3. DETAILED SYLLABI OF THE
COURSES.
C1. General
Topology
Part I (9
weeks)
1. Topological spaces, open and closed sets, basis, closure,
interior and boundary. Subspace topology, Hausdorff spaces.
2. Continuous maps: properties and constructions; Pasting Lemma.
Homeomorphisms. Product topology and Quotient topology (emphasising universal
properties).
3.
Connected, path-connected
and locally connected spaces.
4.
Lindelof and Compact spaces,
Locally compact spaces and one-point compactification. Tychonoff's
theorem.
5. Countability and separation axioms. Urysohn's lemma, Tietze
extension theorem and applications.
6. Completion of metric spaces. Baire Category Theorem and
applications.
If time permits:
(i)
Convergence, nets and filters
(ii)
Urysohn embedding lemma and metrization theorem for second
countable spaces.
(iii)
Stone-Cech compactification. Paracompactness.
Part II (5
weeks)
1.
Constructions of topological
manifolds. Projective spaces.
2.
Group actions and examples
of important orbit spaces. Examples and basic properties of classical
groups.
3.
Homotopy of paths. The
Fundamental Group.
4.
Covering spaces, path
lifting and homotopy lifting theorems.
5.
Fundamental groups of
circle, torus, Mobius band etc.
6.
Van Kampen theorem and
applications.
References:
J. R. Munkres, Topology: a first course. Prentice-Hall,
Inc., 1975.
J. Dugundji, Topology. Allyn and Bacon Series in
Advanced Mathematics. Allyn and Bacon, Inc., 1978.
W. S. Massey, A basic course in algebraic topology.
Graduate Texts in Mathematics, 127. Springer-Verlag, 1991.
I. M. Singer and J. A.
Thorpe, Lecture notes on elementary
topology and geometry. Undergraduate Texts in Mathematics. Springer-Verlag,
1976.
K. K. Mukherjea, Unpublished Notes (Chapter
1).
C2. Complex
Analysis
A
review of basic Complex Analysis: Cauchy-Riemann equations, Cauchy's theorem and
estimates. power series expansions, maximum modulus principle, Classification of
singularities and calculus of residues. Normal families, Arzela's theorem.
Product developments, functions with prescribed zeroes and poles, Hadamard's
theorem. Conformal mappings, the Riemann mapping theorem, the linear fractional
transformations.
Depending on time available,
some of the following topics may be done:
(i)
Subharmonic functions, the
Dirichlet problem and Green's functions
(ii)
An introduction to elliptic
functions
(iii)
Introduction to functions of
several complex variables.
References:
L. V. Ahlfors, Complex analysis. An introduction to the
theory of analytic functions of one complex variable. McGraw-Hill Book Co.,
1978.
J. B. Conway, Functions of one complex variable. II.
Graduate Texts in Mathematics, 159. Springer-Verlag, 1995.
W. Rudin, Real
and complex analysis. McGraw-Hill Book Co., 1987.
C3. Measure
Theory
s-algebras of sets, measurable
sets and measures, extension of measures, construction of Lebesgue measure,
integration, convergence theorems, Radon-Nikodym theorem, product measures,
Fubini's theorem, differentiation of integrals, absolutely continuous functions
(as e.g., in Royden, Chapter 5), Lp-spaces, Riesz
representation theorem for the space C[0, 1].
References:
J. Nevue, Mathematical foundations of the calculus of
probability. Holden-Day, Inc., 1965.
I. K. Rana, An introduction to measure and
integration. Narosa Publishing House, 1997.
P. Billingsley, Probability and measure. John Wiley
& Sons, Inc., 1995.
W. Rudin, Real
and complex analysis. McGraw-Hill Book Co., 1987.
K. R. Parthasarathy, Introduction to probability and measure.
The Macmillan Co. of India, Ltd., 1977.
C4. Algebra I
1.
Groups [3-4 weeks]
(A)
Review: normal subgroups and
quotient groups, homomorphism theorems, direct product, direct sum and free
abelian groups (including infinite index) emphasising universal properties.
(Categories and functors including universal objects and adjoints, free groups
may be introduced).
(B)
Group actions on sets and
applications (including Sylow theorems and applications).
(C)
Permutation groups, simple
groups, composition series, solvable and nilpotent groups.
(D)
Exact sequences,
automorphism and semi-direct product.
2.
Rings and Modules [11-12
weeks]
(A) Review: Universal properties
of quotient rings; Noether's isomorphism theorems and applications to
non-trivial examples; noetherian rings.
(B) Basic concepts: submodules,
quotients, homomorphisms, isomorphism theorems, generators, annihilators,
torsion, direct product and sum, direct summand, free modules, finitely
generated modules, noetherian modules. Algebras, finitely generated algebras.
Exact and split exact sequences.
(C) Tensor product of modules
and algebras. Tensor, symmetric and exterior algebras.
(D) Finitely generated modules
over a PID: structure theorem and applications to abelian groups.
(E)
Review of topics in Linear
Algebra: Matrices and Linear Transformations, Trace, Rank, Determinant, Minimum
polynomial, Characteristic Roots and Polynomials. Rational and Jordan Canonical
forms. Inner Product Spaces. Unitary, Hermitian and Orthogonal Transformations,
Quadratic Forms.
Time permitting, additional
topics can be selected from
(i)
Snake lemma, complexes,
homology sequences.
(ii)
Projective and flat modules.
