4.5 Mathematical Sciences
MTH405: Homological algebra


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Course Outline

Categories
and
functors.

Exact
sequences
and
complexes.

Hom
and
tensor
product,
Projective
and
injective
modules,
Resolutions
of
modules.

Derived
functors
of
additive
functors,
Ext
and
Tor,
Homological
dimension.

(Co)Homology
of
groups.

Exact
couples
and
spectral
sequences,
Applications
in
commutative
algebra
and
theory
of
groups.
Recommended Reading

P.
J.
Hilton
and
U.
Stammbach,
A
Course
in
Homological
Algebra,
Springer
(1997).

Joseph
J.
Rotman,
An
Introduction
to
Homological
Algebra,
(
2nd
edition),
Springer
(2008).

H.
Cartan
and
S.
Eilenberg,
Homological
Algebra,
Princeton
University
Press
(1999).

Sergei
I.
Gelfand
and
Yuri
I.
Manin,
Methods
of
Homological
Algebra,
Springer
(2010).

S.
MacLane,
Homology,
Springer
(Classics
in
Mathematics)
(1995).

D.
G.
Northcott,
A
First
Course
in
Homological
Algebra,
Cambridge
University
Press
(2009).

L.
R.
Vermani,
An
Elementary
Approach
to
Homological
Algebra,
Chapman
&
Hall/CRC
(2003).

M.
Scott
Osborne,
Basic
Homological
Algebra,
Springer
(Graduate
Texts
in
Mathematics
196)
(2000).
MTH406: Fourier analysis


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Course Outline

Fourier
series
of
a
periodic
function,
Cesàro
and
Abel
summability.

Poisson
Kernel
and
Dirichlet
problem
in
the
unit
disc.

Pointwise
and
L^{2}convergence
of
Fourier
series.

Some
application
of
Fourier
series.

Fourier
transformation
on
ℝ,
convolution,
Plancherel
theorem,
inversion
formula.

Weierstrass
approximation
theorem,
Poisson
summation
formula,
Hiesenberg
uncertainty
principle.

Fourier
transformation
on
ℝ^{d}.
Additional Topics

Fourier
analysis
on
finite
Abelian
group.
Recommended Reading

E.
M.
Stein
and
R.
Shakarchi,
Fourier
Analysis,
Princeton
University
Press
(2003).

Walter
Rudin,
Real
and
Complex
Analysis,
McGrawHill
International
Edition
(1987).

G.
B.
Folland,
Fourier
Analysis
and
Its
Applications,
American
Mathematical
Society
(1992).

Anton
Deitmar
and
Siegfried
Echterhoff,
Principles
of
Harmonic
Analysis,
Springer
(2009).
MTH407: Algorithms and complexity


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Course Outline

Review
of
Turing
machines
and
algorithms
as
implemented
using
them.

Sorting
and
searching
algorithms
(including
data
structures
like
balanced
search
trees,
heaps
etc).

Tree
and
graph
traversal:
depthfirst
and
breadth
first
search
with
applications
(e.g.
finding
biconnected
components
of
a
graph
in
linear
time).

Algorithm
paradigms
like
divideandconquer,
the
greedy
method
(minimum
spanning
tree
algorithms,
some
matroid
theory),
dynamic
programming
algorithms,
backtracking
and
branchandbound.

Complexity
of
algorithms.
NPComplete
and
NPHard.
Recommended Reading

Alfred
V.
Aho,
J.E.
Hopcroft
and
Jeffrey
D.
Ullman:
The
Design
and
Analysis
of
Computer
Algorithms,
AddisonWesley
(1974).

Thomas
H.
Cormen,
Charles
E.
Leiserson,
Ronald
L.
Rivest,
Clifford
Stein,
Introduction
to
Algorithms,,
The
MIT
Press
(2001).

J.
Kleinberg
and
E.
Tardos,
Algorithm
Design,
Addison
Wesley
(2005).
MTH408: Riemannian geometry


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Course Outline

Riemannian
manifolds,
Model
spaces
of
Riemannian
geometry,
Connections,
Christoffel
symbols,
Covariant
derivatives,
Geodesics,
Existence
and
uniqueness
of
geodesics.

Parallel
translations,
Riemannian
connections,
Exponential
maps,
Normal
coordinates,
Geodesics
of
the
model
spaces.

Geodesics
and
minimizing
curves,
First
variation
formula,
Gauss
lemma,
HopfRinow
theorem.

Riemannian
curvature
tensor,
Symmetries
of
curvature
tensor,
Riemannian
submanifolds,
Second
fundamental
form.

GaussBonnet
theorem
(local
and
global
form),
Jacobi
fields,
Second
variation
formula,
Curvature
and
topology.
Recommended Reading

John
M.
Lee,
Riemannian
Manifolds;
An
introduction
to
curvature,
GTM176,
Springer
(1997).

J.A.
Thorpe,
Elementary
topics
in
Differential
Geometry,
UTM,
Springer
(1979).

Marcel
Berger,
A
Panoramic
view
of
Differential
Geometry,
Springer
(2002).

Issac
Chavel,
Riemannian
Geometry;
A
modern
introduction,
Cambridge
University
Press,
Cambridge
(1993).

M.P.
do
Carmo,
Riemannian
Geometry,
Birkhauser,
Boatan
(1992).

S.
Gallot,
D.
Hulin,
J.
Lafontiane,
Riemannian
Geometry,
SpringerVerlag
(1987).

Peter
Petersen,
Riemannian
Geometry,
SpringerVerlag
(1998).
MTH409: Computational methods


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Course Outline

Numerical
Computation:
Number
representaion,
machine
precision,
roundoff
errors,
truncating
errors,
random
numebr
generation,
avail
abel
numerical
software.

