Quoteints
and
momomorphisms
of
modules,
simple
modules.
Modules
over
principal
ideal
domains.
Invariant
subspaces
for
a
linear
transformation,
simultaneous
triangulation
and
diagonalization,
Jordan
decomposition
of
a
linear
transformation.
Rational
and
Jordan
canonical
forms
of
matrices.
Inner
product
spaces,
The
Gram-Schmidt
orthogonalization.
orthogonal
complements.
The
adjoint
of
a
linear
operator,
normal
and
self-adjoint
operators,
Unitary
and
orthogonal
operators,
orthogonal
projections
and
spectral
Theorem.
Recommended Reading
M.
Artin,
Algebra,
Prentice-Hall
of
India,
New
Delhi
(1994).
K.
R.
Hoffman
and
R.
A.
Kunze,
LinearAlgebra,
Pearson
Education
(1971).
C.
Musili,
IntroductiontoRingsandModules,
Narosa
Publishing
House
(1994).
N.
S.
Gopalakrishnan,
UniversityAlgebra,
New
Age
International
(1986).
Nathan
Jacobson,
BasicAlgebra
Vol.
I,
Dover
Publications
(2009).
I.
S.
Luthar
and
I.
B.
S.
Passi,
Algebra
Vol.
II
&
III,
Narosa
Publishing
House,
New
Delhi
(2002).
MTH303: Ordinary and partial differential equations
[Cr:4,Lc:3,Tt:1,Lb:0]
Course Outline
Fundamental
existence
and
uniqueness
theorem,
dependence
of
solution
on
initial
conditions
and
parameters.
System
of
first
order
linear
equations,
general
solution
of
homogeneous
linear
systems,
fundamental
matrix,
non-homogeneous
linear
systems,
linear
systems
with
constant
coefficients.
Sturm
theory,
Comparison
theorem,
boundary
value
problems,
Green’s
function,
adjoint
system,
regular
Sturm-Liouville
systems,
eigenvalues
and
eigenfunctions.
First
order
partial
differential
equations
(PDEs),
linear
and
quasi-linear
equations,
general
first
order
PDE
for
a
function
of
two
variables.
Second
order
PDEs,
characteristic
for
linear
and
quasi-linear
second
order
equations.
The
Cauchy
problem,
Cauchy-Koualevski
theorem.
The
method
of
separation
of
variables,
the
Laplace
equation,
the
heat
equation,
the
wave
equation.
Recommended Reading
Earl
A.
Coddington
and
Norman
Levinson,
TheoryofOrdinaryDifferentialEquations,
Tata
McGraw-Hill
Publishing
Company
(1998).
Shepley
L.
Ross,
DifferentialEquations,
Wiley
(1984).
Fritz
John,
PartialDifferentialEquations,
Springer
(1981).
Yehuda
Pinchover
and
Jacob
Rubinstein,
AnIntroductiontoPartialDifferentialEquations,
Cambridge
University
Press
(2005).
Ravi
P.
Agarwal
and
Donal
O’Regan,
OrdinaryandPartialDifferentialEquations,
Springer
(2008).
MTH304: Topology
[Cr:4,Lc:3,Tt:1,Lb:0]
Course Outline
Basic
set
theory,
countable
and
uncountable
sets,
axiom
of
choice,
well-ordering,
Zorn’s
Lemma
and
their
equivalence.
Metric
spaces,
neighbourhoods
and
continuity.
Topological
spaces,
open
sets,
closed
sets,
examples,
order
topology,
subspace
topology,
product
topology,
quotient
topology.
Connectedness
and
compactness,
compact
and
connected
subspaces
of
ℝ,
local
connectedness
and
local
compactness.
Limit
points,
convergence
of
nets
in
topological
spaces.
Lars
V.
Ahlfors,
ComplexAnalysis,
McGraw-Hill
(1979).
John
B.
Conway,
FunctionsofOneComplexVariable,
Springer
(Graduate
Texts
in
Mathematics
Vol.
11)
(1978).
Theodore
W.
Gamelin,
ComplexAnalysis,
Springer
(2003).
Reinhold
Remmert,
TheoryofComplexFunctions,
Springer,
(Graduate
Texts
in
Mathematics/Reading
in
Mathematics
Vol.
122)
(1998).
Elias
Stein
and
Rami
Shakarchi,
ComplexAnalysis,
Princeton
University
Press
(Princeton
Lectures
in
Analysis)
(2003).
W.
Tutschke
and
H.
L.
Vasudeva,
AnIntroductiontoComplexAnalysis:ClassicalandModernApproaches,
Chapman
&
Hall/CRC
(2005).
