Sir Parashurambhau College (Autonomous), Pune Department of Mathematics Courses for M.A./M.Sc. (Mathematics)

Semester-I Semester-II (All courses are compulsory) (All courses are compulsory) Course Code Title of Course Course Code Title of Course MTH11701 Real Analysis MTH12701 Complex Analysis MTH11702 Advanced Calculus MTH12702 Genaral MTH11703 Theory MTH12703 Theory MTH11704 Linear Algebra MTH12704 Numerical Analysis MTH11705 Ordinary Differential MTH12705 Partial Differential Equations Equations

Semester-III Semester-IV Compulsory Courses Compulsory Courses (Department will run any two of these) (Department will run any two of these) Course Code Title of Course Course Code Title of Course MTH23701 Functional Analysis MTH24701 Differential Geometry MTH23702 Theory MTH24702 Number Theory MTH23703 Discrete Mathematics MTH24703 Mathematical Modeling Optional Courses Optional Courses Course Code Title of Course Course Code Title of Course MTH23704 Fourier Analysis MTH24704 Differential Manifolds MTH23705 Measure Theory MTH24705 Approximation Theory MTH23706 Algebraic Topology MTH24706 Commutative Algebra MTH23707 Representation Theory MTH24707 MTH23708 Lattice Theory MTH24708 Coding Theory MTH23709 Optimization Techniques MTH24709 Integral Transforms MTH23710 Data Science MTH24710 Financial Mathematics MTH23711 Statistical Inference MTH24711 Machine Learning MTH23712 Topics in Analysis MTH24712 Topics in Analysis MTH23713 Topics in Algebra MTH24713 Topics in Algebra MTH23714 Topics in Applied MTH24714 Topics in Applied Mathematics Mathematics MTH23715 Topics in Geometry MTH24715 Topics in Geometry MTH23716 Online Courses MTH24716 Online Courses MTH23717 Project MTH24717 Project

Note: From optional courses department will run at least four courses and students will select any three.

1 Aims and Objectives:

• Aims:

1. Strengthening the understanding of the students and substantiating the conceptual framework of the Graduates in Mathematics for furthering their potential and capabilities in the subject. 2. Introducing advanced theories in the subject in an orderly manner with a clearly defined path of interdependence. 3. Introducing the specializations in different areas of Mathematics and at the same time emphasizing the underlying interconnections in different branches of Mathematics. 4. Generating more interest in the subject and motivating students for self learning beyond the realm of syllabi and examinations. 5. Inculcating the spirit of inquiry among the students and preparing them to take up the research in Mathematics. 6. Exhibiting the wide range of applications of Mathematics and preparing students to apply their knowledge in diverse areas such as Physics, Astronomy, Biology, Social Sciences etc.

• Objectives:

1. A student should be able to understand the proof techniques in Math- ematics and importance of theorems for sorting out typical examples. 2. A student should acquire sufficient technical competence to solve the problems of varying difficulty levels and high notational complexity. 3. A student should be able to make observations, experimentation and pattern recognition which would stimulate the research potential 4. A student should acquire the communication skill to present technical Mathematics so as to take up a career in Teaching Mathematics at various levels including schools, colleges, universities, etc.

Examination:

Continuous Internal Assessment: Total 50 Marks.

Two Internal exams(each of 20 Marks) + other component/s of 10 Marks = total 50 Marks.

End semester Exam: 50 Marks.

2 Subject : Real Analysis (MTH11701) Total credits = 4, Total number of contact hours = 60

Syllabus

• The Real Numbers, Sets, Sequences, Functions(Revision): [03 Hours] With main focus on sections - Countable and uncountable sets, Borel sets of real numbers. Reference Book 1: Chapter 1 - sections 1.3 to 1.6.

• Lebesgue Measure: [17 Hours] Lebesgue Outer Measure, σ−algebra of Lebesgue Measurable Sets, Outer and Inner Approximation of Lebesgue Measurable Sets, Countable Addi- tivity, Continuity, Borel-Cantelli Lemma, Non-measurable Sets, Cantor Set, Cantor-Lebesgue Function. Reference Book 1: Chapter 2 - sections 2.1 to 2.7.