Linear
Algebra:
Solving
systems
of
linear
equations,
finding
inverses
of
matrices,
GaussJordan
elimination,
Gaussian
elimination,
LU
decom
position,
illconditioned
systems,
iterative
methods
(Jacob’s
method,
GaussSeidel,
Relaxation
methods)
and
convergence;
eigen
values
and
eigen
vectors,
characteristic
polynomial,
power
methods,
Jacobi’s
method,
QR
method.

Curve
Fitting:
Interpolation
techniqes
(Newton,
Lagrange),
difference
formulas,
cubic
splines,
method
of
least
squares,
twodimensional
in
terpolation.

Root
Finding:
Bisection,
False
position,
NewtonRaphson
methods,
contraction
mapping
methods,
roots
of
polynomials.

Numerical
Differentiation
and
Integration:
numerical
differentiation,
NewtonCotes
integration
formulas,
Romberg
integration,
Quadratures,
improper
integrals,
multiple
integrals.

Differential
Equations:
Euler’s
method,
RungeKutta
methods,
multi
step
methods,
BulirschStoer
extrapolation
methods,
boundary
value
problems.

PDEs:
Elliptic
equations,
onedimensional
and
twodimensional
parabolic
and
hyperbolic
equations.

RealLife
Examples:
Google
search
engine,
1D
and
2D
simulations,
weather
forecasting.
Recommended Reading

Robert
J.
Schilling
and
Sandra
L.
Harris,
Applied
Numerical
Methods
for
Engineers,
ThomsonBrooks/Cole
(1999).
MTH410: Algebraic topology


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Course Outline

Homotopy,
retract,
deformation
retract,
contractible
spaces
and
homotopy
type,
fundamental
group
and
its
properties,
The
fundamental
group
of
circle,
van
Kampen
theorem
(statement
without
proof),
applications
of
van
Kampen
theorem.

Simplicial
complexes,
polyhedra
and
triangulations,
barycentric
subdivision
and
simplicial
approximation
theorem.

Orientation
of
simplicial
complexes,
simplicial
chain
complex
and
homology,
properties
of
integral
homology
groups,
induced
homomorphisms.
degree
of
a
map
from
nsphere
to
itself
and
its
applications

Invariance
of
simplicial
homology
groups.
Lefschetz
fixed
point
theorem,
BorsukUlam
theorem.

Definition
and
examples
of
covering
spaces.
path
lifting
and
homotopy
lifting
property.

Covering
homomorphisms,
deck
transformations,
classification
of
Coverings,
existence
of
universal
covering
(statement
without
proof).

Graphs,
coverings
of
graphs
and
their
fundamental
groups.
Recommended Reading

Satya
Deo,
Algebraic
Topology:
A
Primer,
Texts
and
Readings
in
Mathematics
Vol.
27,
Hindustan
Book
Agency
(2003).

William
Massey,
Algebraic
Topology:
An
Introduction,
Springer
(Graduate
Texts
in
Mathematice
Vol.
127),
(1977).

Allen
Hatcher,
Algebraic
Topology,
Cambridge
University
Press
(2002).

M.
Greenberg
and
J.
Harper,
Algebraic
Topology:
A
First
Course,
AddisionWesley
(1981).

H.
Seifert
and
W.
Threlfall,
A
Textbook
of
Topology,
Academic
Press
(1980).
MTH411: Commutative and homological algebra


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Course Outline

Recapitulation:
Ideals,
factorization
rings,
prime
and
maximal
ideals,
modules.

Nilradical
and
Jacobson
radical,
extensions
and
contractions
of
ideals.

Localization
of
rings
and
modules.

Integral
dependence,
integrally
closed
domains,
going
up
and
going
down
theorem,
valuation
rings.

Noetherian
and
Artinian
rings,
chain
conditions
on
modules.

Exact
sequences
of
modules,
tensor
product,
projective
and
injective
modules.

Basics
of
categories
and
functors.

Exact
sequences
and
complexes
in
categories,
additive
functors,
derived
functors
EXT
and
TOR
functors.

Discrete
valuation
rings
and
Dedekind
domains.
Recommended Reading

M.
F.
Atiyah
and
I.
G.
Macdonald,
Introduction
to
Commutative
Algebra,
Addison
Wesley
(1969).

Nathan
Jacobson,
Basic
Algebra
Vol.
II,
Dover
Publications
(2009).

Joseph
J.
Rotman,
An
Introduction
to
Homological
Algebra,
(
2nd
edition),
Springer
(2008).

L.
R.
Vermani,
An
Elementary
Approach
to
Homological
Algebra,
Chapman
&
Hall/CRC
(2003).
MTH412: Structure of algebras


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Course Outline

Tensor
product
of
modules
and
algebras.

Simple
modules,
Schur’s
lemma,
semisimple
modules,
isotypical
components.

Artinian
rings,
semisimple
rings,
Jacobson
radical,
nil
ideals
and
nilpotent
ideals,
Wedderburn
Artin
structure
theory.

Complex
group
algebras
and
representation
theory
of
finite
groups.

Primitive
and
semiprimitive
rings,
Jacobson
density
theorem.

Structure
of
algebras,
Burnside’s
theorem,
Finite
dimensional
central
simple
algebras,
Brauer
group.
Recommended Reading

Nathan
Jacobson,
Basic
Algebra
Vol.
II,
Dover
Publications
(2009).

T.
Y.
Lam,
A
first
course
in
noncommutative
rings,
Springer
(2001).

TIFR
Notes
on
semisimple
rings.
MTH413: Advanced probability


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Course Outline

Radon
Nikodym
Theorem.
Conditional
Expectation.
Regular
conditional
probability.
Relevant
measure
theoretic
development.

