University Courses
Semester III Semester IV
Compulsory Courses
MT-701. Functional Analysis MT-801. Field Theory
MT-702. Ring Theory MT-802. Combinatorics
Optional Courses (Any one out of three)
MT-703. Mechanics MT-803. Differential Manifolds
MT-704. Measure and Integration MT-804. Algebraic Topology
MT-705. Graph Theory MT-805. Lattice Theory
Departmental Courses (Any two)
MT 706 Topics in Analysis-I MT 806 Topics in Analysis-II
MT 707 Topics in Topology-I MT 807 Topics in Topology-II
MT 708 Topics in Operator Theory-I MT 808 Topics in Operator Theory-II
MT 709 Operations Research-I MT 809 Operational Research-II
MT 710 Topics in Mechanics-I MT 810 Topics in Mechanics-II
MT 711 Topics in Relativity-I MT 811 Topics in Relativity-II
MT 712 Complex Analysis-II MT 812 Fourier Analysis on Finite Groups
MT 713 Representation Theory of Groups MT 813 Advanced Topics in Group Theory
MT 714 Cryptography MT 814 Coding Theory
MT 715 Differential Geometry MT 815 Computational Geometry
MT 716 Projective Geometry MT 816 Algebraic Geometry
MT 717 Algebraic Number Theory MT 817 Commutative Algebra
MT 718 Topics in Matroid Theory MT 818 Advanced Topics in Matroid Theory
MT 719 Banach Algebra MT 819 Advanced Topics in Ring Theory
MT 720 Boundary Value Problems MT 820 Computational Mathematics
2
Paper MT-701
FUNCTIONAL ANALYSIS
1. Banach Spaces
The definition and some examples.
Continuous linear transformations.
The Hahn-Banach theorem.
The natural imbedding of N in N**.
The open mapping theorem.
The conjugate of an operator.
2. Hilbert Spaces
The definition and some simple properties.
Orthogonal complements.
Orthonormal sets.
The conjugate space H*
The adjoint of an operator.
Self-adjoint operators.
Normal and unitary operators.
Projections.
3. Finite-Dimensional spectral Theory
Matrices.
Determinants and the spectrum of an operator.
The spectral theorem.
A survey of the situation.
Text Book :
G. F. Simmons , Introduction to Topology and Modern Analysis, McGraw Hill.
Chapters : 9, 10, 11.
Reference Books :
1. B. V. Limaye, Functional Analysis, Wiley Eastern Ltd.
2. Bachman and Narici, Functional Analysis.
3. W. Rudin, Functional Analysis, Tata Mc Graw Hill Edition.
3
Paper MT -702
RING THEORY
1. Preliminaries: Rings- Definition, Examples, Ring Homomorphism, Ideals, Ring of
Fractions.
2. Euclidean Domains, P.I.D.’s, U.F.D.’s.
3. Polynomial Rings: Definition, properties, Polynomial Rings over Fields, Polynomial
Rings that are U.F.D.’s, Irreducibility Criteria.
4. Noetherian Rings and Affine Algebraic sets, Radicals.
5. Artinian Rings, Discrete Valuation Rings and Dedikind Domains.
Text Book:
Dummit and Foote, Abstract Algebra, second edition (Wiley India).
Sections: 7.1 - 7.5, 8.1 - 8.3, 9.1 - 9.5, 10.1 -10.5, 15.1 –15.2, 16.1 –16.3 .
Reference Books:
1. C. Musili, Rings and Modules, 2nd Revised Edition, Narosa Publishing House.
2. Luther and Passi, Algebra II, Narosa Publishing House.
3. Jain and Bhattacharya, Basic Abstract Algebra, Second Edition, Cambridge
University Press.
4
Paper MT -703
MECHANICS
Elementary principles: Mechanics of a particle, Mechanics of a system of particles,
Constraints, D’Alembert’s principle and Lagrange’s equations. Simple applications of the
Lagrangian formulation.
