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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 e ect from the academic year 2017-2018
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MA MSc Mathematics Sem I II Syllabus Mumbai University by munotes
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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 o ered 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 o ered by the Department of Mathematics of the respective college.
The skill-based course shall be o ered 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.
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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 Di erentiable 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
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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
Di erential Equations
Course Code Unit Topics Credits L/W
PSMT204,PAMT204Unit I Picard's theorem
Unit II Ordinary di erential equations 5 4
Unit III Sturm-Liouville theory
Unit IV First order Partial
Di erential 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
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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 in nite
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, ak dimensional subspace of an n dimensional vector space is intersection of n k
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)
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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;;Nk 1ugwhereNis 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 1 1
2 2 1
2 2 01
A;0
@1 1 1
1 1 1
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 ,Di erential Equations, Dynamical Sys-
tems, Linear Algebra , Elsevier.
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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 ( ;de nition). A function
f:A !Rmis continuous if and only if for every open subset VRmthere is an open
subsetUofRnsuch thatf 1(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
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Theorem and applications.
Reference for Unit II: M. Spivak ,Calculus on Manifolds .
Unit III. Di erentiable functions (15 Lectures)
Di erentiable functions on Rn, the total derivative (Df)pof a di erentiable function f:
U !Rmatp2UwhereUis open in Rn;uniqueness of total derivative, di erentiability
implies continuity. (ref: [1] C.C. Pugh or [2] A. Browder )
Chain rule. Applications of chain rule such as:
1. Let
be a di erentiable curve in an open subset UofRn:Letf:U !Rbe a di erentiable
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 di erentiable at p2Rnif and only if each
fjis di erentiable 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 di erentiable.
(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 di erentiable 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)kMkq pk:(ref: [1] C.C. Pugh or [2] A. Browder )
Mean Value Theorem: Let f:U !Rmis di erentiable 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(q p)i:(ref: [3] T. Apostol )
Iff:U !Rmis di erentiable 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 de ned 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 de ned 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 di erentiable function de ned 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 di erentiable 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(z z0)nbe a power series, of radius of convergence R > 0:Then the function f
de ned byf(z) =Pan(z z0)nis holomorphic on the open ball jz z0j< R andf0(z) =P
n1nan(z z0)n 18 jz z0j< 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 C fz2C:z0gis a holomorphic function and l0(z) = 1=z:
Reference for Unit I:
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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
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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.
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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 in nite 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.
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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, Un the 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
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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 in nite 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 1 1corre-
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.
De nition of eld, characteristic of a eld, sub eld of a eld. A eld contains a sub eld
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
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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).
De nition 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 sub eld 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 sub eld
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[p 5]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)
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Compact spaces, limit point compact spaces, continuity and compactness, tube lemma, com-
pactness and product topology ( nite factors only), local compactness, one point compacti ca-
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
(de nition 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(bi ai));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) =b a:
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 sde ned on the measure space
(X;;)and properties, integral of a non-negative measurable function, Monotone convergence
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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 fde ned on the
measure space (X;;)is integrable (or summable) ifR
Xf+d<1andR
Xf d<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 :
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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 Di erential 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 Di erential 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 Di erential Equations and PDE stands for Partial Di erential
Equations.
Recommended Text Books :
1. Units I and II:
(a)E.A. Codington, N. Levinson ,Theory of ordinary di erential Equations ,
Tata McGraw-Hill, India.
(b)Hurewicz W. ,Lectures on ordinary di erential equations , M.I.T. Press.
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(c)Morris W. Hirsch and Stephen Smale ,Di erential Equations, Dynamical
Systems, Linear Algebra , Elsevier.
2. Unit III: G.F. Simmons ,Di erential equations with applications and historical notes ,
McGraw-Hill international edition.
3. Unit IV: Fritz John ,Partial Di erential 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 Scienti c.
3.Kai Lai Chung, Farid AitSahlia :Elementary Probability Theory , Springer Verlag.
Scheme of Examination
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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
?????????
20