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UNIVERSITY OF MUMBAI

Syllabus

for

M.A./M.Sc. Semester I & II (CBCS)

Program: M.A/M.Sc.

Course: Mathematics

with eect from the academic year 2017-2018

1

UNIVERSITY OF MUMBAI

Syllabus

for

M.A./M.Sc. Semester I & II (CBCS)

Program: M.A/M.Sc.

Course: Mathematics

with eect from the academic year 2017-2018

1

Preamble

The Board of Studies in Mathematics has prepared the syllabus of M.A./M.Sc. Semester I

& II (w.e.f. 2017-18) in the subject of Mathematics under the

Choice Based Credit System (CBCS).

A course on Complex Analysis is oered in the Semester VI of B.A./B.Sc. from the Academic

year 2016-17. Consequently, in this revised syllabus of

Semester I of M.A./M.Sc. Mathematics, there is a Course on Complex Analysis and a course

on Measure theory (Analysis II) in Semester II.

Skill-based Course: Each student of M.A./M.Sc. Mathematics (CBCS) Programme shall

complete a skill-based course oered by the Department of Mathematics of the respective college.

The skill-based course shall be oered on holidays. The skill-based course shall be of 100 hours

duration and attendance of 75% shall be compulsory for this course. There shall be internal

assessment and a student shall be given grades as A,B,C,D (A being the highest grade and D

being the lowest grade). A student shall be required to get minimum C grade to qualify for

the M.A./M.Sc. Degree. However the marks skill-based course shall not be considered for the

CGPA of M.A./M.Sc. degree. Separate fees shall be collected for the skill-based courses, the

quantum of which shall depend on the nature of the skill-based course.

The curriculum retains the current workload of Mathematics Departments.

2

M.A./M.Sc. Semester I and II

Choice Based Credit System (CBCS)

Semester I

Algebra I

Course Code Unit Topics Credits L/W

PSMT101,PAMT101Unit I Dual spaces

Unit II Determinants 5 4

Unit III Characteristic polynomial

Unit IV Bilinear forms

Analysis I

Course Code Unit Topics Credits L/W

PSMT102,PAMT102Unit I Euclidean space Rn

Unit II Riemann integration 5 4

Unit III Dierentiable functions

Unit IV Inverse function theorem , Implicit

Unit IV function theorem

Complex Analysis

Course Code Unit Topics Credits L/W

PSMT103,PAMT103Unit I Holomorphic functions

Unit II Contour integration, Cauchy-Gursat theorem 5 4

Unit III Properties of holomorphic functions

Unit IV Residue Calculus and Mobius transformations

Discrete Mathematics

Course Code Unit Topics Credits L/W

PSMT104,PAMT104Unit I Number Theory

Unit II Advanced Counting 5 4

Unit III Recurrence relations

Unit IV Polya's theory of counting

Set Theory and Logic

Course Code Unit Topics Credits L/W

PSMT105,PAMT105Unit I Introduction to logic

Unit II Sets and functions 4 4

Unit III Partial order

Unit IV Lattices

3

Semester II

Algebra II

Course Code Unit Topics Credits L/W

PSMT201,PAMT201Unit I Groups, group homomorphisms

Unit II Groups acting on sets and Syllow theorems 5 4

Unit III Rings, elds

Unit IV Divisibility in integral domains

Topology

Course Code Unit Topics Credits L/W

PSMT202,PAMT202Unit I Topological spaces

Unit II Connected Topological spaces 5 4

Unit III Compact Topological spaces

Compact metric spaces, Complete

Unit IV metric spaces

Analysis II

Course Code Unit Topics Credits L/W

PSMT203,PAMT203Unit I Measures

Unit II measurable functions and 5 4

integration of non-negative functions

Unit III Dominated convergence theorem

andL1;L2spaces.

Unit IV Signed measures, Radon-Nykodym theorem

Dierential Equations

Course Code Unit Topics Credits L/W

PSMT204,PAMT204Unit I Picard's theorem

Unit II Ordinary dierential equations 5 4

Unit III Sturm-Liouville theory

Unit IV First order Partial

Dierential Equations

Probability Theory

Course Code Unit Topics Credits L/W

PSMT205,PAMT205Unit I Basics of Probability

Unit II Probability measure 4 4

Unit III Random variables

Unit IV Limit theorems

4

Teaching Pattern for Semester I and II

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 ve courses.

SEMESTER I

All Results have to be done with proof unless otherwise stated.

PSMT101,PAMT101 ALGEBRA I

Unit I. Dual spaces (15 Lectures)

Para 1 and 2 of Unit Iare to be reviewed without proof (no question be asked).

1. Vector spaces over a eld, linear independence, basis for nite dimensional and innite

dimensional vector spaces and dimension.

2. Kernel and image, rank and nullity of a linear transformation, rank-nullity theorem (for nite

dimensional vector spaces), relationship of linear transformations with matrices, invertible linear

transformations. The following are equivalent for a linear map T:V!Vof a nite dimensional

vector space V:

1.Tis an isomorphism.