Variational principles and Lagrange’s equations : Hamilton’s principle, Some
techniques of the calculus of variations, Derivation of Lagrange’s equations from
Hamilton’s principle, Conservation theorem and symmetry properties.
The two-body central force problem : Reduction to the one-body equivalent problem,
Equations of motion and first integrals. The virial theorem, The differential equation for
the orbit, and integrable power-law potentials, The Kepler problem: Inverse square law of
force, The motion in time in the Kepler problem.
The kinematics of rigid body motion : The independent co-ordinates of a rigit body,
Orthogonal transformations, The Euler angles, Euler’s theorem on the motion of a rigid
body, Finite rotations, Infinitesimal rotations.
The Hamilton equations of motion : Legendre transformations and the Hamilton
equations of motion, Cyclic co-ordinates and conservation theorems. Derivation of
Hamilton’s equations from a variational principle.
Canonical transformations : The equations of canonical transformation, Examples of
canonical transformations, Poisson brackets.
Text Book :
Goldsten H., Classical Mechanics, Addition-Wesley, Second Edition, Narosa
Publishing House, 2002.
Chapter 1 : 1-1, 1-2, 1-3, 1-4, 1-6, Chapter 2 : 2-1, 2-2, 2-3, 2-4, 2-6.
Chapter 3 : 3-1, 3-2, 3-4, 3-5, 3-7, 3-8. Chapter 4: 4-1, 4-2, 4-4, 4-6, 4-7, 4-8.
Chapter 8 : 8-1, 8-2, 8-5, Chapter 9: 9-1, 9-2, 9-4.
Reference Books :
1. Tiwari, R.N. and Thakur, B.S., Classical Mechanics, Prentice-Hall of India, New
Delhi, 2007.
2. Gregory, R. Dougals, Classical Mechanics, Cambridge University Press, 2006.
3. Pars L. A., A treatise on analytical dynamics, London : Heinemann, 1965.
5
Paper MT -704
MEASURE AND INTEGRATION
Review : Lebesgue Measure and Lebesgue Integration.
1. Measure spaces, Measurable functions, Integration, General convergence
theorems, Signed measure, The Radon-Nikodym Theorem, The LP- spaces.
2. Outer measure and measurability, The extension theorem, The Lebgesuge-
Stieltjes integral, Product measure, Inner measure, Extension by sets of measure
zero, Caratheodory outer measure, Hausdorff measure.
3. Measure and Topology :
Baire sets and Borel sets, The regularity of Barie and Borel measures, The
construction of Borel measures.
Text Book :
H. L. Royden, Real Analysis (Pearson Education).
Chapters : 11, 12 (except 12.5), 13.1, 13.2, 13.3.
Reference Books :
1. P.R. Halmos , Measure Theory, Reprint (Springer-Verlag, 1974).
2. W. Rudin , Real and Complex Analysis, 3rd Edition, (Mc-Graw Hill).
3. C. D. Aliprants, O. Burkinshaw, Principles of Real Analysis ,(Harcourt Asia Pvt.
Ltd.).
4. G. de Barra, Measure Theory and Integration, Wiley Eastern Ltd , 1981.
6
Paper MT -705
GRAPH THEORY
1. Fundamental Concepts :
Graphs, Matrices and isomorphism decomposition, connection in Graphs, bipartite
graphs, Eulerian circuits, vertex degrees, and Graphic sequences.
2. Trees and Distance :
Trees, Distance in trees and Graphs, Enumeration of trees Caycley’s formula, Spanning
trees in graphs, minimum spanning trees, Kruskal’s algorithm, shortest paths, Dijkstra’s
Algorithm.
3. Matchings :
Maximum Matchings, Hall’s matching condition, Min-Max Theorems, Maximum
bipartite Matching, weighted bipartite matching.
4. Connectivity and Paths :
Connectivity, edge-connectivity, blocks, 2-connected graphs, k-connected and k-edgeconnected
graphs, Menger’s Theorem, Maximum Network flow, Max-flow min-cut
Theorem.
Text Book :
West D.B. Introduction to Graph Theory (Second edition), Prentice Hall of India,
New Delhi (2009).
