Year of Teaching 2010 Level PG Credit 04
Syllabus:
M503. Discrete Mathematics
Unit 1: Mathematical Logic and Relations: Statements, Logical connectives, Truth tables, Equivalence, Inference and deduction, Predicates, Quantifiers. Relations and their compositions, Equivalence relations, Closures of relations, Transitive closure and the Warshall’s algorithm, Partial ordering relation, Hasse diagram, Recursive functions.
Unit 2: Semigroups & Monoids: Semigroups, Monoids, Subsemigroups/monoids, Congruence and quotient semigroups/monoids, Homomorphism, isomorphism and the basic isomorphism theorem. Boolean Algebra: Boolean algebra and their various identities, Homomorphisms and isomorphisms, Atoms and the Stone’s theorem (finite case), Boolean functions, their simplification and their applications to combinational circuits.
Unit 3: Combinatorics & Recurrence Relations: Permutation, Combination, Principle of inclusion and exclusion, Recurrence relations, Generating functions
Unit 4: Graph Theory: Basic concepts of graphs, directed graphs and trees, Adjacency and incidence matrices, Spanning tree, Kruskal’s and Prim’s algorithms, Shortest Path, Dijkstra’s algorithm, Planar Graphs, Graph Coloring, Eulerian and Hamiltonian graphs.
References:
1. J.P. Trembley and R.P. Manohar, Discrete Mathematical structures with Applications to Computer Science, McGraw Hill.
2. L.L. Dornhoff and E. F. Hohn, Applied Modern Algebra, McMillan Publishing Co., 1978. .
3. N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice Hall of India, 1980.
4. R. Johnsonbaugh, Discrete Mathematics, Pearson Education, 2001.
5. R. P. Grimaldi, Discrete and Combinatorial Mathematics, Pearson Education, 1999.
6. C.L. Liu, Elements of Discrete Mathematics, McGraw-Hill, 1977
I. Rosen, Discrete Mathematics, Tata McGraw Hill.
B. Kolman, R. Busby, S.C. Ross, Discrete Mathematical Structures, Prentice Hall of India, 2008
Years of Teaching 2010, 2012, 2014, 2015, 2016, 2017, 2023 Level PG Credit 04
Current Syllabus
UNIT I
System of linear equations, Linear transformation, Null space, range space, Matrix representation of a linear transformations, Effect of change of basis on Matrix representation, Similarity of matrices, rank and nullity of linear transformations.
Unit II
Eigen values, Eigen vectors, Characteristic polynomials of a linear transformation, Diagonalization of a matrix; Cayley Hamilton Theorem; Minimal polynomial; Invariant subspaces; Jordan Canonical forms.
.
Unit III
Bilinear forms on a vector space and examples, Matrix of a Bilinear from, Symmetric and Skew-symmetric bilinear forms, Definition of a Quadratic form, matrix of a quadratic form, Reduction to normal form, Orthogonal and congruent reduction., Sylvester’s Law of Inertia. positive definiteness.
Unit IV
Inner product space: Definition and Examples, Norm of a Vector, orthogonally, Orthonormal set, Gram Schmidt orthogonalization, Orthogonal complement, adjoint of a linear transformation. Self-adjoint operator, Unitary operator, Orthogonal, Unitary, Hermitian, skew-Hermitian, symmetric and skew-symmetric matrices. Orthogonal reduction of symmetric matrices, Unitary reduction of Hermitian matrices. Polar and Singular value decomposition.
Texts/References
K. Hoffman and R. A. Kunze, Linear Algebra, 3rd edition, Prentice Hall, 2002.
T. S. Blyth and E. F. Robertson, Further Linear Algebra, Springer, 2002.
M. Artin, Algebra, Prentice Hall of India, 1991.
G. Strang, Linear Algebra and its Applications, Thomas Brooks/Cole, 2006.
Promode Kumar Saikia, Linear Algebra, Pearson, Education, 2009.
Years of Teaching 2011, 2012, 2013, 2017, 2019, 2020, 2021, 2022 Level PG Credit 04
Current Syllabus
UNIT I:
Review of Permutation groups, Dihedral groups, simplicity of , Internal and External direct products and their relationship, Semi direct product, Subnormal and normal series, Zassenhaus’ lemma, Schreier’s refinement theorem, Composition series, Jordan-Hölder’s theorem.