2. kerT=f0g:

3. Im (T) =V:

3. Linear functionals, dual spaces of a vector space, dual basis (for nite dimensional vector

spaces), annihilator Win the dual space Vof a subspace Wof a vector space Vand dimension

formula, akdimensional subspace of an ndimensional vector space is intersection of nk

many hyperspaces. Double dual Vof a Vector space Vand canonical embedding of Vinto

V:Vis isomorphic to VwhenVis of nite dimension. (ref:[1] Hoffman K and Kunze

R)

4. Transpose Ttof a linear transformation T:For nite dimensional vector spaces: rank

(Tt)=rankT;range(Tt) is the annihilator of kernel ( T), matrix representing Tta rank of a

matrix. (ref:[1] Hoffman K and Kunze R )

Unit II. Determinants (15 Lectures)

Determinants as alternating n-forms, existence and uniqueness , Laplace expansion of determi-

nant, determinants of products and transposes, determinants and invertible linear transforma-

tions, determinant of a linear transformation.

Reference for Unit II: [1] Hoffman K and Kunze R ,Linear Algebra .

Unit III. Characteristic polynomial (15 Lectures)

5

Eigen values and Eigen vectors of a linear transformation, Characteristic polynomial, Cayley-

Hamilton theorem, Minimal polynomial, Triangulable and diagonalizable linear operators, invari-

ant subspaces and simple matrix representation (for nite dimension). (ref: [5] N.S. Gopalkr-

ishnan & [3] Serge Lang )

Nilpotent linear transformations on nite dimensional vector spaces, index of a Nilpotent linear

transformation. Linear independence of fu;Nu;;Nk1ugwhereNis a nilpotent linear

transformation of index k2of a vector space Vandu2VwithNu6= 0:(Ref: [2]

I.N.Herstein )

For a nilpotent linear transformation Nof a nite dimensional vector space Vand for any

subspaceWofVwhich is invariant under N;there exists a subspace V1ofVsuch that

V=WV1:(Ref:[2] I.N.Herstein )

Computations of Minimum polynomials and Jordan Canonical Forms for 33-matrices through

examples of matrices such as0

@3 11

2 21

2 2 01

A;0

@1 1 1

111

1 1 01

A:(Ref:[6] Morris W. Hirsch

and Stephen Smale )

Unit IV. Bilinear forms (15 Lectures)

Para 1 of Unit IVis to be reviewed without proof (no question be asked).

1. Inner product spaces, orthonormal basis.

2. Adjoint of a linear operator on an inner product space, unitary operators, self adjoint opera-

tors, normal operators, spectral theorem for a normal operator on a nite dimensional complex

vector inner product space (ref:[1] Hoffman K and Kunze R ).

3. Bilinear form, rank of a bilinear form, non-degenerate bilinear form and equivalent statements

(ref:[1] Hoffman K and Kunze R ).

4. Symmetric bilinear forms, orthogonal basis and Sylvester's Law, signature of a Symmetric

bilinear form (ref:[4] Michael Artin ).

Recommended Text Books

[1 ]Hoffman K and Kunze R :Linear Algebra , Prentice-Hall India.

[2 ]I.N.Herstein :Topics in Algebra , Wiley-India.

[3 ]Serge Lang :Linear Algebra , Springer-Verlag Undergraduate Text in Mathematics.

[4 ]Michael Artin :Algebra , Prentice-Hall India.

[5 ]N.S. Gopalkrishnan :University Algebra , New Age International, third edition,

2015.

[6 ]Morris W. Hirsch and Stephen Smale ,Dierential Equations, Dynamical Sys-

tems, Linear Algebra , Elsevier.

6

PSMT102/PAMT102 ANALYSIS I

Unit I. Euclidean space Rn(15 Lectures)

Euclidean space Rn:inner producthx;yi=Pn

j=1xjyjofx= (x1;;xn);y= (y1;;yn)2Rn

and properties, norm kxk=qPn

j=1x2

jofx= (x1;;xn)2Rn;Cauchy-Schwarz inequality,

properties of the norm function kxkonRn:ref: [4] W. Rudin or [5] M. Spivak )

Standard topology on Rn: open subsets of Rn;closed subsets of Rn;interiorAand boundary

@Aof a subset AofRn:(ref: [5] M. Spivak )

Operator normkTkof a linear transformation T:Rn!Rm(kTk= supfkT(v)k:v2

Rn&kvk 1g) and its properties such as: For all linear maps S;T :Rn!RmandR:

Rm!Rk

1.kS+TkkSk+kTk;

2.kRSkkRkkSk;and

3.kcTk=jcjkTk(c2R):

(Ref: [1] C.C. Pugh or [2] A. Browder )

Compactness: Open cover of a subset of Rn, Compact subsets of Rn(A subsetKofRnis

compact if every open cover of Kcontains a nite subcover), Heine-Borel theorem (statement

only), the Cartesian product of two compact subsets of Rnis compact (statement only), every

closed and bounded subset of Rnis compact. Bolzano-Weierstrass theorem: Any bounded

sequence in Rnhas a converging subsequence.