Discrete
parameter
martingales
with
various
applications.

Path
properties
of
continuous
parameter
martingales.

Introduction
to
Brownian
Motion.
Recommended Reading

David
Williams,
Probability
with
Martingales,
Cambridge
University
Press
(1991).

Patrick
Billingsley,
Probability
and
Measure,
John
Wiley
&
Sons,
Inc.,
New
York
(1995).

Leo
Breiman,
Probability,
Society
for
Industrial
and
Applied
Mathematics
(1968).
MTH414: Advanced complex analysis


[Cr:4,
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Course Outline

Examples
of
conformal
mappings.
Schwarz
reflection
principle

Univalent
functions.
Spaces
of
Analytic
functions.
Riemann
mapping
theorem.

Infinite
products
and
Weierstrass
factorisation
theorem.

Gamma
function,
Riemann
Zeta
function,
prime
number
theorem.

Elliptic
functions.
Recommended Reading

Lars
Ahlfors,
Complex
Analysis,
McGraw
Hill
(1979).

John
B.
Conway,
Functions
of
one
Complex
Variable,
Springer
(GTM),
(1979).

W.
Tutschke
and
H.
L.
Vasudeva
An
Introduction
to
Complex
Analysis:
Classical
and
Modern
Approaches,
Chapman
&
Hall/CRC
Press
(2005).
MTH415: Enumerative problems in geometry


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Tt:1,
Lb:0]
Course Outline

Examples
of
counting
problems
in
geometry:
Grassmanians
and
intersections.
Chasles
problems
on
conics.
Fixed
points
of
transformations.
Bezout’s
theorem
and
9point
circles.

How
to
“count
properly”:
Transversal
intersections.
Fundamental
theorem
of
algebra
and
multiplicity.

Grassmanians
and
Projective
space:
Cell
decompositions.
Schubert
cells.
Schubert
calculus.

Moving
and
blowingup:
Resultants
and
plane
curve
intersections.
Singular
intersections
by
moving.
Singular
intersections
by
blowup.

The
Hilbert
Polynomial:
Interpretation
of
coeffs
of
the
Hilbert
polynomial

Intersection
theory
on
Algebraic
surfaces/4manifolds.
Divisors
and
their
intersections.
NeronSeveri
group
Hodge
Index
theorem

Introductory
Ktheory
Lambda
rings
Chern
classes
for
lambda
rings
Formal
GrothendieckRiemannRoch
theorem
Recommended Reading

W.
Fulton
and
R.
Lazarsfeld,
Interesection
Theory,
Memoirs
of
AMS.

J.
W.
Milnor,
Topology
from
a
differentiable
viewpoint
Princeton
Univ.
Press
(1965).

J.
W.
Milnor
and
J.
D.
Stasheff,
Characteristic
Classes
Princeton
Univ.
Press
(1974).

W.
Fulton,
Intersection
theory
2nd
ed.
Springer
(1998).

M.
F.
Atiyah
and
I.
G.
McDonald,
Commutative
Algebra
Oxford
University
Press
(1978).

A.
Beauville,
Complex
Algebraic
Surfaces
I
London
Math.
Society
(1996).

V.
Srinivas,
Algebraic
KTheory
Birkhauser
(2008).
MTH416: Arithmetic of elliptic curves


[Cr:4,
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Lb:0]
Course Outline
The course aims to introduce elliptic curves and their moduli with an emphasis on curves
over finite and number fields. Statements of theorems will be explained in detail and
some relevant proofs will be given. Some examples of classical problems that can be
studied using elliptic curves will be taken up and the use of the SAGE system to make
calculations on elliptic curves will be introduced.

Analytic
theory:
Doubly
periodic
functions
and
the
Weierstrass
form.

Modular
theory:
Lattices
in
complex
numbers
and
their
classification.

Algebraic
theory:
TateWeierstrass
eqation
and
group
law.

Conversions
between
different
forms
of
elliptic
curves:
Recognising
ellptic
curves
hidden
in
various
problems.

Elliptic
curves
over
finitefields:
Endomorphisms
and
Frobenius.

Elliptic
Curves
over
numberfileds:
MordellWeil
theorem.

Calculations:
Calculating
points
on
elliptic
curves,
calculating
rank
of
an
ellptic
curve,
calculating
modular
forms.
The following advanced topics could also be addressed:

Lfunctions
of
elliptic
curves:
The
terms
of
the
Lfunction.
The
statement
of
the
Birch
and
SwinnertonDyer
conjecture
and
its
similarity
with
the
Dirichlet
unit
theorem.

Modularity
of
Elliptic
curves:
ShimuraTaniyamaWeil
conjecture
and
the
statement
of
the
Theorem
of
WilesTaylor.
Recommended Reading

R.
V.
Gurjar
et
al,
Elliptic
Curves
Narosa/NBHM
(2006).

J.
H.
Silverman,
The
Arithmetic
of
Elliptic
Curves
Springer
GTM
106
(1986).
MTH418: Fuchsian groups


[Cr:4,
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Tt:1,
Lb:0]
Course Outline

Several
models
of
the
hyperbolic
space:
the
upperhalf
space
model
and
the
Poincaré
disc
model.

Hyperbolic
distance,
area
and
geodesics.

The
Möbius
group:
action
of
PSL(2, ℝ)
on
the
hyperbolic
space.

Classifying
different
types
of
isometries.

Hyperbolic
triangles,
hyperbolic
trigonometry,
hyperbolic
polygons.

Fuchsian
groups:
discrete
subgroups
of
PSL(2, ℝ).

