special cyclic codes: BCH codes, Definitions, Parameters of BCH codes,
Decoding of BCH codes.
10. Design Theory
Course Outcomes:

Balanced Incomplete Block Designs, Basic Definitions and Properties, I incidence
Matrices, Isomorphisms and Automorphisms, Constructing BIBDs with Specified
Automorphisms, New BIBDs from Old, Fishers Inequality.

Symmetric BIBDs An Intersection Property, Residual and Derived BIBDs, Projec
tive Planes and Geometries, The BruckRyserChowla Theorem. Finite affine and
and projective planes.

Difference Sets and Automorphisms, Quadratic Residue Difference Sets, Singer Dif
ference Sets, The Multiplier Theorem, Multipliers of Difference Sets, The Group
Ring, Proof of the Multiplier Theorem, Difference Families, A Construction for
Difference Families.

Hadamard Matrices, An Equivalence Between Hadamard Matrices and BIBDs,
Conference Matrices and Hadamard Matrices, A Product Construction, Williamsons
Method, Existence Results for Hadamard Matrices of Small Orders, Regular Hadamard
Matrices, Excess of Hadamard Matrices, Bent Functions.