MTH307: Discrete mathematics
[Cr:4,Lc:3,Tt:1,Lb:0]
Course Outline
Sets,
relations,
disjunctive
and
conjunctive
normal
forms,
well-ordering
principle,
representation
of
sets,
relations
and
numbers
on
the
computer.
Posets,
trees,
Boolean
algebras,
lattices,
representation
of
relations
as
digraphs
and
Boolean
matrices.
Floors
and
ceilings,
applications,
recurrences,
sums
involving
floor
and
ceiling
functions,
summation
formula,
polynomial
functions
with
with
integer
values.
Finite
fields
and
mod
p-arithmetic.
Probability
generating
functions,
flipping
coins,
hashing.
Combinatorics:
Counting
principles,
permutation
groups
and
applications,
Ramsey
theory
by
examples,
Recurrence
relations,
difference
equations.
Additional Topics
Graph
Theory:
Graphs,
digraphs,
trees;
Euler’s
formula
and
graph
colouring,
transitive
closure
and
connectedness,
Warshall’s
algorithm,
Eulerian
and
Hamiltonian
circuits,
Algorithms
for
tree
traversing.
Generating
Functions,
Generating
functions,
solving
recurrences,
specific
generating
functions
(from
number
theory
and
combinatorics),
convolutions,
exponential
generating
functions,
Dirichlet
generation
functions.
Recommended Reading
Ronald
L.
Graham,
Donald
E.
Knuth
and
Oren
Patashnik,
ConcreteMathematics,AFoundationforComputerScience,
Addison
Wesley
(1994).
John
Truss,
DiscreteMathematicsforComputerScientists,
Addison
Wesley
(1998).
R.
G.
Dromey,
HowtoSolveitbyComputer,
Prentice-Hall
(1982).
Kees
Doets
and
Jan
vak
Eijck,
TheHaskellRoadtoLogic,MathsandProgramming,
College
Publications
(2004).
John
O’Donnell,
DiscreteMathematicsUsingaComputer,
Springer
(2006).
MTH308: Groups and fields
[Cr:4,Lc:3,Tt:1,Lb:0]
Course Outline
Recapitulation:
Cayley’s
Theorem,
Class
equations,
Group
of
prime
power
order.
Sylow
Theorems,
Groups
of
order
≤ 8.
Free
groups,
generators
and
relations,
commutators,
and
derived
and
central
series
of
subgroups.
Free
Abelian
groups,
further
examples
of
groups
of
small
order.
Field
extensions,
algebraic
extensions,
perfect
fields,
separable
and
normal
extensions.
Finite
fields,
algebraically
closed
fields.
Automorphisms
of
extensions,
cyclotomic
extensions.
Galois
extensions
and
Galois
group,
fundamental
theorem
of
Galois
theory.
Solutions
of
polynomial
equations
by
radicals,
constructibile
numbers.
Additional Topic
Infinite
Galois
extensions.
Recommended Reading
M.
Artin,
Algebra,
Prentice-Hall
of
India,
New
Delhi
(1994).
Paul
J
McCarthy,
AlgebraicExtensionsofFields,
Dover
Publications
Inc.,
New
York
(1991).
S.
Lang,
Algebra,
Third
Edition,
Springer
(India)
(2004).
Thomas
W.
Hungerford,
Algebra,
Springer-Verlag,
GraduateTextsinMathematics
73
(1974).
I.
S.
Luthar
and
I.
B.
S.
Passi,
Algebra
Vol.
I
&
IV,
Narosa
Publishing
House,
New
Delhi
(2004).
MTH309: Measure and probability
[Cr:4,Lc:3,Tt:1,Lb:0]
Course Outline
Outer
measures
and
Carathéodory
extension,
Lebesgue
measure,
Measurable
function,
Integration.
Absolute
continuous
function
on
ℝ,
fundamental
theorem
of
integral
calculus
for
Lebesgue
Integral.
Measure
on
product
spaces
and
Fubini’s
theorem.
Complex
measures,
Radon-Nikodyn
theorem.
Independence
of
events,
Borel-Cantelli
lemma.
Random
variables,
distribution
functions,
moment
generating
functions.
Conditional
expectation,
independence
of
random
variables
and
Kolmogorov’s
zero-one
law
Joint
distributions.
Convergence
of
andom
variables,
law
of
large
numbers.
Characteristic
function,
central
limit
theorem.
Additional Topics
Kolmogorov
consistency
theorem.
Markov
chains,
Markov
processes.
Stationary
distributions,
limit
theorems.