Unit II
Group action; Cayley's theorem, orbit decomposition; counting formula; class equation, consequences for p-groups; Sylow’s theorems (proofs using group actions). Applications of Sylow’s theorems, structure theorem for finite abelian groups (Statements only).
Unit III
Basic properties and examples of ring, domain, division ring and field; direct products of rings, characteristic of a domain, field of fractions of an integral domain, ring homomorphisms, ideals, factor rings prime and maximal ideals, principal ideal domain.
Unit IV
A Euclidean domain, unique factorization domain; brief review of polynomial rings over a field, reducible and irreducible polynomials, Gauss’ theorem for reducibility of polynomial, Eisenstein’s criterion for irreducibility polynomial, roots of polynomials.
Texts/ References
N. Jacobson, Basic Algebra I, 3rd edition, Hindustan Publishing corporation, New Delhi, 2002.
Ramji Lal, Algebra 1, Springer, 2017
I. N. Herstein, Topics in Algebra, 4th edition, Wiley Eastern Limited, New Delhi, 2003.
J. B. Fraleigh, A First Course in Abstract Algebra, 4th edition, Narosa Publishing House, New Delhi, 2002.
D. S. Dummit and R.M. Foote, Abstract Algebra, John Wiley & Sons, 2003.
M. Artin, Algebra, Prentice Hall of India, 1994.
Year of Teaching: 2013 Level PG Credit 04
Syllabus
UNIT I
Definition and examples of Lie Algebra, examples of classical Lie Algebras, derivation of Lie
Algebras, abelian Lie Algebra, Lie subalgebras, ideals and homomorphisms, normalizers and centralizers of a Lie subalgebras, representation of Lie algebras (definition and some examples), automorphisms of a Lie algebra, solvable algebra, solvable radical, nilpotent algebra, Engel’s Theorem.
UNIT II
Semi-simple Lie algebra, Lie’s Theorem, Jordan-Chevalley decomposition (existence and uniqueness) Cartan’s trace criterion for solvability, Killing form and criterion for semi-simplicity, Simple ideals, inner derivations, abstract Jordan-Chevalley decomposition, definition and examples of Lie algebra modules, Schur’s Lemma, Casimir elements of representation, Weyl’s Theorem preservation of Jordan decomposition.
UNIT III
Representation of sl(2,C), weights, highest weight, maximal vectors, classification of irreducible modules, toral and maximal toral subalgebra, root space decomposition and properties of roots.
UNIT IV
Abstract root system (definition, examples and basic properties), Weyl group, root strings bases and their existence, Weyl chambers, classification of rank 2 root systems.
Texts/References
1. J. E. Humphreys, Lie algebra and Representation Theory, GTM 9, Springer, New York 1978.
2. K. Erdmann and M.J. Wildon Introduction to Lie Algebras, Springer Undergraduate series, Springer-Verlag, London 2006.
3. N. Jacobson, Lie algebras, Dover, New York, 1962.
Year of Teaching: 2011, 2012, 2013, 2014, 2015, 2016 Level PG Credit 04
Syllabus:
Unit I
Graph and level sets, vector fields, the tangent space, surfaces, orientation, the Gauss map, geodesics, parallel transport, the Weingarten map.
Unit II
Curvature of plane curves, arc length and line integrals, curvature of surfaces, parametrized surfaces, surface area and volume, surfaces with boundary, the Gauss-Bonnet Theorem.
Unit III
Riemannian geometry of surfaces, Parallel translation and connections, structural equations and curvature, interpretation of curvature.
Unit IV
Geodesic Coordinate systems, isometries and spaces of constant curvature.