Shanuel lemma.
References:
J. J. Rotman, An introduction to the theory of groups.
Graduate Texts in Mathematics, 148. Springer-Verlag, 1995.
N. Jacobson, Basic algebra.
Vol. I. W. H. Freeman and Company, 1985.
S. Lang, Algebra. Graduate Texts in Mathematics,
211. Springer-Verlag, 2002.
N. S. Gopalakrishnan, University algebra. Wiley Eastern Ltd.,
1986.
N. S. Gopalakrishnan, Commutative algebra. Oxonian Press Pvt.
Ltd., 1984.
C5.
Algebra II
1.
Rings and ideals (Review):
operations on ideals (sum, product, quotient and radical); Chinese remainder
theorem; nilradical and Jacobson radical. Localisation and local rings. Results
on prime ideals like prime avoidance, prime ideals under localisation and
theorems of Cohen and Isaac.
2.
Modules over local rings.
Cayley-Hamilton, NAK lemma and applications. Examples of local-global
principles.
3.
Polynomial and power series
rings: properties and non-trivial applications. Hilbert basis theorem.
4.
Algebraic extensions: finite
and algebraic field extensions, field automorphisms, existence and uniqueness of
algebraic closure, splitting fields and normal extensions, separable,
inseparable and purely inseparable extensions, separable and purely inseparable
closure, theorem of primitive elements, finite fields.
5.
Galois theory: Galois
extension and Galois groups, Artin's theorem, fundamental theorem, roots of
unity, cyclotomic extensions, linear independence of characters, traces and
norms, cyclic extensions, Hilbert theorem 90, Artin-Schreier theorem, algebraic
independence of homomorphisms, normal basis theorem.
6.
Transcendental extensions:
transcendence degree, separating transcendental bases. Derivations, separable
extensions, linear disjointness.
7.
Integral extensions:
integral closure, normalisation and normal rings, Cohen-Seidenberg theorems,
Krull dimension, Noether's normalisation, Hilbert's Nullstellensatz and
applications, algebraic sets, finiteness of integral closure.
If time permits, topics can
be selected from
(i)
Galois Cohomology, Kummer
Extension.
(ii)
Applications and
computations: constructions with straight-edge and compasses, solvable and
radical extensions; computation of Galois groups, polynomials of degree 3 and 4.
(iii)
Real fields: Ordered fields,
real closed fields, Sturm theorem, real zeros and homomorphisms.
(iv)
Review of PID and UFD.
Nagata's criterion for UFD and applications (including Gauss' Theorem);
equivalence of PID and one-dimensional UFD.
(v)
Weierstrass preparation
theorem.
References:
I. Kaplansky, Commutative rings. The University of
Chicago Press, 1974.
S. Lang, Algebra. Graduate Texts in Mathematics,
211. Springer-Verlag, 2002.
M. Nagata, Field theory. Pure and Applied
Mathematics, No. 40. Marcel Dekker, Inc., 1977.
H. Matsumura, Commutative ring theory. Cambridge
Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1989.
E. Kunz, Introduction to commutative algebra and
algebraic geometry. Birkher Boston, Inc., 1985.
N. S. Gopalakrishnan, University algebra. Wiley Eastern Ltd.,
1986.
N. S. Gopalakrishnan, Commutative algebra. Oxonian Press Pvt.
Ltd., 1984.
C6.
Functional Analysis
Normed linear spaces, Banach
spaces. Bounded linear operators. Dual of a normed linear space. Hahn-Banach
theorem, uniform boundedness principle, open mapping theorem, closed graph
theorem. Computing the dual of well-known Banach spaces. Weak and weak*
topologies, Banach-Alaoglu Theorem. The double dual, Goldsteins Theorem,
reflexivity.
Hilbert spaces, adjoint
operators, self-adjoint and normal operators, spectrum, spectral radius,
analysis of the spectrum of a compact operator on a Banach space, spectral
theorem for bounded self-adjoint, normal, and unitary operators.
References:
W. Rudin, Functional analysis. McGraw-Hill, Inc.,
1991.
J. B. Conway, A
course in functional analysis. Graduate Texts in Mathematics, 96.
Springer-Verlag, 1990.
K. Yosida, Functional analysis. Grundlehren der
Mathematischen Wissenschaften, 123. Springer-Verlag, 1980.
C7. Algebraic
Topology
Review of C1, Part II, if
necessary.
1.
Singular homology functors
and its axiomatic properties. Relations between fundamental group and first
homology. Mayer-Vietoris sequence, computation of homology of spheres. Degree of
maps with applications to spheres.
2.
Simplicial CW-complexes,
cellular description of homology, comparison with singular theory. Computation
of homology of projective spaces.
3.
Definition of singular
cohomology, its fundamental properties, statement of universal coefficient
theorem, Betti number and Euler characteristic, cup product, Poincare duality.
Reference:
M. J. Greenberg, Lectures on algebraic topology. W. A.
Benjamin, Inc., 1967.
J. R. Munkres, Elements of algebraic topology.
Addison-Wesley Publishing Company, 1984.
C8.
Differential Geometry
Differentiable manifolds,
tangent bundle, vector bundles, vector fields, flows and the fundamental theorem
of ODE's. Differential forms and integration, Immersion, submersion,
submanifolds and transversality, Riemannian metrics. Riemannian connection on
Riemannian manifolds, Gauss-Bonnet Theorem. Parallel transport, geodesics and
geodesic completeness, the theorem of Hopf-Rinow.