Fundamental
domains
and
Dirichlet
regions.

Limit
sets.
Elementary
and
nonelementary
Fuchsian
groups.

Poincaré’s
theorem
and
groups
generated
by
sidepairing
transformations.
Recommended Reading

S.
Katok,
Fuchsian
Groups,
Chicago
Lectures
in
Mathematics,
University
of
Chicago
Press.

J.
Anderson,
Hyperbolic
Geometry,
Springer
Undergraduate
Mathematics
Series,
SpringerVerlag,
1999.

Alan
F.
Beardon,
The
Geometry
Of
Discrete
Groups,
Graduate
Texts
in
Mathematics
91,
SpringerVerlag,
1983.

John
G.
Ratcliffe,
Foundation
Of
Hyperbolic
manifolds,
Graduate
Texts
in
Mathematics
149,
SpringerVerlag,
1994
MTH419: Number theory


[Cr:4,
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Lb:0]
Course Outline

Congruences,
theorems
of
Chevalley
and
Warning,
absolutely
irreducible
polynomials
modulo
primes.

padic
numbers,
convergence
in
padic
metric,
Ostrowski’s
theorem.

Valuations,
local
fields,
structure
of
multiplicative
group
of
local
fields.

Quadratic
forms
over
padic
numbers,
Hilbert
symbol,
HasseMinkowski
theorem.

Characters
of
finite
Abelian
groups,
Dirichlet
series,
Zeta
function
and
Lfunctions,
Dirichlet
density
theorem,
Dirichlet’s
Theorem
on
primes
in
an
arithmetic
progression.
Recommended Reading

Z.
I.
Borevich
and
I.
R.
Shafarevich,
Number
Theory,
Academic
Press
(1966).

JeanPierre
Serre,
A
course
in
Arithmetic,
SpringerVerklag
Graduate
Texts
in
Mathematics,
Vol.
7
(1973).

Kenkichi
Iwasawa,
Local
Class
Field
Theory,
Oxford
University
Press
(1986).
MTH420: Linear operators in Hilbert spaces


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Recapitulation:
Bounded
linear
operator
and
their
spectrum.
Compact
operator
and
their
spectral
theory,
Example
of
compact
operators,
Fredholm
alternative.

Commutative
Banach
algebras:
spectrum,
resolvent,
maximal
ideals
and
Gelfand
Theorem.

Gelfand
theory
for
commutative
C^{∗}
algebras.

Resolutions
of
identity
operator
and
integration
with
respect
to
projection
valued
measure.

Decomposition
of
spectrum:
absolutely
continuous,
singular
and
point
spectra.

Functional
calculus
for
bounded
self
adjoint
operator.

Spectral
resolution
of
normal
and
unitary
operators.

Square
root
of
positive
operators
and
polar
decomposition
of
bounded
operators.
Additional Topics:

Introduction
to
theory
of
unbounded
Operators.
Recommended Reading

Walter
Rudin,
Functional
analysis
Second
edition.
International
Series
in
Pure
and
Applied
Mathematics.
McGrawHill,
Inc.
(1991).

Peter
D.
Lax
Functional
Analysis,
Wiley
Interscience
(2002).

K
R
Davidson,
C^{∗}Algebras
by
Example
Text
and
Readings
in
mathematics
11,
Hindustan
Book
Agency
(1996).

M.
Reed
and
B.
Simon,
Methods
of
Modern
mathematical
physics
I,
Functional
Analysis,
Academic
press
(1975).

W.
Arveson,
An
invitation
to
C^{∗}algebras,
Graduate
Texts
in
Mathematics,
No.
39.
SpringerVerlag
(1976).
MTH421: Combinatorial group theory


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Free
Groups:
groups
defined
by
generators
and
relations,
the
graph
of
a
group.

Factor
groups
and
subgroups:
word
groups,
reduced
free
groups,
presentations,
subgroups
of
a
free
group.

Nielsen
transformations:
Reduction
process,
automorphisms
of
free
groups.

Free
products
and
amalgamation:
Universal
property
of
a
product.
Categorical
definitions.
HNN
extensions.

Commutator
calculus.
Lower
central
series.
Lie
elements.
BakerHausdorff
formula.
Additional Topics

Discrete
subgroups
of
Lie
groups.

Mapping
Class
groups.

Groups
and
Trees.
Recommended Reading

W.
Magnus,
A.
Karrass,
D.
Solitar
Combinatorial
Group
Theory,
Dover
Books
on
Mathematics,
2004.

R.
C.
Lyndon,
P.
E.
Schupp,
Combinatorial
Group
Theory,
Classics
in
Mathematics,
Springer,
1977.

D.
J.
S.
Robinson,
A
course
in
the
Theory
of
Groups,
Graduate
Texts
in
Mathematics
80,
SpringerVerlag,
1995.
Suggested Reading

M.
S.
Raghunathan,
Discrete
Subgroups
of
a
Lie
Group,
Springer
Verlag.

J.
Birman,
Mapping
Class
groups,
Princeton
University
Press.

J.P.
Serre,
Trees,
Wiley
Press.
MTH422: Representations of finite groups
Course Outlie

Introduction
to
multilinear
algebra,
tensor
algebra,
symmetric
algebra
and
exterior
algebra.

Representations
and
basic
examples,
direct
sum
and
tensor
product
of
representations.

Irreducible
representations,
complete
reducibility
and
Maschke’s
theorem,
Schur’s
lemma.

Character
theory
of
representations,
orthogonality
relations,
decomposition
of
the
regular
representation.

Character
tables
of
Abelian,
dihedral
and
small
groups.

Dimension
theorem,
Burnside’s
pqtheorem.