• Lebesgue Measurable Functions: [12 Hours] Definition and algebra of Lebesgue Measurable Functions, Sequential Point- wise Limits and Approximations by Simple Functions, Littlewood’s Three Principles, Egoroff,s Theorem, Lusin’s Theorem. Reference Book 1: Chapter 3 - sections 3.1 to 3.3.

• Lebesgue Integration: [15 Hours] The Riemann Integral (Revision), The Lebesgue Integral of a Bounded Mea- surable Function over a Set of Finite Measure, The Lebesgue Integral of a Measurable Non-negative Function, The General Lebesgue Integral, Count- able Additivity and Continuity of Integration, Uniform Integrability, The Vitali Convergence Theorem. Reference Book 1: Chapter 4 - sections 4.1 (Revision), 4.2, 4.3 (Linearity and monotonicity: statement only), 4.4 (Linearity and monotonicity: statement only), 4.5, 4.6.

• Differentiation and Integration: [13 Hours] Continuity of Monotone Functions (Statements and definitions only), Lebesgue’s Differentiation Theorem (Statements and definitions only), Functions of Bounded Variation, Jordan’s Theorem, Absolutely Continuous Functions, Integration of Derivatives, Differentiation of Indefinite Integral, Fundamental Theorem of Calculus. Reference Book 1: Chapter 6 - sections 6.1 to 6.5.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

3 1. Real Analysis (4th Edition) - H. L. Royden, P. M. Fitzpatrick. (Prentice Hall, 2010)

2. Real Analysis - Elias M. Stein, Rami Shakarchi. ( Press)

3. Basic Real Analysis - Anthony W. Knapp. (Birkhauser, Springer)

4. P rinciples of Real Analysis - Charalambos D. Aliprantis, Owen Burkin- shaw.

4 Subject : Advanced Calculus (MTH11702) Total credits = 4, Total number of contact hours = 60

Syllabus

• Multivariate Differentiation, Continuously Differentiable Functions, Chain Rule, Maxima and Minima for Multivariate Real Valued Functions, Appli- cation: Lagrange’s Multipliers, Inverse Function Theorem, Implicit Function Theorem. [27 Hours] Reference Book 1: Chapter 2 - sections 5 to 9. Reference Book 2: Chapter 9 - Sections 9 to 15.

• Integral over Rectangle, Integral over Bounded Set, Fubini’s Theorem. [18 Hours] Reference Book 1: Chapter 3 - sections 10 to 14.

• Partitions of Unity (statement only), Diffeomorphisms, Theorem on Change of Variables (Statement Only). [15 Hours] Reference Book 1: Chapter 4 - sections 16 to 19.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. Analysis on Manifolds - James R. Munkres. (CRC Press)

2. Calculus V olume II (2nd Edition) - T. M. Apostol. (Wiley)

3. P rinciples of Mathematical Analysis (3rd Edition) - Walter Rudin. (Mc- Grew Hill)

4. Calculus on Manifolds (2nd Edition) - M. Spivak. (CRC Press)

5. Mathematical Analysis (2nd Edition) - T. M. Apostol. (Addison Wesley Publishing company)

5 Subject : Group Theory (MTH11703) Total credits = 4, Total number of contact hours = 60

Syllabus

• Introduction to Groups: [5 Hours] Definition of group and basic examples. Reference Book 1: Chapter 1 - sections 1.1 to 1.7.

• Subgroups: [12 Hours] Subgroups, Centralizers, Normalizers of group, Cyclic Groups. Reference Book 1: Chapter 2 - sections 2.1 to 2.5.

• Quotient Groups and Homomorphisms: [13 Hours] Quotient Groups, Homomorphisms, Isomorphism Theorems, Alternating Groups. Reference Book 1: Chapter 3 - sections 3.1 to 3.3 and section 3.5.

• Group Actions: [15 Hours] Group Actions, Class Equation, Automorphisms, Sylow Theorems. Reference Book 1: Chapter 4 - sections 4.1 to 4.5.