References:
F. W. Warner, Foundations of differentiable manifolds and
Lie groups. Graduate Texts in Mathematics, 94. Springer-Verlag, 1983.
S. Helgason, Differential geometry, Lie groups, and
symmetric spaces. Graduate Studies in Mathematics, 34. American Mathematical
Society, 2001.
C9. Fourier
Analysis
Fourier and
Fourier-Stieltjes' series, summability kernels, convergence tests. Fourier
transforms, the Schwartz space, Fourier Inversion and Plancherel theorem.
Maximal functions and boundedness of Hilbert transform. Paley-Wiener Theorem.
Poisson summation formula, Heisenberg uncertainty Principle, Wiener's Tauberian
theorem. (An introduction to harmonic analysis on locally compact abelian groups
may be given
while discussing Fourier
transforms.)
Introduction to wavelets and multi-resolution
analysis.
Suggested Texts:
Y. Katznelson, An introduction to harmonic analysis.
Dover Publications, Inc., New York, 1976.
E. Hernez and G. Weiss, A first course on wavelets. Studies in
Advanced Mathematics. CRC Press, 1996.
C10. Partial Differential
Equations-I
Theory of Schwartz distributions and Sobolev spaces; local
solvability and Lewys example; existence of fundamental solutions for constant
coefficient differential operators; Laplace, heat and wave equations,
hypoelliptic and analytic hypoelliptic operators, elliptic boundary value
problems interior regularity, local existence.
Suggested
books:
G. B. Folland, Introduction to partial differential
equations. Princeton University Press, 1995.
F. Trs, Basic linear partial differential
equations. Pure and Applied Mathematics, Vol. 62. Academic Press, 1975.
J. Rauch, Partial differential equations. Graduate
Texts in Mathematics, 128. Springer-Verlag, 1991.
E. DiBenedetto, Partial differential equations. Birkher
Boston, Inc., 1995.
L. C. Evans, Partial differential equations. Graduate
Studies in Mathematics, 19. American Mathematical Society, 1998.
L. Hnder, The analysis of linear partial differential
operators. I. Distribution theory and Fourier analysis. Grundlehren der
Mathematischen Wissenschaften, 256. Springer-Verlag, 1990.
C11. Graph Theory and
Combinatorics
Graphs and digraphs,
connectedness, trees, degree sequences, connectivity, Eulerian and hamiltonian
graphs, matchings and SDR's, chromatic numbers and chromatic index, planarity,
covering numbers, flows in networks, enumeration, Inclusion-exclusion, Ramsey's
theorem, recurrence relations and generating functions.
If time permits, some of the
following topics may be done: (i) strongly regular graphs, root systems, and
classification of graphs with least eigenvalue, (ii) Elements of coding theory
(Macwilliams identity; BCH, Golay and Goppa codes, relations with
designs).
Suggested texts:
C12. Advanced
Probability
Independence, Kolmogorov
Zero-one Law, Kolmogorov Three-series theorem, Strong law of large Numbers.
LevyCramer Continuity theorem, CLT for i. i. d. components, Infinite Products
of probability measures, Kolmogorovs Consistency theorem, RadonNikodym
Theorem, Conditional expectations.
Discrete parameter
martingales with applications.
References:
J. Nevue, Mathematical foundations of the calculus of
probability. Holden-Day, Inc., 1965.
P. Billingsley, Probability and measure. John Wiley
& Sons, Inc., 1995.
Y. S. Chow and H. Teicher,
Probability theory. Independence,
interchangeability, martingales. Springer Texts in Statistics.
Springer-Verlag, 1997.
C13. Representations of
Groups
Structure theory of
semisimple rings and modules. Representation of a finite group: Youngs
Tableaux, examples, Maschkes theorem, sums, products, exterior and symmetric
powers of representations. Applications to group rings,
characters.
Topological Groups, basic
properties like subgroups, quotients and products, fundamental systems of
neighbourhoods, open subgroups, connectedness and compactness. Existence of Haar
measure on locally compact groups, properties of Haar measures.
Group actions on topological
spaces, the space X/G in the topological as also in the analytical case assuming
regularity conditions of the group action..
Representation of a locally
compact group on a Hilbert space, the associated representation of group
algebra, invariant subspaces and irreducibility, Schurs
lemma.
Compact groups: Unitarity of
finite dimensional representations, Peter-Weyl theory, Representations of
SU(2,C)
Induced representation and
Frobenius reciprocity theorem, Principal series representations of SL(2,
R).
Suggested texts:
TIFR Lecture Notes on
Semisimple rings (Unpublished), Chapters
1 & 4.
T. Y. Lam, A first course in noncommutative rings.
Graduate Texts in Mathematics, 131. Springer-Verlag, 2001.
P. J. Higgins, Introduction to topological groups.
London Mathematical Society Lecture Note Series, No. 15. Cambridge University
Press, 1974.
L. H. Loomis, An introduction to abstract harmonic
analysis. D. Van Nostrand Company, Inc., 1953.
G. B. Folland, A course in abstract harmonic analysis.
Studies in Advanced Mathematics. CRC Press, 1995.
Op1. Algebra III:
Commutative Algebra
1.
Free Modules. Projective
Modules. Shanuel Lemma. Tensor Product of Modules and Algebras. Tensor,
Symmetric and Exterior Algebras. Flat, Faithfully Flat Modules and Finitely
Presented Modules.