Restriction
of
a
representation,
induced
representations,
Frobenius
reciprocity,
Mackey’s
irreducibility
criterion.

Representation
theory
of
symmetric
groups.
Recommended Texts

Benjamin
Steinberg,
Representation
Theory
of
Finite
Groups,
Springer
(Universitext),
2012.

Gordon
James
and
Martin
Liebeck,
Representations
and
Characters
of
Groups,
Cambridge
University
Press,
2001.
Suggested reading

William
Fulton
and
Joe
Harris,
Representation
Theory:
A
First
Course,
Springer
(Graduate
Texts
in
Mathematics
129),
1991.

JeanPierre
Serre,
Linear
Representations
of
Finite
Groups,
Springer
(Graduate
Texts
in
Mathematics
42),
1977.
MTH423: Structure of finite groups


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Sylow
theory.
Fitting
subgroup
F(G).

Subnormal
subgroups.
Characteristic
subgroups.
Wielandt
zipper
lemma.
Baers
theorem
on
subgroups
of
F(G).

Commutators;
solvable
and
nilpotent
groups.
Schur
multiplier.

Semidirect
products;
central
products
and
wreath
products.

Automorphism
group.
Horosevskiis
theorem
on
the
orders
of
elements
in
Aut(G).

Hallsubgroups.
SchurZassenhaus
Theorem.

Coprime
action.
Fitting
theorem.

Transfer.
Burnsides
normal
pcomplement
theorem.
Focal
subgroup.

Frobenius
actions.

The
Thompson
subgroup.

Burnsides
paqbtheorem.

Permutation
groups.
Simple
groups.
Recommended Reading

M.
Aschbacher,
Finite
Group
Theory.
Cambridge
Studies
in
Advanced
Mathematics,
Cambridge
University
Press,
2000.

B.
Huppert,
Endliche
Gruppen
I.
Grudlehren
der
mathematischen
Wissenschaften,
Volume
134.
Springer
1967.

Martin
Isaacs,Finite
Group
Theory.
Graduate
Studies
in
Mathematics,
Volume
92.
American
Mathematical
Society.
2008.

Derek
J.
S.
Robinson,
A
Course
in
the
Theory
of
Groups,
GTM
Vol.
80.
Springer,
1996.

Harvey
E.
Rose,
A
Course
on
Finite
Groups,
Universitext.
Springer,
2009.

Joseph
J.
Rotman,
An
Introduction
to
the
Theory
of
Groups,
Springer,
1995.

Bertram
A.
F.
Wehrfritz,A
Second
Course
on
Group
Theory,
World
Scientific,
1999.
MTH424: Introduction to Lie algebras


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Lie
algebras:
definition
and
examples,
ideals
and
homomorphisms,
quotient
algebras,
low
dimensional
Lie
algebras.

Universal
enveloping
algebras,
PoincareBirkhoffWitt
theorem.

Solvable
and
nilpotent
Lie
algebras,
Engels
theorem,
Lies
theorem.

Representations
of
Lie
algebras,
Schurs
lemma,
representations
of
sl(2,C).

Killing
form,
Cartans
criteria
for
solvability
and
semisimplicity,
derivations
of
semisimple
Lie
algebras.

Cartan
subalgebras,
root
space
decomposition,
Cartan
subalgebras
as
inner
product
spaces.

Root
systems,
Weyl
group
of
a
root
system,
Dynkin
diagrams.

Classical
Lie
algebras
sl(n,C),so(n,C),sp(n,C).

Classification
of
root
systems,
irreducible
root
systems
and
complex
simple
Lie
algebras.
Recommended Reading

Karin
Erdmann
&
Mark
J.
Wildon,Introduction
to
Lie
Algebras,
Springer
Undergraduate
Texts
in
Mathematics,
Springer,
2006.

James
E.
Humphreys,
Introduction
to
Lie
Algebras
and
Representation
Theory,
SpringerVerlag,
1980.
MTH425: Geometric group theory


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Free
groups,
group
presentations,
Cayley
graphs.

Amalgamated
free
products
and
HNN
extensions.

Structure
of
a
group
acting
on
a
tree.

Ends
of
a
group.

Group
actions
and
quasiisometries.

Hyperbolic
spaces.
Hyperbolic
groups.

Growth
of
groups.
Polynomial,
subexponential,
exponential
growth
of
groups.
Gromovs
theorem
on
groups
of
polynomial
growth.

Grigorchuk
group.

Subgroup
growth
of
free
groups.
Recommended Reading

Bowditch,
B.
H.:
A
course
on
geometric
group
theory,
MSJ
Memoirs,
Mathematical
Society
of
Japan,
Volume
16,
2006.

Bridson,
M.
R.;
Haefliger,
A.:
Metric
spaces
of
nonpositive
curvature,
Grundlehren
der
mathematischen
Wissenschaften,
Volume
319,
Springer,
1999.

Ghys,
E.;
de
la
Harpe,
P.;
Editors:
Sur
les
Groupes
Hyperboliques
dapr‘es
Mikhael
Gromov,
Progress
in
mathematics,
Birkhäuser,
1990.

Pierre
de
la
Harpe:
Topics
in
Geometric
Group
Theory,
Chicago
Lectures
in
Mathematics,
The
University
of
Chicago
Press,
2000.

Alexander
Lubotzky,
Dan
Segal.
Subgroup
Growth.
Birkhuser,
2003.

Mann,
A.:
How
groups
grow,
London
Mathematical
Society
Lecture
Note
Series
395,
Cambridge
University
Press,
2012.
MTH426: Algebraic curves


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Noetherian
rings
and
Noetherian
modules,
Hilbert
basis
theorem,
affine
space
and
algebraic
sets,
ideal
of
a
set
of
points,
algebraic
subsets
of
a
plane,
Hilbert’s
Nullstellensatz,
integral
and
algebraic
extensions.