Brief review of following three topics:

1. Functions and Continuity: Notation: ARnarbitrary non-empty set. A function f:

A!Rmand its component functions, continuity of a function ( ;denition). A function

f:A!Rmis continuous if and only if for every open subset VRmthere is an open

subsetUofRnsuch thatf1(V) =A\U:

2. Continuity and compactness: Let KRnbe a compact subset and f:K!Rmbe any

continuous function. Then fis uniformly continuous, and f(K)is a compact subset of

Rm:

3. Continuity and connectedness: Connected subsets of Rare intervals. If f:E!Ris

continuous where ERnandEis connected , then f(E)Ris connected.

Unit II. Riemann Integration (15 Lectures)

Riemann Integration over a rectangle in Rn;Riemann Integrable functions, Continuous func-

tions are Riemann integrable, Measure zero sets, Lebesgues Theorem (statement only), Fubini's

7

Theorem and applications.

Reference for Unit II: M. Spivak ,Calculus on Manifolds .

Unit III. Dierentiable functions (15 Lectures)

Dierentiable functions on Rn, the total derivative (Df)pof a dierentiable function f:

U!Rmatp2UwhereUis open in Rn;uniqueness of total derivative, dierentiability

implies continuity. (ref: [1] C.C. Pugh or [2] A. Browder )

Chain rule. Applications of chain rule such as:

1. Let
be a dierentiable curve in an open subset UofRn:Letf:U!Rbe a dierentiable

function and let g(t) =f(
(t)):Theng0(t) =h(rf)(
(t));
0(t)i:

2. Computation of total derivatives of real valued functions such as

(a) the determinant function det (X), (X2Mn(R)),

(b) the Euclidean inner product function hx;yi;((x;y)2RnRn):

(ref: (ref: [5] M. Spivak & [4] W. Rudin & )

Results on total derivative:

1. Iff:Rn!Rmis a constant function, then (Df)p= 08p2Rn:

2. Iff:Rn!Rmis a linear map, then (Df)p=f8p2Rn:

3. A function f= (f1;f2;;fm) :Rn!Rmis dierentiable at p2Rnif and only if each

fjis dierentiable at p2Rn;and(Df)p= ((Df 1)p;(Df 2)p;;(Dfm)p):

(ref: [5] M. Spivak )

Partial derivatives, directional derivative (Duf)(p)of a function fatpin the direction of the

unit vector, Jacobian matrix, Jacobian determinant. Results:

1. If the total derivative of a map f= (f1;;fm) :U!Rm(Uopen subset of Rn) exists

atp2U;then all the partial derivatives@fi

@xjexist atp:

2. If all the partial derivatives@fi

@xjof a mapf= (f1;;fm) :U!Rm(Uopen subset of

Rn) exist and are continuous on U;thenfis dierentiable.

(ref:[4] W. Rudin )

Derivatives of higher order, Ck-functions, C1-functions. (ref: [3] T. Apostol )

Unit IV. Inverse function theorem, Implicit function theorem (15 Lectures)

Theorem (Mean Value Inequality): Suppose f:U!Rmis dierentiable on an open subset U

ofRnand there is a real number such that k(Df)xkM8x2U:If the segment [p;q]is

contained in U;thenkf(q)f(p)kMkqpk:(ref: [1] C.C. Pugh or [2] A. Browder )

Mean Value Theorem: Let f:U!Rmis dierentiable on an open subset UofRn:Letp;q2U

such that the segment [p;q]is contained in U:Then for every vector v2Rnthere is a point

x2[p;q]such thathv;f(q)f(p)i=hv;(Df)x(qp)i:(ref: [3] T. Apostol )

Iff:U!Rmis dierentiable on a connected open subset UofRnand(Df)x= 08x2U;

thenfis a constant map.

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Taylor expansion for a real valued Cm-function dened on an open subset of Rn;stationary

points (critical points), maxima, minima, saddle points, second derivative test for extrema at

a stationary point of a real valued C2-function dened on an open subset of Rn:(ref: [3] T.

Apostol )

Contraction mapping theorem. Inverse function theorem , Implicit function theorem. (ref: [2]

A. Browder )

Recommended Text Books

[1 ]C.C. Pugh :Real mathematical analysis , Springer UTM.

[2 ]A. Browder :Mathematical Analysis, An Introduction , Springer.

[3 ]T. Apostol :Mathematical Analysis , Narosa.

[4 ]W. Rudin :Principles of Mathematical Analysis , Mcgraw-Hill India.

[5 ]M. Spivak :Calculus on Manifolds , Harper-Collins Publishers.

PSMT103/PAMT103 COMPLEX ANALYSIS

Unit I. Holomorphic functions (15 Lectures)

Note: A complex dierentiable function dened on an open subset of Cis called a holomor-

phic function .

Review: Complex Numbers, Geometry of the complex plane, Riemann sphere, Complex se-

quences and series, Sequences and series of functions in C;, Weierstrass's M-test, Uniform

convergence, Complex dierentiable functions, Cauchy-Riemann equations (no questions be

asked).