Chapters : 1, 2, 3.1, 3.2, 4.
Reference Books :
1. J. Clark, D.A. Holton, A First Look at Graph Theory, Allied Publishers.
2. R. J. Wilson, Introduction to Graph Theory, (Fourth Edition), Pearson Education,
Singapore (2003).
7
Paper MT -801
FIELD THEORY
1. Algebraic extensions of fields :
Irreducible polynomials and Eisenstein criterion.
Adjunction of roots.
Algebraic extensions.
Algebraically closed fields, Existence and Uniqueness of algebraic closure (without
proof).
2. Normal and separable extensions :
Splitting fields.
Normal extensions.
Multiple roots.
Finite fields.
Separable extensions.
3. Galois Theory :
Automorphism groups and fixed fields.
Fundamental theorem of Galois theory.
Fundamental theorem of algebra.
4. Applications of Galois Theory to classical problems.
Polynomials solvable by radicals.
Ruler and compass constructions.
Text Book : P. Bhattacharya and S. Jain, Basic Abstract Algebra, Second Edition,
Cambridge University Press.
Chapters : 15, 16, 17, 18.3, 18.5.
Reference Books :
1. John M. Howie, Fields and Galois Theory, Springer Undergraduate Mathematics
Series.
2. Dummit and Foote, Abstract Algebra, 2nd Edition, Wiley Eastern Ltd.
3. M. Nagata, Theory of Field, Marcel Dekker.
4. O. Zariski and P. Sammuel, Commutative Algebra, Vol. 1, Van Nostrand.
8
Paper MT -802
COMBINATORICS
1. General Counting Methods :
Counting Principles, Arrangements and selections, Arrangements and selections with
Repetitions, Distributions, Binomial Identities, Generating permutations and
combinations.
2. Generating Functions :
Generating Function Models, Calculating coefficient of generating functions,
Partitions, Exponential Generating Functions, A Summation Method.
3. Recurrence Relations:
Recurrence Relation Models, Divide and conquer Relations, Solution of Linear
Recurrence Relations, Solution of Inhomogeneous Recurrence Relations, Solutions
with Generating Functions.
4. Inclusion-Exclusion :
Counting with venn diagrams, Inclusion-Exclusion Formula, Restricted Positions and
Rook polynomials.
5. Polya’s Enumeration Formula :
Equivalence and Symmetry Groups, Burnside’s Theorem. The Cycle Index, Polya’s
Formula.
Text Book :
Alan Tucker : Applied Combinations Fourth Edition (John Wiley and Sons, Inc).
Sections : 5.1 to 5.6, 6.1 to 6.5, 7.1 to 7.5, 8.1 to 8.3, 9.1 to 9.4, A4.
Reference Books :
1. V.K. Balkrishnan : Schaum’s outline series. Theory and Problems of
Combinations (Ms Graw Hill).
2. K.D. Joshi: Foundations of Discrete Mathematics (Wiley Eastern Limited).
3. Marshal Hall Jr.: Combinatorial Theory, Second Edition (Wiley Inter science
Publications).
9
Paper MT -803
DIFFERENTIAL MANIFOLDS
Chapter 1 : Differential Manifolds
1. The volume of a Parallelopiped.
2. The volume of a Parametrized – Manifold.
3. Manifolds in Rn.
4. The Boundary of a Manifold.
5. Integrating a Scalar Function over a Manifold.
Chapter 2 : Differential Forms
1. Multilinear Algebra.
2. Alternating Tensors.
3. The Wedge Product.
4. Tangent Vectors and Differential Forms.
5. The Differential Operator.
6. The Action of a Differentiable Map.
Chapter 3 : Stoke’s Theorem
1. Integrating Forms over Parametrized-Manifolds.
2. Orientable Manifolds.
3. Integrating Forms over Oriented Manifolds.
4. The Generalized Stoke’s Theorem.
Text Book :
James R. Munkres, Analysis on Manifolds, (Addision-Wesley Publishing Company).
Chapters : 5, 6, 7.