• Direct Product and Abelian Groups: [15 Hours] Direct Products, Fundamental Theorem of Finitely Generated Abelian Groups (statement and examples), Commutator Subgroup, Elementary Classifica- tion of Groups. Reference Book 1: Chapter 5 - sections 5.1 to 5.4.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. (2nd Edition) - David S. Dummit and Richard M. Foote. (John wiley and Sons)

2. Contemporary Abstract Algebra (7th Edition) - J. A. Gallian. (D. C. Heath and Company)

3. T opics in Algebra (2nd Edition) - I. N. Herstein. (Ginn and Company)

4. Algebra - Michael Artin. (Prentice Hall)

5. A F irst Course in Abstract Algebra (7th Edition) - J. B. Fraleigh. (Pearson)

6 Subject : Linear Algebra (MTH11704) Total credits = 4, Total number of contact hours = 60

Syllabus

• Vector Spaces (Revision): [3 Hours] Definition and Examples, Subspaces, Basis and Dimension Reference Book 1: chapter 4

• Linear mappings and matrices: [12 Hours] Linear Mappings, Quotient Spaces, Vector of Linear Mappings, Lin- ear Mappings and Matrices, Change of Basis, Rank of a Linear Mapping, Decomposition of a Reference Book 1: chapter 5

• Reduction of matrices to canonical forms: [22 Hours] Diagonalization of a Matrix, Triangularization of a matrix, Jordan canonical form, Singular Value Decomposition (SVD) with application to data com- pression. Reference Book 1 : Chapter 6, Reference Book 2 : Section 9.4, 9.5.

• Metric vector spaces: [15 Hours] Bilinear forms (definition only), Representation of a Bilinear Form by a Ma- trix, Hermitian Forms, Matrix of a Hermitian Form, Adjoint of linear opera- tor, Unitary Operators, Euclidean Vector Space (Except Principal Axis The- orem), Isometry, Spectral Theorem for Self-adjoint Operators, Normal Oper- ators, Spectral Theorem for Normal Operators (statement only), Canonical Representation of Unitary Operators (statement only). Reference Book 1 : Chapter 7 - Sections 1 (definition only), 4, 5 (except Principal Axis Theorem and Simultaneous Diagonalization), 6.

• Applications of Linear Algebra [8 Hours] (At least three applications to be taken.) Reference Book 2 : Chapter 10.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. F irst Course in Linear Algebra - P.B. Bhattacharya, S.R. Nagpaul, S.K. Jain, (New Age International)

2. Elementary Linear Algebra (Applications V ersion) (11th Edition) - Howard Anton, Chris Rorres. (Wiley)

7 3. Linear Algebra - K. Hoffman and Ray Kunje (Prentice -Hall of India private Ltd.)

4. Linear Algebra - Vivek Sahai, Vikas Bist (Narosa Publishing House)

5. Introduction to Linear Algebra - Gilbert Strang

6. Linear Algebra - A. Ramachandra Rao, P. Bhimasankaram (Hindustan Book Agency)

8 Subject : Ordinary Differential equations (MTH11705) Total credits = 4, Total number of contact hours = 60

Syllabus

• Real Life Applications of Differential Equations: [10 Hours] Reference Book 3: Projects for various sections in the book.

• Power Series Solutions and Special Functions: [6 Hours] Introduction, A Review of Power Series, Series Solutions of First Order Equa- tions, Second Order Linear Equations, Ordinary Points, Regular Singular Points. Reference Book 1: Sections 26-30.

• Fourier Series: [4 Hours] The Fourier Coefficients, Orthogonal Functions, Fourier Theorem. Reference Book 1: Section 33, 37, Appendix A- last theorem- only statement.

• Sturm-Liouville Problems: [7 Hours] Regular Sturm-Liouville Problems, modifications, orthogonality of eigen- functions, a temperature problem, a vibrating string problem, periodic bound- ary conditions. Reference Book 2: Sections 59-61, 31, 32, 43.

• Legendre polynomials: [6 Hours] Solutions of Legendres equations, Legendre polynomials, orthogonality of Legendre polynomials, Rodrigues formula and norms, Legendre series. Reference Book 2 Sections 86-90.

• Bessel functions: [6 Hours] Bessel functions Jn(x), general solutions of Bessels equation, recurrence re- lations, Bessels integral form, orthogonal sets of Bessel functions, Fourier- Bessel series. Reference Book 2: Sections 71-74, 78, 81.

• Systems of First Order Equations: [5 Hours] General Remarks on Systems, Linear Systems, Homogeneous Linear Systems with Constant Coefficients. Reference Book 1: Sections 54-56.