Ratio test and root test for convergence of a series of complex numbers. Complex Power series,

radius of convergence of a power series, Cauchy-Hadamard formula for radius of convergence

of a power series. Examples of convergent power series such as exponential series, cosine series

and sine series, and the basic properties of the functions ez;cosz;sinz:Abel's theorem: LetP

n0an(zz0)nbe a power series, of radius of convergence R > 0:Then the function f

dened byf(z) =Pan(zz0)nis holomorphic on the open ball jzz0j< R andf0(z) =P

n1nan(zz0)n18 jzz0j< R: Applications of Abel's theorem such as exp0(z) =

expz;cos0(z) =sinz;sin0(z) = cosz;(z2C):

Chain Rule. A basic result: Let

1;

2be open subsets of C:Supposef:

1!Cis a

holomorphic function with f0(z)6= 08z2

1andg:

2!Cbe a continuous function

such thatg(

2)

1andf(g(w)) =w8w2

2:Thengis a holomorphic function on

2

andg0(w) =1

f0(g(w))8w2

2:Application: The logarithm as the inverse of exponential (i.e.

8w6= 0 inC;log(w) :=fz2Cjez=wg);branches of logarithm, the principle branch l(z)of

the logarithmic function on Cfz2C:z0gis a holomorphic function and l0(z) = 1=z:

Reference for Unit I:

9

1.A. R. Shastri :An introduction to complex analysis , Macmillan.

2.Serge Lang :Complex Analysis .

3.L. V. Ahlfors :Complex analysis , McGraw Hill .

4.R. Remmert :Theory of complex functions , Springer.

Unit II. Contour integration, Cauchy-Gursat theorem (15 Lectures)

Contour integration, Cauchy-Goursat Theorem for a rectangular region or a triangular region.

Primitives. Existence of primitives: If fis holomorphic on a disc U;then it has a primitive on

Uand the integral of falong any closed contour in Uis0:Local Cauchy's Formula for discs

(without proof), Power series representation of holomorphic functions (without proof), Cauchy's

estimates, entire functions, Liouville's theorem, Morera's theorem, the Fundamental theorem of

Algebra.

Reference for unit II:

1.A. R. Shastri :An introduction to complex analysis , Macmillan.

2.Serge Lang :Complex Analysis .

3.L. V. Ahlfors :Complex analysis , McGraw Hill .

4.R. Remmert :Theory of complex functions , Springer.

Unit III. Holomorphic functions and their properties (15 Lectures)

Cauchys theorem (homotopy version or homology version) The index (winding number) of a

closed curve, Cauchy integral formula.

Zeros of holomorphic functions, Identity theorem. Counting zeros; Open Mapping Theorem,

Maximum modulus theorem, Schwarz's lemma. Every automorphism of unit disc with center 0

inCis a rotation.

Isolated singularities: removable singularities and Removable singularity theorem, poles and

essential singularities. Laurent Series development. Casorati-Weierstrass's theorem.

Reference for Unit III:

1.J. B. Conway ,Functions of one Complex variable , Springer.

2.L. V. Ahlfors :Complex analysis , McGraw Hill .

3.R. Remmert :Theory of complex functions , Springer.

Unit IV Residue calculus and Mobius transformations (15 Lectures)

Residue Theorem and evaluation of standard types of integrals by the residue calculus method.

Argument principle. Rouch e 's theorem.

Conformal mappings. If f:G!Cis a holomorphic function on the open subset GofCand

f0(z)6= 08z2G;thenfis a conformal map. Mobius transformations (fractional linear

transformation or linear transformation). Any Mobius transformation which xes three distinct

10

points is necessarily the identity map. Cross ratio (z1;z2;z3;z4)of four points z1;z2;z3;z4:

A Mobius transformation preserves cross ratio. Cross ratio (z1;z2;z3;z4)is real if and only if

the four points z1;z2;z3;z4lie on a circle. A Mobius transformation takes circles onto circles.

Symmetry, symmetry principle and applications to the construction of Cayley map.

Reference for Unit IV:

1.J. B. Conway ,Functions of one Complex variable , Springer.

2.L. V. Ahlfors :Complex analysis , McGraw Hill .

3.R. Remmert :Theory of complex functions , Springer.

PSMT104/PAMT104 DISCRETE MATHEMATICS

Unit I. Number theory (15 Lectures)

Divisibility, Linear Diophantine equations, Cardano's Method, Congruences, Quadratic residues,

Arithmetic functions,

Types of occupancy problems, distribution of distinguishable and indistinguishable objects into

distinguishable and indistinguishable boxes (with condition on distribution) Stirling numbers of

second and rst kind. Selections with Repetitions.

Unit II. Advanced counting (15 Lectures)

Pigeon-hole principle, generalized pigeon-hole principle and its applications, Erdos- Szekers the-

orem on monotone subsequences, A theorem of Ramsey. Inclusion Exclusion Principle and its

applications. Derangement. Permutations with Forbidden Positions, Restricted Positions and

Rook Polynomials.

Unit III.Recurrence Relations (15 Lectures)

The Fibonacci sequence, Linear homogeneous recurrence relations with constant coecient.