10
Paper MT -804
ALGEBRAIC TOPOLOGY
Homotopy; Homotopy type and Retractions, Paths, Path connectedness, The
Fundamental group ( Homotopy group), Fundamental group of the circle, Covering
spaces; Fibrations, simplexes and complexes, Simplicial Homotopy Theory.
Text Book:
1. B. K. Lahiri , A First course in Algebraic Topology (Second Edition), Narosa
Publishing House, (2005).
Chapters : 3-11.
References:
1. M.A. Armstrong, Basic Topology, Springer Verlag (2004)(Chapters 5 and 8).
2. Munkres J. R., Topology, Prentice Hall (1975).
11
Paper MT -805
LATTICE THEORY
1. Lattices :
(a) Equivalence of two definitions.
(b) Homomorphisms
(c) Sublattices, ideals and congruence relation.
(d) Product of lattices
(e) Polynomial identities in lattices.
(f) Distributive and Modular lattices
(g) Special elements in lattices.
2. Characterization and Representation :
(a) Dedekinds modularity ceiterion.
(b) Bickhoff’s distributivity criterion.
(c) Stone representation theorem.
(d) Machbin Theorem.
(e) Hahimoto’s Theorem.
3. Modular and Semimodular lattices:
(a) Isomorphism Theorems.
(b) Upper and lower covering conditions.
(c) Semimodular lattices.
(d) Jordan-Holder chain condition.
4. Complete Lattice :
(a) Closure Operations
(b) Embedding in complete lattices.
(c) Conditional Completeness.
(d) Fixpoint Theorem.
Text Books :
1. G. Gruatzer, General Lattice Theory, Academic Press, 1978.
Chapters : 1, 2 (Section 1), 3 (section 1).
Reference Books :
1. G. Baskhoff , Lattice Theory, 3rd Edition, American Mathematical Society, 1940.
2. D. E. Rutherford, Introduction to Lattice Theory, Oliver and Boyd, London,
1965
3. G. Szasz, Introduction to Lattice Theory, Academic Press, New York, 1963.
Semester III Semester IV
Compulsory Courses
MT-701. Functional Analysis MT-801. Field Theory
MT-702. Ring Theory MT-802. Combinatorics
Optional Courses (Any one out of three)
MT-703. Mechanics MT-803. Differential Manifolds
MT-704. Measure and Integration MT-804. Algebraic Topology
MT-705. Graph Theory MT-805. Lattice Theory
Departmental Courses (Any two)
MT 706 Topics in Analysis-I MT 806 Topics in Analysis-II
MT 707 Topics in Topology-I MT 807 Topics in Topology-II
MT 708 Topics in Operator Theory-I MT 808 Topics in Operator Theory-II
MT 709 Operations Research-I MT 809 Operational Research-II
MT 710 Topics in Mechanics-I MT 810 Topics in Mechanics-II
MT 711 Topics in Relativity-I MT 811 Topics in Relativity-II
MT 712 Complex Analysis-II MT 812 Fourier Analysis on Finite Groups
MT 713 Representation Theory of Groups MT 813 Advanced Topics in Group Theory
MT 714 Cryptography MT 814 Coding Theory
MT 715 Differential Geometry MT 815 Computational Geometry
MT 716 Projective Geometry MT 816 Algebraic Geometry
MT 717 Algebraic Number Theory MT 817 Commutative Algebra
MT 718 Topics in Matroid Theory MT 818 Advanced Topics in Matroid Theory
MT 719 Banach Algebra MT 819 Advanced Topics in Ring Theory
MT 720 Boundary Value Problems MT 820 Computational Mathematics
2
Paper MT-701
FUNCTIONAL ANALYSIS
1. Banach Spaces
The definition and some examples.
Continuous linear transformations.
The Hahn-Banach theorem.
The natural imbedding of N in N**.
The open mapping theorem.
The conjugate of an operator.
2. Hilbert Spaces
The definition and some simple properties.
Orthogonal complements.
Orthonormal sets.
The conjugate space H*
The adjoint of an operator.
Self-adjoint operators.