• Nonlinear Equations: [7 Hours] Nonlinear Systems, Autonomous Systems, The Phase Plane and Its Phe- nomena, Types of Critical Points, Stability, Critical Points and Stability for Linear Systems. Reference Book 1: Sections 57-60.

• Qualitative Properties of Solutions: [4 Hours] Oscillations and the Sturm Separation Theorem, The Sturm Comparison Theorem. Reference Book 1: Sections 24-25.

9 • The Existence and Uniqueness of Solutions: [5 Hours] The Method of Successive Approximations, Picards Theorem. Reference Book 1: Sections 68-69.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. Differential Equations with Applications and Historical notes (2nd Edi- tion) - George F. Simmons, Tata McGraw-Hill

2. F ourier series and Boundary value P roblems (7th Edition) - J.R. Brown, R.V. Churchill, Tata McGraw-Hill

3. A F irst Course in Differential Equations (with Modelling Applications) (10th Edition) - Dennis G. Zill

10 Subject : Complex Analysis (MTH12701) Total credits = 4, Total number of contact hours = 60

Syllabus

• Preliminaries to Complex Analysis: [10 Hours] Functions on complex plane, Holomorphic functions, C-R equations, Power series and radius of convergence, Integration along curves. Reference book 1 : chapter 1

• Cauchys Theorem and its Applications: [25 Hours] Goursat’s theorem, Local existence of primitives, Cauchy’s theorem for disc, Evaluation of some real indefinite integrals, Cauchy integral formulas, Liou- ville’s theorem, Identity theorem, Morera’s theorem, Sequences of holomor- phic functions, Holomorphic functions defined in terms of integrals, Schwarz Reflection Principle, Runge’s approximation theorem. Reference book 1 : chapter 2

• Meromorphic Functions and the Logarithm: [25 Hours] Zeros and poles, Residue formula, Singularities and meromorphic functions, Riemann’s theorem on removable singularities, Casorati-Weierstrass’ theo- rem, Argument principle and applications: Rouche’s theorem, Open map- ping theorem, Maximum modulus principle, Homotopy version of Cauchy’s theorem, Complex logarithm, Fourier series and harmonic functions. Reference book 1 : chapter 3

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. Complex analysis - Elias M. Stein, Rami Shakarchi (Overseas Press (India) Ltd.,Princeton Lectures in Analysis)

2. Complex V ariables and Applications - Ruel V. Churchill / James Ward Brown (McGraw Hill)

3. F unctions of one complex variable (2nd Edition) - John B. Conway (Narosa Publishing house)

4. Complex Analysis - Lar’s V. Ahlfors (McGraw Hill)

5. Complex variables with Applications - S. Ponnusamy, Herb Silverman (Birkhauser)

11 Subject : General Topology (MTH12702) Total credits = 4, Total number of contact hours = 60

Syllabus

• Prerequisites (statements only): [04 Hours] Cartesian Products, Finite Sets, Countable and Uncountable Sets, Infinite Sets and Axiom of Choice, Well Ordered Sets. Reference Book 1: Chapter 1 - Sections 5, 6, 7, 9, 10.

• Topological Spaces and Continuous Functions: [19 Hours] Topological Spaces, Basis for Topology, Order Topology, Product Topology on X × Y , Subspace Topology, Closed Sets and Limit Points, Continuous Functions, Product Topology, Metric Topology, Quotient Topology. Reference Book 1: Chapter 2 - Sections 12 to 22.

• Connectedness and Compactness: [25 Hours] Connected Spaces, Components and Local Connectedness, Compact Spaces, Limit Point Compactness, Local Compactness, One Point Compactification, Tychonoff Theorem. Reference Book 1: Chapter 3 - Sections 23 to 29, Chapter 5 - sections 37.

• Countability and Separation Axioms: [12 Hours] The Countability Axioms, Separation Axioms, Normal Spaces, The Urysohn Lemma (statement only), The Urysohn Metrization Theorem(statement only), The Tietze Extension Theorem (statement only). Reference Book 1: Chapter 4 - Sections 30 to 32, 33 to 35 (Statements only).

