The Annual Foundation School (AFS)-II



Funded by
National Board for Higher Mathematics

Jointly organized by
Bhaskaracharya Pratishthana, Pune and Dept. of Mathematics, Univ. of Pune
8 June - 4 July 2009.

Syllabus:

Algebra-II

(1) Group Theory: Group Actions. Prime-power Groups. Nilpotent Groups. Soluble Groups. Matrix Groups. Groups and Symmetry. (6 lectures)
(2) 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 Null- stellensatz, structure of artinian rings, Dedekind domains. (12 lectures)
(3) Introduction to Algebraic Number Theory: (6 lectures)
Text/References:
1. Algebra-I, I.S. Luther, I.B. S. Passi
2. M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra.
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.
(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. S. G. Krantz, Comlex Analysis: The Geometric View Points, Second edition, Carus Math. Monographs, MAA.
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.
(2) 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.
3. J. R. Munkres, Elements of Algebraic Topology,