2.
Local-Global Methods.
Projective and locally free modules. Patching up of Localisations.
3.
Noetherian Modules.
Associated Primes and Primary Decomposition. Artininan Modules. Modules of
Finite Length.
4.
Graded and Filtered Modules.
Artin-Rees Theorem.
5.
Completion. I-adic
Filtrations. Krull Intersection Theorem. Hensel's Lemma and applications.
Weierstrass Preparation Theorem.
6.
Valuations, Discrete
Valuation Rings. Dedekind Domains. Local property of Normal Domains, Normality
and DVR at height one primes, Intersection of DVRs. Finiteness of Normalisation.
Krull-Akizuki Theorem.
7.
Homological Algebra:
Complexes, Homology Sequences. Projective Resolution. The functors Tor and Ext.
8.
Dimension Theory:
Hilbert-Samuel Polynomial. Dimension theorem.
9.
Regular Local Rings:
Jacobian criterion. UFD criteria: principality of height one primes; Nagata's
criterion and applications.
10. (Time permitting)
Homological Dimension. Injective Modules and Injective Resolution. Injective
Dimension and Global Dimension. Global Dimension of Noetherian Local Rings.
Properties of Regular Local Ring. Homological Characterisation of Regular Local
Rings. Regular Local Ring is UFD.
Note: The following topics,
already included in Algebra I and II, have not been mentioned above. However, if
any of these topics have not been covered thoroughly during the previous
semesters, they should be covered in this semester.
Operations on Ideals (sum, product, quotient and radical);
Chinese remainder theorem; nilradical and Jacobson radical. Localisation and
local rings. Results on prime ideals like prime avoidance, prime ideals under
localisation and theorems of Cohen and Isaac. Modules over local rings.
Cayley-Hamilton and Determinant trick, NAK lemma and applications. Integral
extensions: integral closure, normalisation and normal rings, Cohen-Seidenberg
theorems.
References:
H. Matsumura, Commutative algebra. W. A. Benjamin,
Inc., 1970
H. Matsumura, Commutative ring theory. Cambridge
Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1989.
D. Eisenbud, Commutative algebra. With a view toward
algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag,
1995.
E. Kunz, Introduction to commutative algebra and
algebraic geometry. Birkher Boston, Inc., 1985.
J.-P. Serre, Local algebra. Springer Monographs in
Mathematics. Springer-Verlag, 2000.
N. S. Gopalakrishnan, Commutative algebra. Oxonian Press Pvt.
Ltd., 1984.
Homological methods in
Commutative Algebra, TIFR Mathematical Pamphlet
No. 5, Oxford University Press, 1975.
M. Reid, Undergraduate commutative algebra.
London Mathematical Society Student Texts, 29. Cambridge University Press, 1995.
M. F. Atiyah and I. G.
Macdonald, Introduction to commutative
algebra. Addison-Wesley Publishing Co., 1969.
Op2.
Number Theory
Finite fields. Existence and uniqueness of fields of prime
power order. Chevalley-Warning theorem on common zeros of systems of polynomial
equations over finite fields. Law of quadratic reciprocity.
p-adic fields. p-adic
equations and Hensels Lemma.
Dirichlet series: abscissa
of convergence and of absolute convergence. Riemann Zeta function and Dirichlet
L-functions. Dirichlets Theorem on
primes in arithmetic progression. Functional equation and Euler product for L-functions.
Modular forms and the
modular group SL (2, R). Eisenstein series. Zeros and
poles of modular functions. Dimensions of the spaces of modular forms. The j-invariant and Picards Theorem. L-function and Ramanujans T-function. Jacobis product formula for
L-congruence relations satisfied by
T.
Suggested text: J.-P. Serre,
A course in arithmetic. Graduate
Texts in Mathematics, No. 7. Springer-Verlag, 1973.
If there is time,
Hasse-Minkowski theorem from the same book could be included.
Op3.
Algebraic Geometry
1.
Polynomial
rings
2.
Hilbert Basis
theorem
3.
Noether normalisation
lemma
4.
Hilbert
Nullstellensatz
5.
Elementary dimension theory
6.
Smoothness
7.
Curves
8.
Divisors on
curves
9.
Bezouts
theorem
10. Abelian
differential
11. RiemannRoch for
curves
References:
C. Musili, Algebraic geometry for beginners. Texts
and Readings in Mathematics, 20. Hindustan Book Agency, 2001.
W. Fulton, Algebraic curves. An introduction to
algebraic geometry. Advanced Book Classics. Addison-Wesley Publishing
Company, Advanced Book Program, 1989.
K. Kendig, Elementary algebraic geometry. Graduate
Texts in Mathematics, No. 44. Springer-Verlag, 1977.
R. Shafarevich, Basic algebraic geometry. 1. Varieties in
projective space. Springer-Verlag, 1994.
J. Harris, Algebraic geometry. A first course.
Graduate Texts in Mathematics, 133. Springer-Verlag, 1995.
M. Reid, Undergraduate algebraic geometry. London
Mathematical Society Student Texts, 12. Cambridge University Press, 1988.
Op4. Algebraic Number
Theory
Dedekind Domains, Fractional
Ideals and Class Group, Prime Decomposition in Number Fields, Finiteness of
Class Number, Minkowski's Bound, Dirichlet's Unit Theorem.