Proof of the solution in case of distinct roots and statement of the theorem giving a general

solution (in case of repeated roots), Iteration and Induction. Ordinary generating Functions,

Exponential Generating Functions, algebraic manipulations with power series, generating func-

tions for counting combinations with and without repetitions, exponential generating function

for bell numbers, applications to counting, use of generating functions for solving recurrence

relations.

Unit IV. Polyas Theory of counting (15 Lectures)

Equivalence relations and orbits under a permutation group action. Orbit stabiliser theorem,

Burnside Lemma and its applications, Cycle index, Polyas Formula, Applications of Polyas For-

mula.

Recommended Text Books

1.D. M. Burton ,Introduction to Number Theory , McGraw-Hill.

11

2.Nadkarni and Telang , Introduction to Number Theory

3.V. Krishnamurthy :Combinatorics: Theory and applications , Aliated East-West

Press.

4.Richard A. Brualdi :Introductory Combinatorics , Pearson.

5.A. Tucker :Applied Combinatorics , John Wiley & Sons.

6.Norman L. Biggs :Discrete Mathematics , Oxford University Press.

7.Kenneth Rosen :Discrete Mathematics and its applications , Tata McGraw Hills.

8.Sharad S. Sane ,Combinatorial Techniques , Hindustan Book Agency, 2013.

PSMT105/PAMT105 SET THEORY AND LOGIC

Unit I. Introduction to logic (15 Lectures)

Statements, Propositions and Theorems, Truth value, Logical connectives and Truth tables,

Conditional statements, Logical inferences, Methods of proof, examples.

Basic Set theory: Union, intersection and complement, indexed sets, the algebra of sets, power

set, Cartesian product, relations, equivalence relations, partitions, discussion of the example

congruence modulo mrelation on the set of integers.

Unit II. Sets and functions (15 Lectures)

Functions, composition of functions, surjections, injections, bijections, inverse functions, Cardi-

nality Finite and innite sets, Comparing sets, Cardinality, jAj

RR:

Unit III. Partial order (15 Lectures)

Order relations, order types, partial order, total order, Well ordered sets, Principle of Math-

ematical Induction, Russells paradox, Statements of the Axiom of Choice, the Well Ordering

Principle, Zorns lemma, applications of Zorns lemma to maximal ideals and to bases of vector

spaces.

Unit IV. Lattices (15 Lectures)

Mobius inversion formula on a partially ordered set, Hasse Diagrams of a partially ordered set,

Lattices, Distributive and Modular Lattices, complements, Boolean Algebra, Boolean expres-

sions, Only elementary Applications.

Recommended Text Books

1.Robert R. Stoll :Set theory and logic , Freeman & Co.

12

2.James Munkres :Topology , Prentice-Hall India;

3.J. F. Simmons ,Introduction to Topology and real analysis .

4.Richard A. Brualdi :Introductory Combinatorics , Pearson.

5.Kenneth Rosen :Discrete Mathematics and its applications , Tata McGraw Hills.

6.Larry J. Gerstein :Introduction to mathematical structures and proofs , Springer.

7.Joel L. Mott, Abraham Kandel, Theodore P. Baker :Discrete mathematics

for Computer scientists and mathematicians , Prentice-Hall India.

8.Robert Wolf :Proof, logic and conjecture, the mathematicians toolbox , W. H.

Freemon.

SEMESTER II

All Results have to be done with proof unless otherwise stated.

PSMT201/PAMT201 ALGEBRA II

Unit I. Groups, group Homomorphisms (15 lectures)

Review : Groups, subgroups, normal subgroups, center Z(G)of a group. The kernel of a

homomorphism is a normal subgroup. Cyclic groups. Lagrange's theorem. The product set

HK :=fhkjh2H&k2Kgof two subgroups of a group G:Examples of groups such as

Permutation groups, Dihedral groups, Matrix groups, Unthe group of units of Zn(no questions

be asked).

Quotient groups. First Isomorphism Theorem and the following two applications (reference:

Algebra byMichael Artin )

1. Let Cbe the multiplicative group of non-zero complex numbers and R>0be the multi-

plicative group of positive real numbers. Then the quotient group C=Uis isomorphic to

R>0:

2. The quotient group GLn(R)=SL n(R)is isomorphic to the multiplicative group of non-zero

real numbers R:

Second and third isomorphism theorems for groups, applications.

Product of groups. The group ZmZnis isomorphic to Zmnif and only if the g.c.d. of m

andnis1:Internal direct product (A groupGis an internal direct product of two normal

subgroupsH;K ifG=HK and everyg2Gcan be written as g=hkwhereh2H;k2Kin

a unique way). If H;K are two nite subgroups of a group, then jHKj=jHjjKj

jH\Kj:IfH;K are

two normal subgroups of a group Gsuch thatH\K=fegandHK =G;thenGis internal

13

direct product of HandK:If a groupGis an internal direct product of two normal subgroups

HandK;thenGis isomorphic to HK:(Ref: Algebra byMichael Artin )

Automorphisms of a group. If Gis a group, then A(G);the set of all automorphisms of G, is a

group under composition. If Gis a nite cyclic group of order r;thenA(G)is isomorphic to Ur;

the groups of all units of Zrunder multiplication modulo r:For the innite cyclic group Z;A(Z)

is isomorphic to Z2:Inner automorphisms of a group. Topics in Algebra byI.N.Herstein ).