Texts/References
1. W. Kuhnel, Differential Geometry-curves-surfaces-Manifolds, AMS 2006.
2. A. Mishchenko and A. Formentko, A course of Differential Geometry and Topology, Mir Publishers Moscow, 1988.
3. A. Pressley, Elementary Differential Geometry, SUMS, Springer, 2004.
4. I. A. Thorpe, Elementary Topics in Differential Geometry. Springer, 2004
Year of Teaching: 2014, 2015, 2018, 2023 Level PG Credit 04
Syllabus:
UNIT I:
Unit I Basic structure of General Linear Group, Special linear group and Projective special linear group, Simplicity of Projective special linear group, Bruhat decomposition in general linear group.
Unit II
Free groups, Generators and relations, Todd Coxeter Algorithm, Semidirect product, Free product of groups, Generalized free products, Presentation of group, Finitely presented group, Central product.
Unit III
Lower and Upper central series, Nilpotent group, $p$-group, Characterizations of finite nilpotent group, Fitting theorem, Fitting subgroup, Frattini subgroup, The Burnside basis theorem, Extra special $p$-groups.
Unit IV
Derived Series, Solvable groups, Properties of Solvable groups. Nilpotent groups are solvable, Solvability of groups of order Solvability of groups of order Solvability of groups of order , Solvability of groups of order less than 60.
Texts/References
J. J. Rotman, Theory of Groups: An Introduction, Allyn and Bacon, 1973.
Michael Artin, Algebra, Prentice- Hall of India, 1991.
D. J. S. Robinson, A course in theory of groups, Springer, 1996.
M. Suzuki, Group Theory-I, Springer, 1986.
J. L. Alperin, R.B. Bell, Groups and Representations, Springer, 1995.
Year of Teaching: 2016, Level PG Credit 04
Syllabus:
UNIT I:
Irreducible and completely reducible modules, Schur’s Lemma, Jacobson density Theorem, Wedderburn Structure theorem for semisimple modules and rings, Group algebra, Maschke’s Theorem.
Representations, Subrepresentations, Characters, Orthogonality relations, Decomposition of regular representation, Number of irreducible representations, Canonical decomposition and explicit decompositions, Subgroups, Product groups, Abelian groups.
Example including cyclic groups, dihedral groups, Quaternion group of order 8, Symmetric
and alternating groups on 3 and 4 symbols. Representations of direct product of two groups, Integrality properties of characters, Burnside's p^aq^b theorem.
Induced representations, The character of induced representation, Frobenius Reciprocity Theorem, Mackey's irreducibility criterion, Examples of induced representations, Statement of Brauer and Artin’s Theorems.
Texts/References
M. Burrow, Representation Theory of Finite Groups, Academic Press, 1965.
L. Dornhoff, Group Representation Theory-I, Marcel Dekker, New York, 1971.
N. Jacobson, Basic Algebra II, Hindustan Publishing Corproation, 1983.
S. Lang, Algebra, 3rd Ed. Springer, 2004.
J. P. Serre, Linear Representation of Groups, Springer-Verlag, 1977.
Year of Teaching: 2012 Level PG Credit 04
Syllabus:
UNIT I:
Preliminaries on rings and ideals, local and semilocal rings, nilradical and Jacobson radical, operations on ideals, extension and contraction ideals, modules and module homomorphisms, submodules and quotient modules, operations on submodules; annihilator of a module, generators for a module, finitely generated modules, Nakayama's lemma,
Unit II:
Exact sequences. Existence and uniqueness of tensor product of two modules, tensor product of n modules, restriction and extension of scalars exactness properties of tensor products flat modules,
Unit III
Multiplicatively closed subsets, saturated subsets; ring of fractions of a ring, localization of a ring, module of fractions and its properties, extended and contracted ideals in a ring of fractions, total ring of fractions of a ring.
Unit IV
Primary ideals, p-primary ideals, Primary decomposition, Minimal primary decomposition, uniqueness theorems, Primary submodules of a module.
Texts/References
1. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison Wesley, 2000.