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. T opology (2nd Edition) - James R. Munkres. (Prentice Hall)

2. Introduction to General T opology - K. D. Joshi. (New Age International)

3. General T opology - J. L. Kelley. (GTM, Springer,1975)

4. Counterexamples in T opology - L. A. Steen and J. A. Seebach Jr. (Holt Rinehart and winston)

12 Subject : Ring Theory (MTH12703) Total credits = 4, Total number of contact hours = 60

Syllabus

• Introduction to Rings, Definition of ring and basic examples, Ring Homo- morphisms, Ideals, Quotient Rings, Rings of Fractions, Chinese Remainder Theorem. [20 Hours] Reference Book 1: Chapter 7 - sections 7.1 to 7.6.

• Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains. [17 Hours] Reference Book 1: Chapter 8 - sections 8.1 to 8.3.

• Polynomial Rings, Irreducibility Criteria. [15 Hours] Reference Book 1: Chapter 9 - sections 9.1 to 9.5.

• Modules, Quotient Modules, Module Homomorphisms. [8 Hours] Reference Book 1: Chapter 10 - sections 10.1, 10.2.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books

1. Abstract Algebra (2nd Edition) - David S. Dummit and Richard M. Foote. (John wiley and Sons)

2. T opics in Algebra (2nd Edition) - I. N. Herstein. (Ginn and Company)

3. Algebra - Michael Artin. (Prentice Hall)

4. Contemporary Abstract Algebra (7th Edition) - J. A. Gallian. (D. C. Heath and Company)

5. A F irst Course in Abstract Algebra (7th Edition) - J. B. Fraleigh. (Pearson)

13 Subject : Numerical Analysis (MTH12704) Total credits = 4, Total number of contact hours = 60 Syllabus Algorithms to be implemented using Mathematical Software. • Preliminaries and Error Analysis: [5 Hours] Convergence, Floating Point Number Systems, Round Up Error, Floating Point Arithmetic, Absolute and Relative Errors. Reference Book 1: Chapter 1 - sections 1.2, 1.3. • Root Finding Methods: [15 Hours] Bisection Method, Fixed Point Iteration Schemes, Newton’s Method, Secant Method, Method of False Position, Error Analysis for Iterative Methods, Accelerating Convergence, Muller’s Method. Reference Book 1: Chapter 2 - sections 2.1 to 2.6. • Differentiation and Integration: [12 Hours] Interpolation using Lagrange’s Polynomial, Numerical Differentiation us- ing Lagranges Interpolating Polynomial, Trapezoidal Rule, Simpson’s Rule, Newton-Cotes Quadrature, Gaussian Quadrature, Composite Quadrature Formulae. Reference Book 1: Chapter 3 - section 3.1, Chapter 4 - sections 4.1, 4.3, 4.4, 4.7. • Initial Value Problems of Ordinary Differential Equations: [8 Hours] Eulers Method, Higher order Taylor methods, Runge-Kutta Methods. Reference Book 1: Chapter 5 - sections 5.1 to 5.4. • System of Equations: [15 Hours] Gaussian Elimination, Pivoting Strategies, Error Estimates and Condition Number, LU Decomposition, Direct Factorization, Iterative Techniques for Linear Systems, Numerical Solution of Nonlinear Systems of Equations using Newton’s Method and Steepest Descent Method. Reference Book 1: Chapter 6 - sections 6.1, 6.2, 6.5, Chapter 7 - sections 7.1, 7.3 to 7.5, Chapter 10 - sections 10.2, 10.4. • Eigenvalues and Eigenvectors: [5 Hours] Power Method, Inverse Power Method. Reference Book 1: Chapter 9 - sections 9.1, 9.3 (except Deflation Methods).

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books: 1. Numerical Analysis (9th Edition) - Richard L. Burden and J. Douglas Faires. (Brooks/Cole)

14 2. A F riendly Introduction to Numerical Analysis - Brian Bradie. (Pearson Education India)

3. An Introduction to Numerical Analysis - K .E. Atkinson. (Wiley)

15 Subject : Partial Differential Equations (MTH12705) Total credits = 4, Total number of contact hours = 60

Syllabus

• First order P.D.E.s : [12 Hours] Linear equations of first order, Integral surfaces through given curve, Quasi- Linear equations, Non-Linear first order P.D.E.s. Reference Book 1: Sections 1.4, 1.9, 1.10, 1.11.