Structure theorem of Abelian groups(statement only) and applications (ref: A rst Course in

Abstract Algebra byJ. B. Fraleigh ,)

Unit II. Groups acting on sets, Syllow theorems

Center of a group, centralizer or normalizer N(a)of an element a2G;conjugacy class C(a)of

ainG:In nite group G;jC(a)j=o(G)=o(N(a))ando(G) =Po(G)

o(N(a))where the summation

is over one element in each conjugacy class, applications such as: (1) If Gis a group of order

pnwherepis a prime number, then Z(G)6=feg:(2) Any group of order p2;wherepis a prime

number, is Abelian (Reference: Topics in Algebra byI.N.Herstein ).

Groups acting on sets, Class equation, Cauchy's theorem: If pis a prime number and pjo(G)

whereGis nite group,then Ghas an element of order p:(Reference: Topics in Algebra by

I.N.Herstein ).

p-groups, Syllow's theorems and applications:

1. There are exactly two isomorphism classes of groups of order 6:

2. Any group of order 15is cyclic

(Reference for Syllow's theorems and applications: Algebra byMichael Artin ).

Unit III. Rings, Fields (15 lectures)

Review: Rings (with unity), ideals, quotient rings, prime ideals, maximal ideals, ring homomor-

phisms, characteristic of a ring, rst and second Isomorphism theorems for rings, correspondence

theorem for rings (If f:R!R0is a surjective ring homomorphism, then there is a 11corre-

spondence between the ideals of Rcontaining the ker fand the ideals of R0). Integral domains,

construction of the quotient eld of an integral domain. (no questions be asked).

For a commutative ring Rwith unity:

1. An ideal MofRis a maximal ideal if and only if the quotient ring R=M is a eld.

2. An ideal NofRis a prime ideal if and only if the quotient ring R=M is an integral

domain.

3. Every maximal ideal is a prime ideal.

Denition of eld, characteristic of a eld, subeld of a eld. A eld contains a subeld

isomorphic to ZporQ:

Polynomial ring F[X]over a eld, irreducible polynomials over a eld. Prime ideals, and maximal

ideals of a Polynomial ring F[X]over a eld F:A non-constant polynomial p(X)is irreducible

14

in a polynomial ring F[X]over a eld Fif and only if the ideal (p(X))is a maximal ideal of

F[X]:Unique Factorization Theorem for polynomials over a eld (statement only).

Denition of eld extension, algebraic elements, minimal polynomial of an algebraic element,

extension of a eld obtained by adjoining one algebraic element. Kronecker's theorem: Let F

be any eld and let f(X)2F[X]be such that f(X)has no root in F:Then there exists a eld

EcontainingFas a subeld such that fhas a root in E:Application of Kronecker's theorem:

LetFbe any eld and let f(X)2F[X]:Then there exists a eld EcontainingFas a subeld

such thatf(X)factorises completely into linear factors in E[X]:(reference: Rings, elds and

Groups, An Introduction to Abstract Algebra byR.B.J.T. Allenby ).

Finite elds: A nite eld of characteristic pcontains exactly pnelements for some n2N:

Existence result for nite elds: For every prime number pand positive integer n;there exists

a led with exactly pnelements (reference: Rings, elds and Groups, An Introduction to

Abstract Algebra byR.B.J.T. Allenby ).

Unit IV. Divisibility in integral domains (15 lectures)

Prime elements, irreducible elements, Unique Factorization Domains, Principle Ideal Domains,

Gauss's lemma, Z[X]is a UFD, irreducibility criterion, Eisenstein's criterion, Euclidean domains.

Z[p5]is not a UFD.

Reference for Unit IV: Michael Artin :Algebra , Prentice-Hall India.

Recommended Text Books

1.Michael Artin :Algebra , Prentice-Hall India.

2.I.N.Herstein :Topics in Algebra , Wiley-India.

3.R.B.J.T. Allenby :Rings, elds and Groups, An Introduction to Abstract Algebra ,

Elsevier (Indian edition).

4.J. B. Fraleigh ,A rst Course in Abstract Algebra , Narosa.

5.David Dummit, Richard Foot :Abstract Algebra , Wiley-India.

PSMT202/PAMT202 TOPOLOGY

Unit I. Topological spaces (15 Lectures)

Topological spaces, basis, sub-basis, product topology (nite factors only), subspace topology,

closure, interior, continuous functions, T1;T2spaces, quotient spaces.

Unit II. Connected topological spaces (15 Lectures)

Connected topological spaces, path-connected topological spaces, continuity and connected-

ness, Connected components of a topological space, Path components of a topological space.

Countability Axioms, Separation Axioms, Separable spaces, Lindelo spaces, Second countable

spaces.