Valuations, Completions,
Product Formula, Decomposition and Inertia Groups, Artin Map.
Distribution of Ideals in a
Number Field, Dedekind Zeta Function and Dirichlet L-functions, Frobenius
Density Theorem.
Group Cohomology of Cyclic
Groups, First and Second Fundamental Inequalities for Cyclic Extensions, Hasse's
Norm Theorem, Artin's Reciprocity Law, Kronecker-Weber Theorem, Existence of the
Hilbert Class Field.
Prerequisite: Galois Theory,
Commutative Algebra.
Suggested Text: G. J.
Janusz, Algebraic number fields. Pure
and Applied Mathematics, Vol. 55. Academic Press, 1973.
Op5. Probability and Stochastic Processes I
1.
(a)Rates of convergence to stationarity, Dirichlet
Form and Spectral gap methods
2.
(b)Some Coupling methods with applications
3.
(c)Random walk on Finite Groups
4.
(d)Poisson Processes
5.
(e) Continuous time Markov Chains , Birth-and-death processes
6.
Books:
7.
S. M. Ross, Stochastic processes. John Wiley & Sons, Inc.,
1996.
8.
R. N. Bhattacharya and E. C. Waymire, Stochastic processes
with applications
9.
E. GinG. R. Grimmett and L. Saloff-Coste, Lectures on
probability theory and statistics
Op6.
Probability and Stochastic Processes II
Selected topics from the
following:
1.
Stationary processes.
2.
Markov processes and
generators
3.
Weak Convergence of
probability measures on polish spaces including C[0, 1].
4.
Brownian motion;
construction, simple properties of paths.
5.
Poisson processes.
6.
Connections between Brownian
Motion / Diffusion and PDEs.
References:
P. Billingsley, Convergence of probability measures.
John Wiley & Sons, Inc., 1999.
K. Ito, Stochastic processes. Lecture Notes
Series, No. 16 Matematisk Institut, Aarhus Universitet, Aarhus
1969.
D. Revuz and M. Yor, Continuous martingales and Brownian
motion. Grundlehren der Mathematischen Wissenschaften, 293. Springer-Verlag,
1999.
Op7.
Ergodic Theory
1.
Measure preserving systems;
examples: Hamiltonian dynamics and Liouvilles theorem, Bernoulli shifts, Markov
shifts, Rotations of the circle, Rotations of the torus, Automorphisms of the
Torus, Gauss transformations, Skew-product.
2.
Poincare Recurrence lemma:
Induced transformation: Kakutani towers: Rokhlins lemma. Recurrence in
Topological Dynamics, Birkhoffs Recurrence theorem
3.
Ergodicity, Weak-mixing and
strong-mixing and their characterisationsErgodic Theorems of Birkhoff and Von
Neumann. Consequences of the Ergodic theorem. Invariant measures on compact
systems, Unique ergodicity and equidistribution. Weyls theorem.
4.
The Isomorphism problem;
conjugacy, spectral equivalence.
5.
Transformations with
discrete spectrum, Halmosvon Neumann theorem.
6.
Entropy. The
Kolmogorov-Sinai theorem. Calculation of Entropy. The Shannon Mc-MillanBreiman
Theorem.
7.
Flows. Birkhoffs ergodic
Theorem and Wieners ergodic theorem for flows. Flows built under a function.
References:
Peter Walters, An introduction to ergodic theory.
Graduate Texts in Mathematics, 79. Springer-Verlag, 1982.
Patrick Billingsley, Ergodic theory and information. Robert
E. Krieger Publishing Co., 1978.
M. G. Nadkarni, Basic ergodic theory. Texts and Readings
in Mathematics, 6. Hindustan Book Agency, 1995.
H. Furstenberg, Recurrence in ergodic theory and
combinatorial number theory. Princeton University Press, 1981.
K. Petersen, Ergodic theory. Cambridge Studies in
Advanced Mathematics, 2. Cambridge University Press, 1989.
Op8. Lie Groups and Lie
algebras
1.
Linear Lie groups: the
exponential map and the Lie algebra of linear Lie group, some calculus on a
linear Lie group, invariant differential operators, finite dimensional
representations of a linear Lie group and its Lie algebra. Examples of linear
Lie group and their Lie algebras e. g Complex groups: GL(n, C), SL(n, C ), SO(n,
C), Groups of real matrices in those complex groups: GL(n, R), SL(n, R ), SO(n,
R), Isometry groups of Hermitian forms SO(m, n), U(m, n), SU(m, n). Finite
dimensional representations of su(2 ) and SU(2) and their connection. Exhaustion
using the lie algebra su(2).
2.
Lie algebras in general,
Nilpotent, solvable, semisimple Lie algebra, ideals, Killing form, Lies and
Engels theorem. Universal enveloping algebra and Poincare-Birkhoff-Witt Theorem
(without proof).
3.
Semisimple Lie algebra and
structure theory: Definition of Linear reductive and linear semisimple groups.
Examples of Linear connected semisimple/ reductive Lie groups along with their
Lie algebras (look back at 2 above and find out which are reductive/semisimple).
Cartan involution and its differential at identity; Cartan decomposition g=k+p,
examples of k and p for the groups discussed above. Definition of simple and
semisimple Lie algebras and their relation, Cartans criterion for
semisimplicity. Global Cartan decomposition, Root space decomposition; Iwasawa
decomposition; Bruhat decomposition (statement only).
4.
