## M.A/M.Sc. (Mathematics)

Teaching Pattern for Semester I, II, III and IV

1. Four lectures per week per course. Each lecture is of 60 minutes duration.
2. In addition, there shall be tutorials, seminars as necessary for each of the five courses.

## PSMT301 / PAMT301: ALGEBRA III

Course Outcomes:

1. Students will learn about classical groups like Simple groups, Solvable groups and Nilpotent groups and applications of these classical groups.
2. 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.
3. Students will learn Finitely generated modules, Free modules, Free module of rank n.
4. 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:

1. Students will learn Hilbert spaces and Banach spaces.
2. 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.
3. Students should know about ℓp, Lp spaces, dual spaces and their properties.
4. Students should understand the fundamental theorems as mentioned in the syllabus.

## PSMT 303/PAMT 303: Differential Geometry

Course Outcomes:

1. Students will be able to grasp parametrization of curves and surfaces.
2. Students will be able to understand the various geometrical aspects like tangent, arc length, curvature, torsion etc of plane and space curves.
3. 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.
4. Students will aware about properties of various special types of curves and surfaces.

## 1. Algebraic Topology

Course Outcomes:

1. Students will learn about homotopy of maps, homotopy of paths and the fundamental group and its applications.
2. Students will learn about universal covering spaces.
3. Students will understand the Singular homology and Excision theorem.

Course Outcomes:

1. Students will learn about monodromy and the monodromy theorem.
2. Students will gain knowledge of Elliptic Functions and Zeta functions.
3. Students will understand Uniform convergence, Ascoli’s theorem, Riemann mapping theorem.

## 3. Commutative Algebra

Course Outcomes:

1. Students will learn about basics of rings and modules, primary decomposition and associated primes.
2. Students will gain knowledge of integral extensions, valuation rings, discrete valuation rings, Dedekind domains.
3. Students will understand the Going up theorem and Going down theorem.

## 4. Algebraic Number Theory

Course Outcomes:

1. Students will learn about algebraic numbers, algebraic integers and further properties of rings of integers.
2. Students will understand the class group.
3. Students will gain knowledge of Ramification theory and Diophantine equations.

## 5. Advanced Partial Differential Equations

Course Outcomes:

1. Students will be able to grasp nature of the differential operator viz parabolic, hyper- bolic and elliptic.
2. Students will be able to understand the solution and various properties of the Laplacian operator, heat operator and wave operator.
3. Students will aware about applications of the Laplacian operator, heat operator and wave operator.

## 6. Integral Transforms

Course Outcomes:

1. Students will be able to grasp the concept of integral transforms and development of corresponding kernels.
2. Students will be able to understand various properties of the Laplace transform, Fourier transform, Mellin transform and Z−transform.
3. Students will aware about applications of the integral transform in the solution of initial and boundary value problems.

## 7. Numerical Analysis

Course Outcomes:

1. Students will be able to grasp the concept of numerical solution of various mathemat- ical problems and corresponding erorrs.
2. Students will be able to understand the approximation of functions by least square method.
3. 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:

1. Overview of Graph theory-Definition 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 problem-Dijkstra’s algorithm, Vertex and Edge connectivity-Result κ ≤ κ′ ≤δ, Blocks, Block-cut point theorem, Construction of reliable communication network, Menger’s theorem.
2. Cut vertices, Cut edges, Bond.
3. Trees, Characterizations of Trees, Spanning trees, Fundamental cycles.
4. Vector space associated with graph, Cayley’s formula, Connector problem- Kruskal’s algorithm, Proof of correctness, Binary and rooted trees, Huffman coding, Searching algorithms-BFS and DFS algorithms.
5. Eulerian Graphs- Characterization of Eulerian Graph, Randomly Eulerian graphs, Chinese postman problem- Fleury’s algorithm with proof of correctness.
6. 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.
7. Matchings-augmenting 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,Matchings-augmenting 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,
8. Independent sets and covering- α + β = p, Gallai’s theorem.
9. Ramsey theorem-Existence 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 numbers-r(k1, k2, . . . , kn), Graph Ramsey Theorem, Evaluation of r(G, H) when for simple graphs G = P3, H = C4.

## 9. Coding Theory

Course Outcomes:

1. Error detection, Correction and Decoding. Communication channels, Maximum likelihood decoding, Hamming distance, Nearest neighbor / minimum distance decoding, Distance of a code.
2. 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.
3. Definition of cyclic codes, Generator polynomials, Generator and parity check ma- trices, Decoding of cyclic codes, Burst-error-correcting codes.
4. Some special cyclic codes: BCH codes, Definitions, Parameters of BCH codes, Decoding of BCH codes.

## 10. Design Theory

Course Outcomes:

1. 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.
2. Symmetric BIBDs An Intersection Property, Residual and Derived BIBDs, Projec- tive Planes and Geometries, The Bruck-Ryser-Chowla Theorem. Finite affine and and projective planes.
3. 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.
4. 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.