2. M. Reid, Undergraduate Commutative Algebra, London Math. Soc. Student Texts, No. 29\, 1995.
3. I. S. Luther and I. B. S. Passi, Algebra (Volume 2: Rings), Narosa Publishing House, New Delhi,1999.
4. I. S. Luther and I. B. S. Passi , Algebra (Volume 3: Modules), Narosa Publishing House, New Delhi, 1999.
5. S. Lang, Algebra, Addison-Wesley Publishing Company, London, 2000.
6. D. Eisenbud, Commutative Algebra.
Year of Teaching: 2023 Level PG Credit 04
Syllabus:
UNIT I (25% Weightage)
Definition and examples of topological spaces (including metric spaces), Open and closed sets, Subspaces and relative topology, Closure and interior, Accumulation points and derived sets, Dense sets, Neighbourhoods, Boundary, Bases and sub-bases, Alternative methods of defining a topology in terms of the Kuratowski closure operator and neighbourhood systems.
UNIT II (25 % Weightage)
Filter and Ultra filter, Continuous functions and homeomorphism, Quotient topology, First and second countability and separabilty, Lindelöf spaces, Separation axioms T0, T1, T2, T3, T3½, and T4 and their characterizations, Urysohn’s lemma, Tietze’s extension theorem.
UNIT III (25 % Weightage)
Compactness, Compactness and the finite intersection property, Local compactness, One-point compactification, Connected spaces, Connectedness of the real line, Components, Locally connected spaces, Path connectedness
UNIT IV (25% Weightage)
Product topology in terms of the standard sub-base and its characterizations, Product topology and separation axioms, connectedness, countability properties and compactness, Tychonoff’s theorem.
Texts/References:
J. L. Kelley, General Topology, Van Nostrand, 1995.
K. D. Joshi, Introduction to General Topology, Wiley Eastern, 1983.
James R. Munkres, Topology, 2nd Edition, Pearson International, 2000.
J. Dugundji, Topology, Prentice-Hall of India, 1966.
George F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1963.
S. Willard, General Topology, Addison-Wesley, 1970.
Year of Teaching: 2019, 2020, 2021, 2022 Level PG Credit 04
Current Syllabus:
UNIT I
Finite and infinite sets, Countable and uncountable sets, Cantor’s theorem, Cardinal numbers, Schröder-Bernstein theorem, Euclidean spaces, Metric spaces, Metric induced by norm, open ball, closed ball, open and closed sets, interior, exterior, closure, boundary points and their properties,
UNIT II
Sequences in metric spaces, Complete Metric spaces, Completion of a metric space; relatively open sets in a subspace, Limit, Continuity and and uniform continuity in Metric spaces. Pointwise and Uniform convergence of sequences of functions, Pointwise and Uniform convergence of series of functions, Uniform convergence and continuity, Uniform convergence and integration, Uniform convergence and differentiation.
UNIT III
Compact spaces; Heine-Borel theorem, Finite intersection property, totally bounded set, Bolzano - Weierstrass theorem, sequentially compactness; Connected sets, connected subsets of real numbers, Intermediate value theorem, connected components, totally disconnected sets, , Cantor’s Intersection Theorem.
UNIT IV
Riemann Integral, Riemann Stielzet Integration and its properties,
References:
R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, 3rd edition, John Wiley & Sons, Inc., New York, 2000.
W. Rudin, Principles of Mathematical Analysis, 5th edition, McGraw Hill Kogakusha Ltd., 2004.
N. L. Carother, Real Analysis, Cambridge University Press, 2000.
T. Apostol, Mathematical Analysis, 5th edition, Addison-Wesley, Publishing Company, 2001.
S. Kumaresan, Topology of Metric Spaces, 2nd edition, Narosa Book Distributors Pvt Ltd, 2011.
Year of Teaching: 2023 Level PG Credit 02
Syllabus:
UNIT I: Introduction to LaTeX (30 % Weightage)
Installation of LaTeX . Understanding Latex compilation, Basic Syntax, Writing equations, Matrix, Tables, Page Layout – Titles, Abstract, Chapters, Sections, References, Equation references, Citation. List making environments, Table of contents, Generating new commands. Figure handling , numbering. List of figures, List of tables, Generating index.
UNIT II: Drawing Pictures and Beamer Presentation (20% Weightage)
Simple pictures with PSTricks, Simple pictures with TikZ , Beamer presentation.
Text Book/References
● Leslie Lamport, LaTeX: A Document Preparation System.
● George Gatzer, More Math into LaTeX.
● Tobias Octiker, The Not So Short Introduction to LaTeX.