• Second Order P.D.E.s [8 Hours] Genesis of second order P.D.E.s, Classification of second order P.D.E.s. Reference Book 1: Sections 2.1, 2.2.

• Some Special Second Order P.D.E.s [25 Hours] d’Alembert’s Solution for Wave Equation, Laplace Equation, The Fourier Method, Boundary Value Problems. Reference Book 1: Sections 2.3 (1, 2, 3), 2.4 (1, 2, 3, 4, 5), 2.5 (1, 2) and Reference Book 2: Chapter 4, 5.

• Applications of O.D.E.s to Second Order P.D.E.s [15 Hours] Applications of Sturm-Liouville problems, Applications of Bessel functions, Applications of Legendre Polynomials, Verification of solutions and unique- ness. Reference Book 2: Sections 65, 67, 70, 83-85, 92-93 and Chapter 11.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. An Elementry Course in P artial Differential Equations (2nd Edition) - T. Amaranath, Narosa Publishing House Pvt. Ltd.

2. F ourier series and Boundary value P roblems (7th Edition) - J.R. Brown, R.V. Churchill, Tata McGraw-Hill

16 Subject : Functional Analysis (MTH23701) Total credits = 4, Total number of contact hours = 60

Syllabus

• Hilbert Spaces: [25 Hours] Hilbert space L2, Orthogonality, Unitary Mappings, Pre-Hilbert Spaces, Fourier Series, Closed Subspaces and Orthogonal Projections, Linear Trans- formations, Linear Functionals, Riesz Representation Theorem, Adjoint of Linear Transformation, Compact Operators. Reference Book 1: Chapter 4 - sections 1 to 8.

• Applications to PDEs: [5 Hours] Weak solutions to Partial Differential Equations with Constant Coefficients (theorem and estimate without proof). Reference Book 1: Chapter 5 - section 3.

• Lp Spaces and Banach Spaces: [30 Hours] Lp Spaces, H¨olderand Minkowski Inequalities, Completeness of Lp, Banach Spaces, Linear Functionals and the Dual of a Banach Space, The Dual Space of Lp, Separation of Convex Sets, Hahn-Banach Theorem and its conse- quences, Complex Lp and Banach Spaces. Reference Book 2: Chapter 1 - sections 1 to 6 (except 5.4), sections 8, 9.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. Real analysis - Elias M. Stein, Rami Shakarchi, Overseas Press (India) Ltd., Princeton Lectures in Analysis

2. F unctional analysis - Elias M. Stein, Rami Shakarchi, Overseas Press (In- dia) Ltd., Princeton Lectures in Analysis

3. F unctional Analysis - B. V. Limaye. (New Age International)

4. Linear F unctional Analysis - Bryan P. Rynne and Martin A. Youngson. (Springer)

17 Subject : Field Theory (MTH23702) Total credits = 4, Total number of contact hours = 60

Syllabus

• Field Extensions: [32 Hours] Basic Theory of Field Extensions, Algebraic Extensions, Classical Straight- edge and Compass Constructions, Splitting Fields and Algebraic Closures, Separable and Inseparable Extensions, Cyclotomic Polynomials and Exten- sions. Reference Book 1: Chapter 13 - sections 13.1 to 13.6.

• Galois Theory: [28 Hours] Basic Definitions, The Fundamental Theorem of Galois Theory, Finite Fields, Galois Groups of Polynomials, Solvable and Radical Extensions - Insolvabil- ity of the Quintic. Reference Book 1: Chapter 14 - sections 14.1 to 14.3, 14.6 (Discussion only), 14.7 (statements only).

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. Abstract Algebra (2nd Edition) - David S. Dummit and Richard M. Foote. (John wiley and Sons)

2. T opics in Algebra (2nd Edition) - I. N. Herstein. (Ginn and Company)

3. Algebra - Michael Artin. (Prentice Hall)

4. Contemporary Abstract Algebra (7th Edition) - J. A. Gallian. (D. C. Heath and Company)

5. A F irst Course in Abstract Algebra (7th Edition) - J. B. Fraleigh. (Pearson)

18 Subject : Discrete Mathematics (MTH23703) Total credits = 4, Total number of contact hours = 60

Syllabus

• Counting: [10 Hours] The Basics of Counting, The Pigeonhole Principle, Permutations and Combi- nations, Binomial Coefficients and Identities, Generalized Permutations and Combinations, Generating Permutations and Combinations Reference Book 1: Chapter 6 - sections 6.1 to 6.6.