If time permits, one of the following topics :
(i)
A brief introduction to
Harmonic Analysis on SL (2, R).
(ii)
Representations of Compact
Lie Groups and Weyl Character Formula
(iii)
Representations of Nilpotent
Lie Groups
Suggested Texts:
J. E. Humphreys, Introduction to Lie algebras and
representation theory. Graduate Texts in Mathematics, 9. Springer-Verlag,
1978.
S. C. Bagchi, S. Madan, A.
Sitaram and U. B. Tiwari, A first course
on representation theory and linear Lie groups. University Press,
2000.
Serge Lang, SL(2, R). Graduate Texts in Mathematics,
105. Springer-Verlag, 1985.
W. Knapp, Representation theory of semisimple groups.
An overview based on examples. Princeton Mathematical Series, 36. Princeton
University Press, 1986.
Op9.
Partial Differential Equations II
Topics to be selected by the
teacher. A possible list of topics is given below:
Cauchy problem:
Cauchy-Kowalevska theorem and Holmgrens uniqueness theorem, properties of
hyperbolic polynomials, Cauchy problem for a hyperbolic equation.
Differential operators of
constant strength, existence when coefficients are continuous, hypoellipticity,
non-uniqueness.
Basic theory of pseudo
differential operators, L2 boundedness, Gardings inequality,
elliptic regularity theorem.
Semi-group theory,
applications to heat, Schrger and wave equations.
Spectra of differential
operators, random Schrger operators.
Suggested
books:
S. Kesavan, Topics in functional analysis and
applications. John Wiley & Sons, Inc., 1989.
L. Hnder, The analysis of linear partial differential
operators. II. Differential operators with constant coefficients.
Grundlehren der Mathematischen Wissenschaften, 257. Springer-Verlag, 1983.
L. Hnder, The analysis of linear partial differential
operators. III. Pseudodifferential operators. Grundlehren der Mathematischen
Wissenschaften, 274. Springer-Verlag, 1985.
M. E. Taylor, Pseudodifferential operators. Princeton
Mathematical Series, 34. Princeton University Press, 1981.
M. W. Wong, An introduction to pseudo-differential
operators. World Scientific Publishing Co., Inc., 1991.
E. B. Davies, Spectral theory and differential
operators. Cambridge Studies in Advanced Mathematics, 42. Cambridge
University Press, 1995.
R. Carmona and J. Lacroix,
Spectral theory of random Schrger
operators. Probability and its Applications. Birkher Boston, Inc., 1990.
Op10. Algebraic
Groups
Review of background
commutative algebra (facts on varieties and morphisms as in chapter 0 of
Humphreys's book1st reference below). Definition of linear algebraic groups and
homomorphisms over algebraically closed fields, examples. Orbit-closures under
actions, linearity of affine groups. Homogeneous spaces and quotients,
Chevalley's theorem. Commutative algebraic groups, diagonalizable groups and
algebraic tori. Definition of weights and roots, Weyl group. Unipotent groups,
Lie-Kolchin theorem, Structure theorem for connected solvable groups. Definition
of reductive and semisimple groups, Borel and parabolic subgroups. Basic facts
on complete varieties, Borel's fixed point theorem. Conjugacy of maximal tori,
Nilpotency of Cartan subgroups. Density theorem and connectedness of
centralisers of tori. Normaliser theorem for parabolics. Regular and singular
tori, Structure theorem for groups of semisimple rank one. Structure theorem for
reductive groups, Bruhat decomposition, semisimple groups. Tits system, standard
parabolics, simplicity proof. Root lattice and weight lattice, semisimple and
adjoint groups. Representations and their highest weights.
References:
J. E. Humphreys, Linear algebraic groups. Graduate Texts
in Mathematics, No. 21. Springer-Verlag, 1975.
T. A. Springer, Linear algebraic groups. Progress in
Mathematics, 9. Birkher, 1981.
R. Steinberg, Conjugacy classes in algebraic groups.
Lecture Notes in Mathematics, Vol. 366. Springer-Verlag, 1974.
Op11.
Algebraic and Differential Topology
Alexander-Lefschetz duality
in topological manifolds. De Rham cohomology of manifolds, de Rham theorem,
Stokes theorem. Computation of Cohomology rings of projective spaces,
Borsuk-Ulam theorem. Higher homotopy groups, fibration, homotopy exact sequence
of a pair and of a fibration. Poincare-Hopf theorem.
References.
R. Bott and L. W. Tu, Differential forms in algebraic
topology. Graduate Texts in Mathematics, 82. Springer-Verlag, 1982.
M. J. Greenberg, Lectures on algebraic topology. W. A.
Benjamin, Inc., 1967
F. W. Warner, Foundations of differentiable manifolds and
Lie groups. Graduate Texts in Mathematics, 94. Springer-Verlag, 1983.
Op
12. Advanced Functional Analysis
Brief introduction to
topological vector spaces (TVS) and locally convex TVS. Linear Operators.
Uniform Boundedness Principle. Geometric Hahn-Banach theorem and applications
(Markov-Kakutani fixed point theorem, Haar Measure on locally compact abelian
groups, Liapounovs theorem). Extreme points and Krein-Milman
theorem.
In addition, one of the
following topics:
(a)
Geometry of Banach spaces:
vector measures, Radon-Nikodym Property and geometric equivalents. Choquet
theory. Weak compactness and Eberlein-Smulian Theorem. Schauder
Basis.