Normal and unitary operators.
Projections.
3. Finite-Dimensional spectral Theory
Matrices.
Determinants and the spectrum of an operator.
The spectral theorem.
A survey of the situation.
Text Book :
G. F. Simmons , Introduction to Topology and Modern Analysis, McGraw Hill.
Chapters : 9, 10, 11.
Reference Books :
1. B. V. Limaye, Functional Analysis, Wiley Eastern Ltd.
2. Bachman and Narici, Functional Analysis.
3. W. Rudin, Functional Analysis, Tata Mc Graw Hill Edition.
3
Paper MT -702
RING THEORY
1. Preliminaries: Rings- Definition, Examples, Ring Homomorphism, Ideals, Ring of
Fractions.
2. Euclidean Domains, P.I.D.’s, U.F.D.’s.
3. Polynomial Rings: Definition, properties, Polynomial Rings over Fields, Polynomial
Rings that are U.F.D.’s, Irreducibility Criteria.
4. Noetherian Rings and Affine Algebraic sets, Radicals.
5. Artinian Rings, Discrete Valuation Rings and Dedikind Domains.
Text Book:
Dummit and Foote, Abstract Algebra, second edition (Wiley India).
Sections: 7.1 - 7.5, 8.1 - 8.3, 9.1 - 9.5, 10.1 -10.5, 15.1 –15.2, 16.1 –16.3 .
Reference Books:
1. C. Musili, Rings and Modules, 2nd Revised Edition, Narosa Publishing House.
2. Luther and Passi, Algebra II, Narosa Publishing House.
3. Jain and Bhattacharya, Basic Abstract Algebra, Second Edition, Cambridge
University Press.
4
Paper MT -703
MECHANICS
Elementary principles: Mechanics of a particle, Mechanics of a system of particles,
Constraints, D’Alembert’s principle and Lagrange’s equations. Simple applications of the
Lagrangian formulation.
Variational principles and Lagrange’s equations : Hamilton’s principle, Some
techniques of the calculus of variations, Derivation of Lagrange’s equations from
Hamilton’s principle, Conservation theorem and symmetry properties.
The two-body central force problem : Reduction to the one-body equivalent problem,
Equations of motion and first integrals. The virial theorem, The differential equation for
the orbit, and integrable power-law potentials, The Kepler problem: Inverse square law of
force, The motion in time in the Kepler problem.
The kinematics of rigid body motion : The independent co-ordinates of a rigit body,
Orthogonal transformations, The Euler angles, Euler’s theorem on the motion of a rigid
body, Finite rotations, Infinitesimal rotations.
The Hamilton equations of motion : Legendre transformations and the Hamilton
equations of motion, Cyclic co-ordinates and conservation theorems. Derivation of
Hamilton’s equations from a variational principle.
Canonical transformations : The equations of canonical transformation, Examples of
canonical transformations, Poisson brackets.
Text Book :
Goldsten H., Classical Mechanics, Addition-Wesley, Second Edition, Narosa
Publishing House, 2002.
Chapter 1 : 1-1, 1-2, 1-3, 1-4, 1-6, Chapter 2 : 2-1, 2-2, 2-3, 2-4, 2-6.
Chapter 3 : 3-1, 3-2, 3-4, 3-5, 3-7, 3-8. Chapter 4: 4-1, 4-2, 4-4, 4-6, 4-7, 4-8.
Chapter 8 : 8-1, 8-2, 8-5, Chapter 9: 9-1, 9-2, 9-4.
Reference Books :
1. Tiwari, R.N. and Thakur, B.S., Classical Mechanics, Prentice-Hall of India, New
Delhi, 2007.
2. Gregory, R. Dougals, Classical Mechanics, Cambridge University Press, 2006.
3. Pars L. A., A treatise on analytical dynamics, London : Heinemann, 1965.
5
Paper MT -704
MEASURE AND INTEGRATION
Review : Lebesgue Measure and Lebesgue Integration.
1. Measure spaces, Measurable functions, Integration, General convergence
theorems, Signed measure, The Radon-Nikodym Theorem, The LP- spaces.