Coordinate
rings,
affine
coordinate
transformations,
discrete
valuation
rings,
ideals
with
finitely
many
zeros,
multiple
points,
tangent
lines
and
local
rings.

Projective
space
and
projective
algebraic
sets,
affine
and
projective
varieties,
product
spaces,
linear
system
of
curves
and
Bézout’s
theorem,
Max
Noether’s
theorem.

The
Zariski
topology,
morphism
of
varieties,
algebraic
function
fields
and
dimension
of
varieties,
rational
maps.

Rational
maps
of
curves,
blowing
up
of
a
point
in
A^{2}
and
ℙ^{2},
quadratic
transformations
and
non
singular
model
curves.

Divisors,
Riemann’s
theorem
and
the
genus
of
a
non
singular
model
curve,
derivations
and
differentials,
canonical
divisors
and
the
RiemannRoch
theorem.
Recommended Reading

William
Fulton
,
Algebraic
Curves,
http://www.math.lsa.umich.edu/ wfulton/.

Igor
R.
Shafarevich,
Basic
Algebraic
Geometry
I,
Springer,
Third
Edition.
Suggested Reading

C.
Musili,
Algebraic
Geometry
for
Beginners,
Hindustan
Book
Agency.
MTH427: Introduction to Global Analysis


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Presheaf,
Sheaf,
tale
space,
Differentiable
Manifolds
sheaf
theoritic
approach.

First
order
Differential
operators,
locally
free
sheaves
and
Vector
Bundles,
theorem
of
Frobenius.

Differential
operators
of
higher
order.

Integration
on
Manifold
and
adjoints
of
Differential
operators.

Local
analysis
of
Elliptic
operators
:
Schwartz
space
and
Densities,
Fourier
transforms,
Distributions,
Sobolevs
theorem,
Interior
regularity
of
Elliptic
solutions,
Rellichs
theorem.

Elliptic
operators
on
Differentiable
Manifolds,
Regularity
theorem,
finiteness
theorem,
Elliptic
Complexes
and
Laplacian.
Additional Topics

PseudoDifferential
operators
on
Manifolds.

Fredholm
operators
and
the
Index
of
a
Fredholm
operator.
Recommended Reading

S.
Ramanan,
Global
Calculus,
American
Mathematical
Society,
Providence,
RI
(2005).

Raghavan
Narasimhan,
Analysis
on
Real
and
Complex
Manifolds,
NorthHolland
Publishing
Co.,
Amsterdam
(1968).
Advanced Reading

R.
O.
Wells,
Differential
analysis
on
complex
manifolds,
Second
edition,
GTM
65,
SpringerVerlag
(1980).

P.
B.
Gilkey,
Invariance
theory,
the
heat
equation,
and
the
AtiyahSinger
index
theorem,
Second
edition,
CRC
Press
(1995).
MTH428: Commutative Algebra and Combinatorics


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Review
of
Primary
decompositions
of
modules,
Associated
primes,
pri
mary
decompositions
in
graded
Modules.

Dimension
theory,
Hilbert
function
of
a
graded
module,
HilbertSamuel
polynomial
of
a
local
ring,
system
of
parameters
and
multiplicity.

Regular
sequences,
Depth
of
a
module,
CohenMacaulay
module,
Macaulay
theorem,
Graded
depth.

StanleyReisner
rings
(or
face
rings
)
of
simplicial
complexes,
Hilbert
series,
hvectors
and
fvectors.
Macaulays
theorem
on
Hilbert
functions.
Shellability
and
CohenMacaulayness.

Partially
ordered
sets,
Mbius
functions,
Mbius
inversion,
Eulerian
posets,
Shellable
posets,
Poset
rings.
Additional Topics

Local
Cohomology
of
StanleyReisner
rings
and
Reisner
crietrion
for
Cohen
Macaulayness.

Upper
bound
theorem.

Free
resolution
of
monomial
ideals.
Recommended Reading

H.
Matsumura,
Commutative
ring
theory,
Cambridge
University
Press,
1986.

W
Bruns
And
J.
Herzog,CohenMacaulay
Rings
(Revised
edition),
Cambridge
University
Press,
1998.

S.
R.
Ghorpade,
A
R
Shastri,
M
K
Srinivasan
and
J
K
Verma(Editors),Combinatorial
Topology
and
Algebra,
Lecture
notes
Series
18,
Ramanujan
Mathematical
Society
2013.

E.
Miller
and
B.
Sturmfels,
Combinatorial
commutative
algebra,
GTM227,
Springer,
2004.

J.
Herzog
and
T.
Hibi,Monomial
Ideals,
GTM260,
Springer,
2011.

Balwant
Singh,Basic
Commutative
Algebra,
World
Scientific,
2013.

R.
H.
Villarreal,
Monomial
Algebra,
Marcel
Dekker,
2001.
MTH601: Topics in algebra


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Groups
acting
on
sets,
Quotients,
Direct
products,
Semidirect
products,
Exact
Sequences,
Automorphism
groups
of
various
objects,
Linear
groups,
Nilpotent
and
Solvable
groups,
Structure
of
finitely
generated
abelian
groups,
Sylow
Theorems,
Free
Groups,
Generators
and
relations,
Amalgamation.

Linear
Transformations,
Eigenvalues
and
eigenvectors,
Triangulation,
Diagonalizable
transformations,
Spectral
Theorems.

Modules,
Direct
Products
and
Direct
Sums,
Modules
over
PIDs,
Various
Canonical
Forms,
Simple
and
semisimple
modules,
Semisimple
rings,
WedderburnArtin
structure
Theory.