(b)
Banach algebras, spectral
radius, maximal ideal space, Gelfand transform
(c)
Unbounded operators,
Domains, Graphs, Adjoints, spectral theorem
References:
N. Dunford and J. T.
Schwartz, Linear operators. Part II:
Spectral theory. Self adjoint operators in Hilbert space. Interscience
Publishers John Wiley & Sons 1963
Walter Rudin, Functional analysis. Second edition.
International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., 1991.
K. Yosida, Functional analysis. Grundlehren der
Mathematischen Wissenschaften, 123. Springer-Verlag, 1980.
J. Diestel and J. J. Uhl,
Jr., Vector measures. Mathematical
Surveys, No. 15. American Mathematical Society, 1977.
Op
13. Operator theory
I.
Compact operators on Hilbert
Spaces.
a)
Fredholm
Theory
b)
Index
II.
C*-algebrasnoncommutative
states and representations, Gelfand-Neumark representation
theorem
III.
Von-Neumann Algebras;
Projections, Double Commutant theorem, L^infinity functional
Calculus
IV.
Toeplitz
operators
References:
W. Arveson, An invitation to C*-algebras. Graduate
Texts in Mathematics, No. 39. Springer-Verlag, 1976.
N. Dunford and J. T.
Schwartz, Linear operators. Part II:
Spectral theory. Self adjoint operators in Hilbert space. Interscience
Publishers John Wiley & Sons 1963
R. V. Kadison and J. R.
Ringrose, Fundamentals of the theory of
operator algebras. Vol. I. Elementary theory. Pure and Applied Mathematics,
100. Academic Press, Inc., 1983.
V. S. Sunder, An invitation to von Neumann algebras.
Universitext. Springer-Verlag, 1987.
Op 14. Set
Theory
Either (a) or (b):
(a) Descriptive Set Theory
A quick review of elementary
cardinal and ordinal numbers, transfinite induction, induction on trees,
Idempotence of Souslin operation.
Polish spaces, Baire
category theorem, Transfer theorems Standard Borel spaces, Borel isomorphism
theorem, sets with Baire property, Kuratowski-Ulam Theorem. The projective
hierarchy and closure properties.
Analytic and coanalytic sets
and their regularity properties, separation and reduction theorems, Von Neumann
and Kuratowski-Ryll Nardzewskis selection theorems, Uniformization of Borel
sets with large and small sections. Kondos uniformization theorem.
References:
S. M. Srivastava, A course on Borel sets. Graduate Texts
in Mathematics, 180. Springer-Verlag, 1998.
S. Kechris, Classical descriptive set theory.
Graduate Texts in Mathematics, 156. Springer-Verlag, 1995.
(b)
Axiomatic Set Theory
A naive review of cardinal
and ordinal numbers including regular and singular cardinals, some large
cardinals like inaccessible and measurable cardinals. Martins Axiom and its
consequences. Axiomatic development of set theory upto foundation axiom, Class
and Class as models, relative consistency, absoluteness, Reflection principle,
Mostowski collapse lemma etc. , non-decidability of large cardinal axioms,
Godels second incompleteness theorem, Godels constructible universe, Forcing
lemma and independence of CH.
References:
K. Kunen, Set theory. An introduction to independence
proofs. Studies in Logic and the Foundations of Mathematics, 102.
North-Holland Publishing Co., 1980.
T. Jech, Set theory. Academic Press, 1978.
Op15.
Mathematical Logic
Propositional Logic,
Tautologies and Theorems of propositional Logic, Tautology Theorem.
First Order Logic: First
order languages and their structures, Proofs in a first order theory, Model of a
first order theory, validity theorems, Metatheorems of a first order theory, e.
g. , theorems on constants, equivalence theorem, deduction and variant theorems
etc. Completeness theorem, Compactness theorem, Extensions by definition of
first order theories, Interpretations theorem, Recursive functions,
Arithmatization of first order theories, Godels first Incompleteness theorem,
Rudiments of model theory including Lowenheim-Skolem theorem and categoricity.
References: J. R.
Shoenfield, Mathematical logic.
Addison-Wesley Publishing Co., 1967
Op16. Theory of
Computation
1.
Automata and Languages:
Finite automata, regular languages, regular expressions, equivalence of
deterministic and non-deterministic finite automata, minimisation of finite
automata, closure properties, Kleenes theorem, pumping lemma and its
applications, Myhill-Nerode theorem and its uses. Context-free grammar,
context-free languages, Chomsky normal form, closure properties, pumping lemma
for CFL, pushdown automata. Contextsensitive languages, Chomsky hierarchy,
Closure properties, linear bounded automata.
2.
Computability: Computable
functions, primitive and partial recursive functions, universality and halting
problem, recursive and recursively enumerable sets, parameter theorem,
diagonalisation and reducibility, Rice' theorem and its application, Turing
machines and its variants, equivalence of different models of computation and
Turing-Church thesis.
3.
Complexity: Time complexity
of deterministic and non-deterministic Turing machines, P and NP,
NP-completeness, Cooks theorem: other NP-complete problems.
Reference:
N. Cutland, Computability. An introduction to recursive
function theory. Cambridge University Press, 1980.
M. D. Davis, Ron Sigal and
E. J. Weyuker, Computability, complexity,
and languages. Fundamentals of theoretical computer science. Academic Press,
Inc., 1994.