2. Outer measure and measurability, The extension theorem, The Lebgesuge-
Stieltjes integral, Product measure, Inner measure, Extension by sets of measure
zero, Caratheodory outer measure, Hausdorff measure.
3. Measure and Topology :
Baire sets and Borel sets, The regularity of Barie and Borel measures, The
construction of Borel measures.
Text Book :
H. L. Royden, Real Analysis (Pearson Education).
Chapters : 11, 12 (except 12.5), 13.1, 13.2, 13.3.
Reference Books :
1. P.R. Halmos , Measure Theory, Reprint (Springer-Verlag, 1974).
2. W. Rudin , Real and Complex Analysis, 3rd Edition, (Mc-Graw Hill).
3. C. D. Aliprants, O. Burkinshaw, Principles of Real Analysis ,(Harcourt Asia Pvt.
Ltd.).
4. G. de Barra, Measure Theory and Integration, Wiley Eastern Ltd , 1981.
6
Paper MT -705
GRAPH THEORY
1. Fundamental Concepts :
Graphs, Matrices and isomorphism decomposition, connection in Graphs, bipartite
graphs, Eulerian circuits, vertex degrees, and Graphic sequences.
2. Trees and Distance :
Trees, Distance in trees and Graphs, Enumeration of trees Caycley’s formula, Spanning
trees in graphs, minimum spanning trees, Kruskal’s algorithm, shortest paths, Dijkstra’s
Algorithm.
3. Matchings :
Maximum Matchings, Hall’s matching condition, Min-Max Theorems, Maximum
bipartite Matching, weighted bipartite matching.
4. Connectivity and Paths :
Connectivity, edge-connectivity, blocks, 2-connected graphs, k-connected and k-edgeconnected
graphs, Menger’s Theorem, Maximum Network flow, Max-flow min-cut
Theorem.
Text Book :
West D.B. Introduction to Graph Theory (Second edition), Prentice Hall of India,
New Delhi (2009).
Chapters : 1, 2, 3.1, 3.2, 4.
Reference Books :
1. J. Clark, D.A. Holton, A First Look at Graph Theory, Allied Publishers.
2. R. J. Wilson, Introduction to Graph Theory, (Fourth Edition), Pearson Education,
Singapore (2003).
7
Paper MT -801
FIELD THEORY
1. Algebraic extensions of fields :
Irreducible polynomials and Eisenstein criterion.
Adjunction of roots.
Algebraic extensions.
Algebraically closed fields, Existence and Uniqueness of algebraic closure (without
proof).
2. Normal and separable extensions :
Splitting fields.
Normal extensions.
Multiple roots.
Finite fields.
Separable extensions.
3. Galois Theory :
Automorphism groups and fixed fields.
Fundamental theorem of Galois theory.
Fundamental theorem of algebra.
4. Applications of Galois Theory to classical problems.
Polynomials solvable by radicals.
Ruler and compass constructions.
Text Book : P. Bhattacharya and S. Jain, Basic Abstract Algebra, Second Edition,
Cambridge University Press.
Chapters : 15, 16, 17, 18.3, 18.5.
Reference Books :
1. John M. Howie, Fields and Galois Theory, Springer Undergraduate Mathematics
Series.
2. Dummit and Foote, Abstract Algebra, 2nd Edition, Wiley Eastern Ltd.
3. M. Nagata, Theory of Field, Marcel Dekker.
4. O. Zariski and P. Sammuel, Commutative Algebra, Vol. 1, Van Nostrand.
8
Paper MT -802
COMBINATORICS
1. General Counting Methods :
Counting Principles, Arrangements and selections, Arrangements and selections with
Repetitions, Distributions, Binomial Identities, Generating permutations and
combinations.
2. Generating Functions :
Generating Function Models, Calculating coefficient of generating functions,
Partitions, Exponential Generating Functions, A Summation Method.
3. Recurrence Relations:
Recurrence Relation Models, Divide and conquer Relations, Solution of Linear
Recurrence Relations, Solution of Inhomogeneous Recurrence Relations, Solutions
with Generating Functions.