Categories,
Groupoids,
Small
categories,
Full
subcategories,
Functors,
Left
and
right
adjoints
of
functors,
Universal
objects,
Natural
transformations
of
functors,
Equivalence
of
categories.

Algebraic
numbers,
Field
extensions,
Constructible
numbers,
Ruler
and
Compass
Constructions,
Splitting
Fields,
Algebraic
Closure
and
Normality,
The
Fundamental
Theorem
of
Algebra,
Separability,
Galois
extensions,
Galois
group
of
a
Galois
extension,
The
Galois
Pairing,
Finite
Fields,
Cyclic
Extensions,
Cyclotomic
Extentions,
Solvability
by
radicals,
The
general
equation
of
degree
n,
Transcendental
extensions
and
transcendence
basis.
Recommended Reading

Nathan
Jacobson,
Basic
Algebra

Vol
II,
W.H.
Freeman
(1989);
Hindustan
Book
Agency
(Indian
Edition).

Serge
Lang,
Algebra,
Springer
(2002).

TsitYuen
Lam,
A
first
course
in
Noncommutative
Rings,
Springer
(2001).
MTH602: Topics in topology


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

The
Fundamental
Group:
Homotopy
and
path
homotopy,
contractible
spaces,
deformation
retracts,
Fundamental
groups,
Covering
spaces,
Lifting
lemmas
and
their
applications,
Existence
of
Universal
covering
spaces,
Galois
covering,
Seifertvan
Kampen
theorem
and
its
application.

Higher
Homotopy
Groups:
Cobrations,
Cober
homotopy
equivalence,
Fibration,
Fiber
homotopy
equivalence.
Cofiber
sequences,
Fiber
sequences,
Higher
homotopy
groups,
long
exact
sequences
associated
to
brations,
CW
complexes,
Homotopy
excision
and
suspension
theorems.

Homology
and
Cohomology:
Simplicial
and
singular
homology:
Simplicial
complexes,
barycentric
subdivision
and
its
uses,
Singular
homology,
Homotopy
invariance,
Excision
theorems,
MayerVietoris
sequences,
Homology
with
arbitrary
coecients,
Singular
cohomology,
cup
products,
cohomology
ring,
Poincaré
duality.
Recommended Reading

J.
Peter
May,
A
concise
course
in
Algebraic
Topology,
Chicago
Lectures
in
Mathematics,
Univ.
Chicago
Press
(1999).

Allen
Hatcher,
Algebraic
Topology,
Cambridge
University
Press
(2002);
online
available
at
the
author’s
webpage:
http
BBC//www.math.cornell.edu/
hatcher/AT/ATpage.html
MTH603: Mathematics seminar course


[Cr:2,
Lc:2,
Tt:0,
Lb:0]
Course Outline
The aim of this course is to make PhD students understand and present technical talks of
their interest. Faculty and Mathematics PhD students will meet once a week. Suitable
topics will be decided with the mutual consent of the participants. Occasionally visitors
of the institute will be asked to give a talk in this course.
MTH604: Homological and commutative algebra


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline
Homological Algebra

Categories
and
functors,
Derived
functors,
Hom
and
⊗
functors,
Flat,
projective
and
injective
modules,
Resolutions
of
modules,
Ext
and
Tor
functors.

Cohomology
of
groups.
Commutative algebra

Recollection
of
rings,
ideals,
Spec
and
MaxSpec
of
rings,
Zariski
topology.

Modules,
Finitely
generated
modules,
Nakayama
lemma,
Localisation
of
rings
and
modules.

Chain
conditions,
Noetherian
rings,
Hilbert
basis
theorem,
Artinian
rings,
Noetherian
and
Artinian
modules.

Associated
primes
and
Primary
decomposition,
Integral
extensions,
Going
up
and
going
down
theorems,
Noether
normalisation
theorem,
Hilbert
Nullstellensatz
and
their
geometric
interpretations.

Valuation
rings
and
Dedekind
domains,
Ideal
class
group.

Direct
and
inverse
limits,
Completions,
Graded
rings
and
modules,
ArtinRees
lemma.

Dimension
theory,
Hilbert
and
Samuel
functions,
Dimension
theorem,
Krull’s
principal
ideal
theorem.

Regular
sequences,
Depth,
CohenMacaulay
rings,
Gorenstein
rings,
Regular
rings.
Recommended Reading

Hideyuki
Matsumura,
Commutative
Ring
Theory,
Cambridge
Series
in
Advanced
Mathematics
8,
Cambridge
University
Press
(1989).

Kenneth
S.
Brown,
Cohomology
of
Groups,
GTM
87,
SpringerVerlag
(1982).

M.F.
Atiyah
and
I.G.
Macdonald,
Introduction
to
Commutative
Algebra,
Perseus
Books
Group
(1994).

David
Eisenbud,
Commutative
Algebra
with
a
view
toward
Algebraic
Geometry,
GTM
150,
SpringerVerlag
(1995).

R.Y.
Sharp,
Steps
in
Commutative
Algebra,
Cambridge
University
Press
(2000).
MTH605: Topics in analysis


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Lebesgue
integral
and
construction
of
Lebesgue
measure
on
ℝ^{n},
Non
measurable
sets,
Fatou’s
Lemma
and
Convergence
Theorems,
L^{p}
spaces
and
their
completeness,
Reisz
representation
Theorem
for
C_{0}(X)
;
X
locally
compact.

Complex
Measure,
RadonNikodym
Theorem,
Differentiation
of
an
integral,
Absolutely
continuous
of
functions,
Hahn
decomposition
Theorem,
Product
measure
and
Fubini
Theorem.

