Organisation : Panjab University
Course Name : M.Sc. Mathematics
Service Name : 2020-21 Syllabi/ Syllabus
Semester : I
Website : https://puchd.ac.in/syllabus.php
PUCHD M.Sc. Mathematics Syllabi
Panjab University Syllabi of M.Sc. Mathematics 2020-21
Syllabus
Semester-I
MATH 601S : Real Analysis-I
Unit-I:
(i) Basic Topology : Finite, countable and uncountable sets. Metric spaces, compact sets. Perfect sets. Connected sets.
(ii) Sequences and series : Convergent sequences (in metric spaces). Subsequences. Cauchy sequences. Upper and lower limits of a sequence of real numbers. Riemann’s Theorem on Rearrangements of series of real and complex numbers.
(iii) Continuity : Limits of functions (in metric spaces). Continuous functions. Continuity and compactness. Continuity and connectedness. Monotonic functions.
Unit- II:
(iv) The Riemann-Stieltjes integral: Definition and existence of the Riemann-Stieltjes integral. Properties of the integral. Integration of vector-valued functions. Rectifiable curves.
(v) Sequences and series of functions: Problem of interchange of limit processes for sequences of functions. Uniform convergence. Uniform convergence and continuity. Uniform convergence and integration. Uniform convergence and differentiation. Equicontinuous families of functions, The Stone- Weierstrass theorem.
Math 602S: Algebra- I
Unit- I:
Review of basic concepts of groups with emphasis on exercises. Permutation groups, Even and odd permutations, Conjugacy classes of permutations, Alternating groups, Simplicity of An, n > 4. Cayley’s Theorem, Direct products, Fundamental Theorem for finite abelian groups, Sylow theorems and their applications, Finite Simple groups [Scope as in chapters 2-4 Modern Algebra by Surjeet Singh and Qazi Zameerudin, Eighth Edition and chapters 11, 24, 25 of Contemporary Abstract Algebra by Gallian, Fourth Edition]
Unit-II:
Survey of some finite groups, Groups of order p2, pq (p and q primes). Solvable groups, Normal and subnormal series, composition series, the theorems of Schreier and Jordan Holder [Scope as in Chapters 6 of Modern Algebra by Surjeet Singh and Qazi Zameerudin, Eighth Edition and Chapter 7 of Algebra, Vol. I by Luther and Passi].
Review of basic concepts of rings with emphasis on exercises. Polynomial rings, formal power series rings, matrix rings, the ring of Guassian Integers. [Scope as in Chapters 7, 8 and 9 of Modern Algebra by Surjeet Singh and Qazi Zameerudin, Eighth Edition , 2006].
Math 603S: Differential Equations
Unit-I:
Differential Equations Existence and uniqueness of solution of first order equations. Boundary value problems and Strum-Liouville theory. ODE in more than 2-variables. [Scope as in Chapter V of the book ‘An introduction to Ordinary Differential Equations’ by E.A.Coddington and Chapters X & XI of the book ‘Elementary Differential Equations and Boundary Value Problems’ by W.E.Boyce and R.C.Diprima.]
Unit-II:
Partial differential equations of first order. Partial differential equations of higher order with constant coefficients. Partial differential equations of second order and their classification. [Scope as in Chapters I, II & III of the book ‘Elements of Partial Differential Equations’ by I.N.Sneddon].
Math 604S : Complex Analysis-I
Unit-I:
Complex plane, geometric representation of complex numbers, joint equation of circle and straight line, stereographic projection and the spherical representation of the extended complex plane. Topology on the complex plane, connected and simply connected sets. Complex valued functions and their continuity. Curves, connectivity through polygonal lines.Analytic functions, Cauchy-Riemann equations, Harmonic functions and Harmonic conjugates.Power series, exponential and trigonometric functions, arg z, log z, az and their continuous branches. (Scope as in “Foundations of Complex Analysis” by Ponnusamy S., Chapter 1, (§1.1-§ 1.5),Chapter 2 (§ 2.2, §2.3), Chapter 3, (§3.1-§3.5), Chapter 4, (§4.9).)
Unit-II:
Complex Integration, line integral, Cauchy’s theorem for a rectangle, Cauchy’s theorem in a disc, index of a point with respect to a closed curve, Cauchy’s integral formula, Higher derivatives, Morrera’s theorem, Liouville’s theorem, fundamental theorem of Algebra. The general form of Cauchy’s theorem. (Scope as in “Foundations of Complex Analysis” by Ponnusamy S., Chapter 4, (§4.1-§ 4.8), Chapter 6 (§ 6.4, §6.6).”Complex Analysis” by L/ V. Ahlfors, Chapter 4 (§1, 2, 4.1 to 4.5and §5.1)
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