Unit III. Compact topological spaces (15 Lectures)

15

Compact spaces, limit point compact spaces, continuity and compactness, tube lemma, com-

pactness and product topology (nite factors only), local compactness, one point compactica-

tion. A compact T2space is regular and normal space.

Unit IV. Compact metric spaces, Complete metric spaces (15 Lectures)

Complete metric spaces, Completion of a metric space, total boundedness , compactness in Met-

ric spaces, sequentially compact metric spaces, uniform continuity, Lebesgue covering lemma.

Recommended Text Books

1.James Munkres :Topology , Pearson.

1.George Simmons :Topology and Modern Analysis , Tata Mcgraw-Hill.

2.M.A.Armstrong :Basic Topology , Springer UTM.

PSMT203/PAMT203 ANALYSIS II

Unit I. Measures (15 lectures)

Outer measure on a setX;-measurable subsets of X( A subsetEof a setXwith outer

measureis said to be -measurable if (A) =(A\E) +(A\(XnE)8AX

(denition due to Carath eodory)), the collection of all-measurable subsets of Xform a

-algebra, measure space (X;;):

Volume(I)of any rectangle in Rd(for the interval I= d

i=1(ai;bi)ofRd;

(I) = d

i=1(biai));Lebesgue's Outer measure minRdand results:

1. Lebesgue's Outer measure mis translation invariant.

2. LetA;B be any two subsets of Rdwithd(A;B)>0:Thenm(A[B) =m(A)+m(B):

3. For any bounded interval I= (a;b)ofR; m(I) =ba:

4. For any interval IofRd; m(I) =(I):

The-algebra Mof all Lebesgue measurable subsets of Rd;the Lebesgue measure m=m

jM

and the measure space (Rd;M;m):Existence of a subset of Rwhich is not Lebesgue measurable.

Reference for unit I: Andrew Browder ,Mathematical Analysis, An Introduction , Springer

Undergraduate Texts in Mathematics.

Unit II. Measurable functions and integration of non-negative functions (15

lectures)

Measurable functions on (X;;), simple functions, properties of measurable functions. If

f0is a measurable function, then there exists a monotone increasing sequence (sn)of

non-negative simple measurable functions converging to pointwise to the function f:

IntegralR

Xsd of a non-negative simple measurable function sdened on the measure space

(X;;)and properties, integral of a non-negative measurable function, Monotone convergence

16

theorem. If f0andg0are measurable functions, thenR

X(f+g)d=R

Xfd +R

Xgd:

Reference for unit II:

1.H.L. Royden ,Real Analysis by , PHI.

2.Andrew Browder ,Mathematical Analysis, An Introduction , Springer Undergraduate

Texts in Mathematics.

Unit III. Dominated convergence theorem and L1(); L2()spaces (15 lectures)

Integrable functions with respect to a measure (A measurable function fdened on the

measure space (X;;)is integrable (or summable) ifR

Xf+d<1andR

Xfd<1) and

linearity properties.

Fatous lemma, Dominated convergence theorem, Completeness of L1();L2():

Lebesgue and Riemann integrals: A bounded real valued function on [a;b]is Riemann integrable

if and only if it is continuous at almost every point of [a;b]; in this case, its Riemann integral

and Lebesgue integral coincide.

Reference for unit III:

1.H.L. Royden ,Real Analysis by , PHI.

2.Andrew Browder ,Mathematical Analysis, An Introduction , Springer Undergraduate

Texts in Mathematics.

Unit IV. Signed measures, Radon-Nykodym theorem (15 lectures)

Borel- algebra of Rd:Any closed subset and any open subset of Rdis Lebesgue measurable.

Every Borel set in Rdis Lebesgue measurable. For any bounded Lebesgue measurable subset

EofRd;given any any >0there exist a compact set Kand open set UinRdsuch that

KEU&m(UnK)<: For any Lebesgue measurable subset EofRd;there exist Borel

setsF;G inRdsuch thatFEG&m(EnF) = 0 =m(GnE):

Complex valued Lebesgue measurable functions on Rd;Lebesgue integral of complex valued

measurable functions, Approximation of Lebesgue integrable functions by continuous functions

with compact support.

Signed measures, positive measurable sets and negatives measurable sets for a signed measure.

Notion of Absolutely continuity << of a signed measure with respect to a positive measure

;Hahn Decomposition theorem, Jordan Decomposition of a signed measure, examples.

Radon-Nykodym theorem and Radon-Nykodym derivative.

Reference for unit IV:

1.H.L. Royden ,Real Analysis by , PHI.

2.Andrew Browder ,Mathematical Analysis, An Introduction , Springer Undergraduate

Texts in Mathematics.

See also Measure, Integral and probability byM. Capinski and E. Kopp , Springer (SUMS)

for a proof of the Radon-Nykodym theorem.

Recommended Text Books :

17

1.Andrew Browder ,Mathematical Analysis, An Introduction , Springer Undergraduate

Texts in Mathematics.

2.H.L. Royden ,Real Analysis , PHI.