Fourier
series
and
Fourier
transform,
L^{2}
theory.

Theory
of
Distributions.
Recommended Reading

Walter
Rudin,
Real
and
Complex
Analysis,
McGrawHill
International
Editions,
Mathematics
Series,
McGrawHill
Education
(1987).

Walter
Rudin,
Functional
Analysis,
Tata
McGrawHill
(1990).
(for
the
Theory
of
Distributions)

Halsey
L.
Royden,
Real
Analysis,
Prentice
Hall,
Third
Edition
(1988).
MTH606: Mathematics seminar course


[Cr:2,
Lc:2,
Tt:0,
Lb:0]
Course Outline

The
aim
of
this
course
is
to
make
PhD
students
understand
and
present
technical
talks
of
their
interest.
Faculty
and
Mathematics
PhD
students
will
meet
once
a
week.
Suitable
topics
will
be
decided
with
the
mutual
consent
of
the
participants.
Occasionally
visitors
of
the
institute
will
be
asked
to
give
a
talk
in
this
course.
MTH607: Euclidean harmonic analysis


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Review
of
basics;
Fourier
Series,
Fourier
Transforms
in
ℝ^{n};
Plancherel
and
Fourier
Inversion
Theorems.
Convolutions.

The
Schwartz
Space
and
Tempered
Distributions.

Poisson
Summation
Formula
and
applications.
Uncertainty
Principles:
Heisenberg,
BenedicksAmreinBerthier,
and
Beurling.
PaleyWiener
Theorems.

Translation
Invariant
Operators
on
L^{p}
spaces.
Interpolation
Theorems
(ReiszThorin
and
Marcinkeiwicz).

Maximal
functions.
Hilbert
Transform
and
convergence
of
Fourier
Series
and
Integrals.
CalderonZygmund
Singular
Integrals.
Additional topics (a subset of the following):

LittlewoodPaley
inequalities;
HormanderMihlin
and
Marcinkeiwicz
Multipliers.

H^{1}BMO

Time
frequency
phase
plane
analysis.
Wavelets.
Recommended Reading

E.
Stein
and
R.
Shakarchi:
Fourier
Analysis,
Princeton
University
Press
(2003).

E.
Stein
and
R.
Shakarchi:
Complex
Analysis,
Princeton
University
Press
(2005).

Javier
Duoandikoetxea:
Fourier
Analysis,
AMS
(2001).
MTH608: Algebraic number theoryI


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Characteristic
and
minimal
polynomial
of
an
element
relative
to
a
finite
extension,
Equivalent
definitions
of
norm
and
trace,
Algebraic
numbers,
algebraic
integers
and
their
properties.

Integral
bases,
discriminant,
Stickelberger’s
theorem,
Brille’s
theorem,
description
of
integral
basis
of
quadratic,
cyclotomic
and
special
cubic
fields.

Ideals
in
the
ring
of
algebraic
integers
and
their
norm,
factorization
of
ideals
into
prime
ideals,
generalised
Fermat’s
theorem
and
Euler’s
theorem.

Dirichlet’s
theorem
on
units,
regulator
of
an
algebraic
number
fields,
explicit
computation
of
fundamental
units
in
real
quadratic
fields.

Dedekind’s
theorem
for
decomposion
of
rational
primes
in
algebraic
number
fields
and
its
application,
splitting
of
rational
primes
in
quadratic
and
cyclotomic
fields.
Recommended Reading

Saban
Alaca
and
Kenneth
Williams,
Introductory
Algebraic
Number
Theory,
Cambridge
University
Press
(2003).

M.
Ram
Murty
and
J.
Esmonde
Problems
in
Algebraic
Numbers
Theory,
SpringerVerlag
(2004).

Wladyslaw
Narkiewicz,
Elementary
and
Analytic
Theory
of
Algebraic
Numbers,
SpringerVerlag
(2004).

Erich
Hecke,
Lectures
on
the
Theory
of
Algebraic
Numbers,
SpringerVerlag
(1981).

Paula
Ribenboim,
Algebraic
Numbers,
John
Wiley
&
Sons
(1972).

Harry
Pollard
and
Harold
Diamond,
The
Theory
of
Algebraic
Numbers,
Dover
Publications
(2010).
MTH609: Algebraic number theoryII


[Cr:4,
Lc:3,
Tt:1,
Lb:0]
Course Outline

Index
of
ramification,
residual
degree,
norm
of
ideals
for
relative
extensions
of
algebraic
number
fields
and
their
properties.
Fundamental
equality.

Different,
Discriminant
and
Dedekind’s
theorem
on
ramified
prime
ideals
in
extensions
of
algebraic
number
fields.

Finiteness
of
class
number
and
determination
of
class
numbers
in
special
cases.

Dirichlet’s
Density
theorem
and
simple
applications.

Dirichlet’s
class
number
formula
and
its
explicit
determination
for
cyclotomic
and
quadratic
fields
in
terms
of
Lseries.
Recommended Reading

M.
Ram
Murty
and
J.
Esmonde,
Problems
in
Algebraic
Number
Theory
SpringerVerlag
(2004).

Saban
Alaca
and
Kenneth
Williams,
Introductory
Algebraic
Number
Theory
Cambridge
University
Press
(2003).

Wladyslaw
Narkiewikz,
Elementary
and
Analytic
Theory
of
Algebraic
Numbers
SpringerVerlag
(2004).

Erich
Hecke,
Lectures
on
the
Theory
of
Algebraic
Numbers
SpringerVerlag,
(1981).

Paulo
Ribenboim,
Algebraic
Numbers
John
Wiley
and
Sons
(1972).