• Advanced Counting Techniques: [22 Hours] Applications of Recurrence Relations, Solving Linear Recurrence Relations, Divide-and-Conquer Algorithms and Recurrence Relations, Generating Func- tions, Inclusion-Exclusion, Applications of Inclusion-Exclusion Reference Book 1: Chapter 8 - sections 8.1 to 8.6.

• Graphs: [15 Hours] Graphs and Graph Models, Graph Terminology and Special Types of Graphs, Representing Graphs and Graph Isomorphism, Connectivity, Euler and Hamil- ton Paths, Shortest-Path Problems, Planar Graphs, Graph Coloring Reference Book 1: Chapter 10 - sections 10.1 to 10.8.

• Trees: [13 Hours] Introduction to Trees, Applications of Trees, Tree Traversal, Spanning Trees, Minimum Spanning Trees Reference Book 1: Chapter 11 - sections 11.1 to 11.5.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books:

1. Discrete Mathematics and Its Applications (7th Edition) - Kenneth H. Rosen, McGraw-Hill Publications

2. Applied Combinatorics (3rd edition) - Alan Tucker, John Wiley & sons

3. Introduction to Graph T heory - R. J. Wilson, Pearson Publications

4. A F irst Look at Graph T heory - John Clarke and D.A. Holton, Allied Publisher

19 Subject : Differential Geometry (MTH24701) Total credits = 4, Total number of contact hours = 60

Syllabus

• Curves in the plane and in space: [2 Hours] What is curve?, Arc-length, Reparametrization, Level curves Vs. Parametrized curves. Reference Book 1: Chapter 1

• How much does a curve curve? [9 Hours] Curvature, Plane curves, Space curves. Reference Book 1: Chapter 2

• Global properties of curves: [6 Hours] Simple closed curves, The isoperimetric inequality, The four vertex theorem. Reference Book 1: Chapter 3

• Surfaces in three dimensions: [9 Hours] What is surface?, Smooth surfaces, Tangents, Normals and Orientability, Examples of surfaces, Quadric surfaces, Triply orthogonal systems, Applica- tions of inverse function theorem. Reference Book 1: Chapter 4

• The first fundamental form: [10 Hours] Lengths of curves on surfaces, Isometric surfaces, Conformal mappings and surfaces, Surface area, Equiareal maps and theorem of Archimedes. Reference Book 1: Chapter 5

• Curvature of surfaces: [9 Hours] The second fundamental form, Curvature of curves on surface, The Normal and Principal curvatures, Geometric interpretation of Principal curvatures. Reference Book 1: Chapter 6

• Gaussian curvature and the Gauss map: [6 Hours] The Gaussian and Mean curvatures, The Pseudosphere, Flat surfaces, Sur- faces of constant mean curvature, Gaussian curvature of Compact surfaces, The Gauss map. Reference Book 1: Chapter 7

• Geodesics [9 Hours] Definition and basic properties, Geodesic equations, Geodesic on surface of revolution, Geodesics as shortest path, Geodesic coordinates. Reference Book 1: Chapter 8

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

20 Reference Books:

1. Elementary Differential Geometry - Andrew Pressley, Springer Interna- tional Edition UTM, Indian Reprint 2004.

2. Differential Geometry - John A. Thorpe, Springer International Edition UTM, Indian Reprint 2004.

3. Differential Geometry of Lightlike Submanifolds - Krishan L. Duggal, Bayram Sahin, (Birkhauser).

4. Differential Geometry of Curves and Surfaces - Manfredo P. do Carmo, Revised and Updated 2nd Edition, (Dover Publications)

21 Subject : Number Theory (MTH24702) Total credits = 4, Total number of contact hours = 60

Syllabus

• Unique factorization in Z: [3 Hours] Divisibility in Integers, Primes, Division Algorithm, G.C.D., Euclid’s Algo- rithm, Fundamental Theorem of Arithmetic. Reference Book 1: Chapter 1 - section 1.