3.Walter Rudin ,Real and Complex Analysis , McGraw-Hill India, 1974.

PSMT204/PAMT204 DIFFERENTIAL EQUATIONS

Unit I. Picards Theorem (15 Lectures)

Existence and Uniqueness of solutions to initial value problem of rst order ODE- both au-

tonomous, non-autonomous (Picard's Theorem), Picard's scheme of successive Approximations,

system of rst order linear ODE with constant coecients and variable coecients, reduction

of ann-th order linear ODE to a system of rst order ODE.

Unit II. Ordinary Dierential Equations (15 Lectures)

Existence and uniqueness results for an n-th order linear ODE with constant coecients and

variable coecients, linear dependence and independence of solutions of a homogeneous n-th

order linear ODE, Wronskian matrix, Lagrange's Method (variation of parameters), algebraic

properties of the space of solutions of a non-homogeneous n-th order linear ODE.

Unit III. Sturm-Liouville theory (15 Lectures)

Solutions in the form of power series for second order linear equations of Legendre and Bessel,

Legendre polynomials, Bessel functions.

Sturm- Liouville Theory: Sturm-Liouville Separation and comparison Theorems, Oscillation prop-

erties of solutions, Eigenvalues and eigenfunctions of Sturm-Liouville Boundary Value Problem,

the vibrating string.

Unit IV. First Order Partial Dierential Equation (15 Lectures)

First order quasi-linear PDE in two variables: Integral surfaces, Characteristic curves, Cauchys

method of characteristics for solving First order quasi-linear PDE in two variables.

First order non-linear PDE in two variables, Characteristic equations, Characteristic strip, Cauchy

problem and its solution for rst order non linear PDE in two variables.

Note : ODE stands for Ordinary Dierential Equations and PDE stands for Partial Dierential

Equations.

Recommended Text Books :

1. Units I and II:

(a)E.A. Codington, N. Levinson ,Theory of ordinary dierential Equations ,

Tata McGraw-Hill, India.

(b)Hurewicz W. ,Lectures on ordinary dierential equations , M.I.T. Press.

18

(c)Morris W. Hirsch and Stephen Smale ,Dierential Equations, Dynamical

Systems, Linear Algebra , Elsevier.

2. Unit III: G.F. Simmons ,Dierential equations with applications and historical notes ,

McGraw-Hill international edition.

3. Unit IV: Fritz John ,Partial Dierential Equations , Springer.

PSMT205/PAMT205 PROBABILITY THEORY

Unit I. Probability basics (15 Lectures)

Modelling Random Experiments: Introduction to probability, probability space, events.

Classical probability spaces: uniform probability measure, elds, nite elds, nitely additive

probability, Inclusion-exclusion principle, -elds,-elds generated by a family of sets, -eld

of Borel sets, Limit superior and limit inferior for a sequence of events.

Unit II. Probability measure (15 Lectures)

Probability measure, Continuity of probabilities, First Borel-Cantelli lemma, Discussion of Lebesgue

measure on -eld of Borel subsets of assuming its existence, Discussion of Lebesgue integral

for non-negative Borel functions assuming its construction.

Discrete and absolutely continuous probability measures, conditional probability, total probability

formula, Bayes formula, Independent events.

Unit III. Random variables (15 Lectures)

Random variables, simple random variables, discrete and absolutely continuous random variables,

distribution of a random variable, distribution function of a random variable, Bernoulli, Binomial,

Poisson and Normal distributions, Independent random variables, Expectation and variance of

random variables both discrete and absolutely continuous.

Unit IV. Limit Theorems (15 Lectures)

Conditional expectations and their properties, characteristic functions, examples, Higher mo-

ments examples, Chebyshev inequality, Weak law of large numbers, Convergence of random

variables, Kolmogorov strong law of large numbers (statement only), Central limit theorem

(statement only).

Recommended Text Books :

1.M. Capinski, Tomasz Zastawniak : Probability Through Problems.

2.J. F. Rosenthal :A First Look at Rigorous Probability Theory , World Scientic.

3.Kai Lai Chung, Farid AitSahlia :Elementary Probability Theory , Springer Verlag.

Scheme of Examination

19

The scheme of examination for the syllabus of Semesters I & II of M.A./M.Sc. Programme

(CBCS) in the subject of Mathematics will be as follows. There shall be a Semester-end Exter-

nal Theory examination with 100 marks to be conducted by the University.

(i) Duration:- Examination shall be of 3 Hours duration.

(ii) Theory Question Paper Pattern:-

1. There shall be ve questions each of 20 marks.

2. On each unit there will be one question and the fth one will be based on entire syllabus.

3. All questions shall be compulsory with internal choice within each question.

4. Each question may be subdivided into sub-questions a, b, c, and the allocation of marks

depend on the weightage of the topic.

5. Each question will be of 30 marks when marks of all the sub-questions are added (including

the options) in that question.

Questions Marks

Q1 Based on Unit I 20

Q2 Based on Unit II 20

Q3 Based on Unit III 20

Q4 Based on Unit IV 20

Q5 Based on Units I,II,III& IV 20

Total Marks 100

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