4. Inclusion-Exclusion :
Counting with venn diagrams, Inclusion-Exclusion Formula, Restricted Positions and
Rook polynomials.
5. Polya’s Enumeration Formula :
Equivalence and Symmetry Groups, Burnside’s Theorem. The Cycle Index, Polya’s
Formula.
Text Book :
Alan Tucker : Applied Combinations Fourth Edition (John Wiley and Sons, Inc).
Sections : 5.1 to 5.6, 6.1 to 6.5, 7.1 to 7.5, 8.1 to 8.3, 9.1 to 9.4, A4.
Reference Books :
1. V.K. Balkrishnan : Schaum’s outline series. Theory and Problems of
Combinations (Ms Graw Hill).
2. K.D. Joshi: Foundations of Discrete Mathematics (Wiley Eastern Limited).
3. Marshal Hall Jr.: Combinatorial Theory, Second Edition (Wiley Inter science
Publications).
9
Paper MT -803
DIFFERENTIAL MANIFOLDS
Chapter 1 : Differential Manifolds
1. The volume of a Parallelopiped.
2. The volume of a Parametrized – Manifold.
3. Manifolds in Rn.
4. The Boundary of a Manifold.
5. Integrating a Scalar Function over a Manifold.
Chapter 2 : Differential Forms
1. Multilinear Algebra.
2. Alternating Tensors.
3. The Wedge Product.
4. Tangent Vectors and Differential Forms.
5. The Differential Operator.
6. The Action of a Differentiable Map.
Chapter 3 : Stoke’s Theorem
1. Integrating Forms over Parametrized-Manifolds.
2. Orientable Manifolds.
3. Integrating Forms over Oriented Manifolds.
4. The Generalized Stoke’s Theorem.
Text Book :
James R. Munkres, Analysis on Manifolds, (Addision-Wesley Publishing Company).
Chapters : 5, 6, 7.
10
Paper MT -804
ALGEBRAIC TOPOLOGY
Homotopy; Homotopy type and Retractions, Paths, Path connectedness, The
Fundamental group ( Homotopy group), Fundamental group of the circle, Covering
spaces; Fibrations, simplexes and complexes, Simplicial Homotopy Theory.
Text Book:
1. B. K. Lahiri , A First course in Algebraic Topology (Second Edition), Narosa
Publishing House, (2005).
Chapters : 3-11.
References:
1. M.A. Armstrong, Basic Topology, Springer Verlag (2004)(Chapters 5 and 8).
2. Munkres J. R., Topology, Prentice Hall (1975).
11
Paper MT -805
LATTICE THEORY
1. Lattices :
(a) Equivalence of two definitions.
(b) Homomorphisms
(c) Sublattices, ideals and congruence relation.
(d) Product of lattices
(e) Polynomial identities in lattices.
(f) Distributive and Modular lattices
(g) Special elements in lattices.
2. Characterization and Representation :
(a) Dedekinds modularity ceiterion.
(b) Bickhoff’s distributivity criterion.
(c) Stone representation theorem.
(d) Machbin Theorem.
(e) Hahimoto’s Theorem.
3. Modular and Semimodular lattices:
(a) Isomorphism Theorems.
(b) Upper and lower covering conditions.
(c) Semimodular lattices.
(d) Jordan-Holder chain condition.
4. Complete Lattice :
(a) Closure Operations
(b) Embedding in complete lattices.
(c) Conditional Completeness.
(d) Fixpoint Theorem.
Text Books :
1. G. Gruatzer, General Lattice Theory, Academic Press, 1978.
Chapters : 1, 2 (Section 1), 3 (section 1).
Reference Books :
1. G. Baskhoff , Lattice Theory, 3rd Edition, American Mathematical Society, 1940.
2. D. E. Rutherford, Introduction to Lattice Theory, Oliver and Boyd, London,
1965
3. G. Szasz, Introduction to Lattice Theory, Academic Press, New York, 1963.
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