J. E. Hopcroft and J. D.
Ullman, Introduction to automata theory,
languages, and computation. Addison-Wesley Publishing Co., 1979.
H. R. Lewis and C. H.
Papadimitriou, Elements of the theory of
computation, Prentice-Hall, 1981(**).
M. Sipser, Introduction to the theory of
computation. (**)
M. R. Garey and D. S.
Johnson, Computers and intractability. A
guide to the theory of NP-completeness. W. H. Freeman and Co., 1979.
Op17. Advanced Fluid
Dynamics
Inviscid Incompressible
fluid:
Two dimensional motion.
stream function, complex potential and velocity, sources, sinks. Doublets and
their images. Circle theorem, Blasiuss theorem, Kutta-Jokowaski theorem.
Axi-symmetric motion, Stokes stream function. Image of a source and a sink with
respect to a sphere.
Vortex motion, vortex lines
and filaments, systems of vortices, rectilinear vortices, vortex pair and
doublets. A single infinite row of vortices, Karmans vortex sheet.
Linearised gravity waves,
progressive waves in deep and shallow water, stationary waves, energy and group
velocity, long waves and their energy, capillary waves.
Inviscid compressible
fluid:
First and second law of
thermodynamics, polytropic gas and its entropy, adiabatic and isentropic flow,
propagation of small disturbances. Mach number, Mach cone, irrotional motion,
Bernoullis Equation, pressure, density and temperature in terms of Mach number.
Area velocity relations in one-dimensional flow, concept of subsonic and
supersonic flows. Normal shock-wave, Rankine-Hugonoit and Prandtls relations in
case of a plane shock wave.
Viscous incompressible
fluid:
Equations of motion of a
viscous fluid, Reynolds number, circulation in a viscous liquid, Flow between
parallel plates, flow through pipes of circular, elliptic and annular section
under constant pressure gradient. Prandtls concept of boundary layer.
Suggested
Texts:
L. M. Milne-Thomson, Theoretical hydrodynamics. The Macmillan
Co., 1960.
L. D. Landau and E. M.
Lifshitz, Fluid mechanics. Course of
Theoretical Physics, Vol. 6 Pergamon Press, 1959
H. Lamb, Hydrodynamics. Cambridge Mathematical
Library. Cambridge University Press, 1993.
W. H. Besant and A. S.
Ramsey, A treatise of
Hydro-mechanics, Part II, ELBS (**).
P. K. Kundu, Fluid mechanics, Academic Press
(**).
Op. 18. Quantum Mechanics I
1.
Physical Basis of Quantum
2.
Schrodinger wave equation (ii) Perturbation theory.
3.
Problem of two or more degrees of freedom without spherical symmetry; Stark effect
4.
Angular momentum, SU(2) algebra
5.
Three-dimensional Schrodinger equation. Problems with spherical symmetry .
6.
Harmonic Oscillator.
7.
Scattering problem , differential cross section, phase shift, variational principle, SW transformation, Regge poles.
8.
WKB approximation
9.
Particles with spin, Pauli matrices, Pauli-Schrodinger equation.
10.
Quantum Statistics.
Op19. Quantum Mechanics
II
1.
Non stationary
problems
2.
Relativistic Dirac equation,
Spinors.
3.
Scattering by a central
force.
4.
Radiation theory.
Quantization of Schrodinger field. Born approximation.
5.
Compton effect ( Klein
Nishina formula)
6.
Bremsstrahlung.
7.
Symmetry and conservation
laws.
8.
Quantum Probability and
quantum Statistics.
9.
Supersymmetric Quantum
Mechanics, SWKB. Path integral method. .
References:
1.
L. I. Schiff, Quantum Mechanics. (**)
2.
P. A. M. Dirac, The Principles of Quantum Mechanics. Oxford, at the
Clarendon Press, 1947.
3.
P. A. M. Dirac, Spinors in Hilbert space. Plenum
Press, 1974.
4.
M. E. Rose,
Elementary theory of angular momentum. John Wiley & Sons, Inc.
5.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and
Path integrals. (**)
6.
L. D. Landau and E. M. Lifshitz, Statistical physics. Course of Theoretical Physics. Vol. 5. Pergamon Press Ltd., 1958.
7.
S. FlPractical quantum mechanics. Classics in Mathematics. Springer-Verlag, Berlin, 1999.
8.
H. Weyl, The theory of groups and Quantum Mechanics. (**)
Op20.
Analytical Mechanics
Generalised coordinates,
Lagranges Equation. Examples of Lagranges equation. Conservation laws. Motion
in a central field. Collision of particles. Small Oscillations. Rotating
Coordinate systems. Inertial forces. Dynamics of a rigid body. Hamiltonian
Mechanics.
Suggested
Texts:
I. Arnold, Mathematical methods of classical
mechanics. Graduate Texts in Mathematics, 60. Springer-Verlag, 1978.
R. Abraham and J. E.
Marsden, Foundations of mechanics.
Second edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program,
1978.
Op21. Advanced Linear
Algebra
The course will cover topics
chosen from the following
Majorization and doubly
stochastic matrices. Matrix Decomposition Theorems
(Polar, QR, LR, SVD etc.)
and their applications. Perturbation Theory.
Nonnegative matrices and
their applications. Wavelets and the Fast Fourier
Transform. Basic ideas of
matrix computations.
Suggested Text: R. Bhatia,
Matrix analysis. Graduate Texts in
Mathematics, 169. Springer-Verlag.