Recommended Reading
S.
R.
Athreya
and
V.
S.
Sunder,
MeasureandProbability,
CRC
Press
(2009).
Kai
Lai
Chung,
ACourseinProbabilityTheory,
Academic
Press,
San
Diego
(2001).
Patrick
Billingsley,
ProbabilityandMeasure,
John
Wiley
&
Sons,
Inc.,
New
York
(1995).
Jacques
Neveu,
MathematicalFoundationsoftheCalculusofProbability,
Holden-Day
Inc.,
San
Francisco
(1965).
K.
R.
Parthasarathy,
IntroductiontoProbabilityandMeasure,
Hindustan
Book
Agency,
New
Delhi
(2005).
Walter
Rudin,
Realandcomplexanalysis,
McGraw-Hill
Book
Co.,
New
York
(1987).
MTH402: Functional analysis
[Cr:4,Lc:3,Tt:1,Lb:0]
Course Outline
Normed
linear
spaces,
Banach
spaces,
examples
and
interesting
dense
subspaces.
Continuous
linear
functionals,
duals.
Hahn-Banach
theorem,
separation
theorems.
Duals
of
classical
spaces
ℓ^{p},
L^{p}.
Bounded
linear
operators,
open
mapping
and
closed
graph
theorems,
uniform
boundedness
principle
and
applications,
spectrum
of
an
operator.
Hilbert
spaces,
orthogonality
and
geometric
structure,
projections,
Reisz
representation
theorem.
Fourier
series
—
L^{2}
theory.
The
Banach
space
()
Adjoint
of
an
operator,
self-adjoint,
normal
and
unitary
operators.
spectral
theorem
for
compact
self-adjoint
operators.
Additional Topics
Weak
and
weak-^{*}
topologies,
Banach
Alaoglu
theorem.
Spectral
theorem
for
general
self-adjoint
and
normal
operators.
Reisz
representation
theorem,
dual
of
C_{0}(X),
X
a
locally
compact
space,
Gelfand
theory.
Unbounded
operators:
definition
and
examples.
Recommended Reading
J.
B.
Conway,
AcourseinFunctionalAnalysis,
Springer
(Graduate
Texts
in
Mathematics
Vol.
96)
(1990).
G.
F.
Simmons,
IntroductiontoTopologyandModernAnalysis,Tata
McGraw
Hill
(2004).
B.
Bollobás,
LinearAnalysis:AnIntroductoryCourse,
Cambridge
University
Press,
Cambridge
(1999).
B.
V.
Limaye,
FunctionalAnalysis,
New
Age
International
Publishers
Limited,
New
Delhi
(1996).
MTH403: Manifolds
[Cr:4,Lc:3,Tt:1,Lb:0]
Course Outline
Topological
and
smooth
manifolds,
examples.
manifolds
with
boundary.
smooth
functions,
maps
between
manifolds.
Lie
groups.
smooth
partition
of
unity.
Tangent
vectors,
tangent
bundle
of
a
manifold
and
vector
fields.
Lie
brackets.
The
Lie
algebra
of
a
Lie
group.
covectors
and
cotangent
bundle.
Submersions,
immersions
and
embeddings,
inverse
and
implicit
function
theorem.
embedded
submanifolds.
level
sets.
Vector
and
covector
fields
on
submanifolds.
Lie
subgroups.
Lie
group
actions,
equivariant
maps,
proper
action.
quotients
of
manifolds
by
group
action.
Homogeneous
spaces.
Additional Topics
The
Whitney
embedding
theorem.
The
Whitney
approximation
theorem.
Connections,
Chern
classes.
Universal
connections
Recommended Reading
John
M.
Lee,
IntroductiontoSmoothManifolds,
Springer
(Graduate
Texts
in
Mathematics
Vol.
218)
(2003).
Michael
Spivak,
CalculusonManifolds,
W.
A.
Benjamin,
New
York
(1965).
N.
J.
Hicks,
NotesonDifferentialGeometry,
Van
Nostrand,
Princeton
N.J
(1965).
James
R.
Munkres,
AnalysisonManifolds,
Addison
Wesley
(1991).
R.
W.
Sharpe,
DifferentialGeometry,CARTAN’SGENERALIZATIONOFKLEIN’SERLANGENPROGRAM,
Springer-Verlag
(Graduate
Texts
in
Mathematics
Vol.
166)
(1997).
I.
M.
Singer
and
J.
A.
Thorpe,
LecturenotesonElementaryTopologyandGeometry
(Undergraduate
Texts
in
Mathematics),
Springer-Verlag
(1967).