Annual Foundation Schools (AFS)
Basic knowledge in algebra,
analysis, discrete mathematics and topology forms the core of all advanced
instructional schools the schools to be organized in this programme.
The
objective of the Annual Foundation Schools, to be offered in Winter and Summer
every year, is two fold:
- To bring up students with diverse backgrounds to a
common level.
- To identify those who are fit for further training.
Any student
who wishes to attend the advanced instructional schools is strongly encouraged
to enroll in the Annual Foundation Schools.
The topics listed in
the syllabi will be quickly covered in the lectures. There will be intensive
problem sessions in the afternoons. The objective will not be to cover the
syllabus prescribed, but to inculcate the habit of problem solving. However, the
participants will be asked to study all the topics in the syllabus at home since
the syllabi of these schools will be assumed in all the advanced instructional
schools devoted to individual subjects.
These schools will
admit 40 students in their first and second years of Ph. D. programme, students
of M. Sc. (II Year), university lecturers and college teachers who lack the
knowledge of basic topics covered in these schools.
A participant who has
attended AFS-I and II will never be allowed to attend these again.
Algebra-II
(1) Basic commutative algebra-I: Prime ideals and maximal ideals, Zariski topology, Nil and Jacobson radicals, Localization of rings and modules, Noetherian rings, Hilbert Basis theorem, modules, primary decomposition, integral dependence, Noether normalization lemma, principal ideal theorem, Hilbert's Nullstellensatz, structure of artinian rings, Dedekind domains. (12 lectures)
(2) Introduction to Algebraic Number Theory: (6 lectures)
(3) Introduction to Algebraic Geometry: (6 lectures)
Text/References:
1. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra.
2. D. Eisenbud, Commutative algebra with a view towards algebraic geometry.
3. P. Samuel, Algebraic Number Theory.
Complex Analysis
(1) Euclidean similarity geometry, inversive geometry, hyperbolic geometry and complex analysis.
(2) Analytic functions, Path integrals, Winding number, Cauchy integral formula and consequences. Hadamard gap theorem, Isolated singularities, Residue theorem, Liouville theorem.4
(3) Casorati-Weierstrass theorem, Bloch-Landau theorem, Picard's theorems, Mobius transformations, Schwartz lemma, Extremal metrics, Riemann mapping theorem, Argument principle, Rouche's theorem.
(4) Runge's theorem, Infinite products, Weierstrass p-function, Mittag-Leffler expansion.
Text/References:
1. Murali Rao & H. Stetkaer, Complex Analysis, World Scientific, 1991
2. L. V. Ahlfors, Complex Analysis, McGraw-Hill, Inc., 1996
3. A. R. Shastri, Complex Analysis
4. Krantz
5. A. F. Beardon, Geometry of Discrete Groups, GTM Springer Verlag.
Algebraic Topology
(1) Basic notion of homotopy; contractibility, deformation etc. Some basic constructions such as cone, suspension, mapping cylinder etc. fundamental group; computation for the circle. Covering spaces and fundamental group. Simplicial Complexes, CW complexes.
(2) Simplicial Complexes, CW complexes. Homology theory and applications: Simplicial homology, Singular homology, Cellular homology of CW-complexes, Jordan-Brouwer separation theorem, invariance of domain, Lefschetz fixed point theorem etc.
(3) Categories and functors; Axiomatic homology theory.
Texts/References:
1. E. H. Spanier, Algebraic Topology, Tata-McGraw-Hill
2. A. Hatcher, Algebraic topology, Cambridge University Press.
Schedule of Lectures
| Lecture |
9.00 - 10.00 |
| Lecture |
10.30 - 11.30 |
| Lecture |
11.45 - 12.45 |
| Tutorial |
2.15 - 4.15 |
| UM Lecture |
4.30 - 5.30 |
Syllabus for Annual Foundation School-II (December 2004)
Algebra II
(1) Homological algebra: Derived functors, projective modules, injective modules, free and
projective resolutions, tensor, exterior and symmetric algebras, injective
resolutions, Ext and Tor. ( 12 lectures)
(2) Basic commutative algebra: Prime
ideals and maximal ideals, Zariski topology, Nil and Jacobson radicals,
Localization of rings and modules, Noetherian rings, Hilbert Basis theorem,
modules, primary decomposition, integral dependence, Noether normalization
lemma, principal ideal theorem, Hilbert's Nullstellensatz, structure of artinian
rings, Dedekind domains. (12 lectures)
Text/References:
1. M. F. Atiyah and I.
G. Macdonald, Introduction to commutative algebra.
2. D. Eisenbud, Commutative
algebra with a view towards algebraic geometry.
Complex analysis
(1) Analytic
functions, Path integrals, Winding number, Cauchy integral formula and
consequences. Hadamard gap theorem, Isolated singularities, Residue theorem,
Liouville theorem.4
(2) Casorati-Weierstrass theorem, Bloch-Landau theorem,
Picard's theorems, Mobius transformations, Schwartz lemma, Extremal metrics,
Riemann mapping theorem, Argument principle, Rouche's theorem.
(3) Runge's theorem, Infinite products, Weierstrass p-function, Mittag-Leffler expansion.
Text: Complex analysis by Murali Rao & H. Stetkaer, World Scientific, 1991
Algebraic topology
(1) Basic notion of homotopy; contractibility, deformation
etc. Some basic constructions such as cone, suspension, mapping cylinder etc.
fundamental group; computation for the circle. Covering spaces and fundamental
group. Simplicial Complexes, CW complexes.
(2) Simplicial Complexes, CW
complexes. Homology theory and applications: Simplicial homology, Singular
homology, Cellular homology of CW-complexes, Jordan-Brouwer separation theorem,
invariance of domain, Lefschetz fixed point theorem etc.
(3) Categories and
functors; Axiomatic homology theory.
Texts/References
1. E. H. Spanier, Algebraic Topology, Tata-McGraw-Hill
2. A. Hatcher, Algebraic topology, Cambridge University Press.
Number Theory Arithmetic functions, congruences, quadratic
residues, quadratic forms, Diophantine approximations, quadratic fields,
Diophantine equations.
Text/References :
1. A. Baker, Theory of numbers.
2. K. Ireland and M. Rosen, A classical introduction to modern number
theory.