• Applications of Unique Factorization: [6 Hours] Infinitely Many Primes in Z, Some Arithmetic Functions - Euler Function φ(n), Divisor Function ν(n), M¨obiusFunction µ(n), M¨obius Inversion The- orem. Reference Book 1: Chapter 2 - sections 1 and 2.

• Congruence: [7 Hours] Congruence in Z, Congruence Classes, Complete Set of Residues, The Con- gruence ax ≡ b(m), Euler’s Theorem, Fermat’s Little Theorem, Chinese Remainder Theorem. Reference Book 1: Chapter 3 - sections 1 to 4.

• Quadratic Reciprocity: [15 Hours] Quadratic Residues, Legendre Symbol, Gauss’ Lemma, Law of Quadratic Reciprocity, Jacobi Symbol, A Proof of the Law of Quadratic Reciprocity. Reference Book 1: Chapter 5 - sections 1 to 3.

• Diophantine Equations: [14 Hours] Linear Diophantine Equations, Solutions of y3 = px + 2, x2 + y2 = z2 and x3 + py3 + p2z3 = 0, The Method of Descent (Insolvability of x4 + y4 = z2 in Positive Integers), Legendre’s Theorem, Sophie Germain’s Theorem, Pell’s Equation, Sums of Two Squares. Reference Book 1: Chapter 17 - sections 1 to 6.

• Algebraic Number Theory: [15 Hours] Algebraic Numbers and Algebraic Integers, Unique Factorization in Alge- braic Number Fields, Dedekind Rings, Ramification and Degree, Chinese Remainder Theorem for rings. Reference Book 1: Chapter 6 - section 1, Chapter 12 - sections 1 to 3.

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books: 1. A Classical Introduction to Modern Number T heory (2nd Edition) - Ken- neth Ireland and Michael Rosen, Springer (Graduate texts in Mathematics).

22 2. Algebraic Number T heory and F ermat0s Last T heorem - Ian Stewart, David Tall, (A. K. Peters Publishers).

3. Introductory Algebraic Number T heory - Saban Alaca and Kenneth Williams, (Cambridge Press).

4. An Introduction to the T heory of Numbers - G. H. Hardy and E. M. Wright, Oxford University Press.

5. An Introduction to Number T heory - Ivan Niven, H. S. Zuckerman and Hugh L. Montgomery, Wiley Eastern Limited.

6. Elementary Number T heory - David M. Burton, Universal Book Stall, New Delhi.

7. An Introduction to Analytical Number T heory - T. M. Apostol, Springer International Student’s Edition.

23 Subject : Mathematical Modeling (MTH24703) Total credits = 4, Total number of contact hours = 60

Syllabus

• What is Modeling: [05 Hours] Models and Reality, Properties of Models, Building a Model. Reference Book 1: Chapter 1.

• Arguments from Scale: [7 Hours]

– Effects of Size Any one illustration from Reference Book 1: Chapter 2 section 2.1. – Dimensional Analysis Any one illustration from Reference Book 1: Chapter 2 section 2.2.

• Graphical Methods: [13 Hours]

– Comprehensive Statics Any one illustration from Reference Book 1: Chapter section 3.2. – Stability Questions Any one illustration from Reference Book 1: Chapter 3 section 3.3.

• Basic Optimization: [12 Hours]

– Optimization by Differentiation Any one illustration from Reference Book 1: Chapter section 4.1. – Graphical Methods Any one illustration from Reference Book 1: Chapter 3 section 4.2.

• Basic Probability: [13 Hours]

– Analytic Models Any one illustration from Reference Book 1: Chapter section 5.1. – Monte Carlo Simulation Any one illustration from Reference Book 1: Chapter 3 section 5.2.

• Miscellaneous: [10 Hours] Any two illustration from Reference Book 1: Chapter 6.

(Reference Books 2, 3 should be used for students’ mini projects.)

Twelve hours of teacher-student contact for continuous internal assessment are included in total 60 contact hours.

Reference Books: 1. An Introduction to Mathematical Modeling - Edward Bender, J. Wiley and Sons.

24 2. Mathematical Modelling : Case Studies and P rojects - Jim Caldwell and Douglas K. S. Ng, Kluwer Academic Publishers.

3. Mathematical Modeling : Models, Analysis and Applications - Sandip Banerjee, CRC Press.

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