Annual Foundation School - Part II (2012)

 2.1. Ring Theory.

(1) Module 1: (6 lectures) Commutative rings, nil radical, Jacobson radical, localization of rings and modules, Noetherian rings, primary decomposition of ideals and modules.
(2) Module 2: (6 lectures) Integral extensions of rings, Going up and going down theorems, finiteness of integral closure, discrete valuation rings, Krull’s normality criterion, Noether normalization lemma,Hilbert’s Nullstellensatz
(3) Module 3 : 6 lectures Semisimple rings, Wedderburn’s Theorem, Rings with chain conditions and Artin’s theorem, Wedderburn’s main theorem,
(4) Module 4 : 6 lectures Semisimplicity and tensor products, SkolemNoether theorem, Wedderburn’s theorem about finite division rings,Frobenious theorem about division algebras over the reals.

References:
(1) S. Lang, Algebra, 3rd edition, Addison-Wesley, (2) D. S. Dummit and R.
M. Foote, Abstract Algebra, 2nd edition John-Wiley (3) N. Jacobson, Basic
Algebra, Vol. 1 and 2, Dover, 2011. (4) A. W. Knapp, Advanced Algebra,
Birkhauser, 2011.

 2.2. Functional Analysis. Chapters refer to the ones in [K].

(1) Module 1 : 6 lectures Normed linear spaces, Continuous linear transformations, application to differential equations,Hahn-Banach theorems-analytic and geometric versions, vector valued integration
(2) Module 2 : 6 lectures Baire’s theorem and applications: Uniform boundedness principle and application to Fourier series, Open map-
ping and closed graph theorems, annihilators, complemented subspaces, unbounded operators and adjoints
(3) Module 3 : 6 lectures Weak and weak* topologies, Applications to reflexive separable spaces, Uniformly convex spaces, Application to calculus of variations
(4) Module 4 : 6 lectures : Chapter 7 of [K] Hilbert spaces, Riesz representation theorem, Lax-Milgram lemma and application to variational inequalities, Orthonormal bases, Applications to Fourier series and examples of special functions like Legendre and Hermite polynomials.

2.3. Algebraic Topology. Numbers in brackets indicate sections from [Sh].

(1) Module 1: 6 lectures: Statement of basic problems in Algebraic Topology: extension problems and lifting problems; homotopy, relative homotopy, deformation, contraction, retracts etc.(1.1),Typical constructions: Adjunction spaces, Mapping cones, Mapping cylinder, Smash-product, reduced cones reduced suspension etc.(1.2,1.3), Categories and Functors. Definition and examples. Equivalence of functors, adjoint functors, examples.(1.4), Computation of fundamental group of the circle and applications. (3.1,3.2).
(2) Module 2: 6 lectures: Spaces obtained by attaching n-cells. Basic topological properties. Tubular neighbourhood theorem. Examples (2.1), CW-complexes and basic topological properties. Products of CW complexes. (2.1), Homotopy theoretic properties of CW compelxes.(2.1), Abstract Simplicial complexes and geometric realization. (2.2), Barycentric subdivision and Simplicial Approximation theorem.(2.3), Cellular Approximation theorem and applications.(2.4)
(3) Module 3: 6 lectures: Covering Spaces. (4.1), Lifting properties.(4.2), Relation with fundamental group. (4.3) Classification (proof of existence may be skipped).(4.4), Computation of Fundamental groups: simpler cases of Van-Kampen theorem. Effect of attaching n-cells.(4.6).
(4) Module 4: 6 lectures: Chain complexes, exact sequences of complexes, snake lemma, four lemma and five lemma. Homology long exact sequence, Axioms for homology, Construction of Singular chain complex, verification of axioms (except homotopy axiom and excisionaxiom), Simplicial and Singular simplicial chain complexes. Statement of equivalence of all these homologies, Computations, Separation theorems, Invariance of Domain. Euler charasteristic.

References:
[H] A Hatcher, Algebraic Topology, Cambridge University Press.
[M] Algebraic Topology, C. R. F. Maunder, Van Nostrand Reinhold Companay. London.
[Sp] E. H. Spanier, Algebraic Topology, Tata McGraw-Hill.
[Sh] A. R. Shastri, Basic Algebraic Topology, http//www.math.iitb.ac.in/ ars

• The selected participants are requested to confirm their participation by 31st March 2012 for booking accommodation.
• Confirmation mail should be sent to sakatre at gmail.com and to "Vinayak Joshi"  vinayakjoshi111 at gmail.com
• Travel by 3rd AC Train fare is allowed to the participants.
• The participants are encouraged to bring along books related to topics of AFS-II.

University of Pune is on the University Road, and it is about 5 km from the Main Railway Station and Main Bus stand. The nearest Railway Station is Shivajinagar Railway Station (3km) and it is most convenient while coming from Mumbai side, but one should make sure that the train stops there. The  autorickshaw fare is about 10 times the meter reading.

From airport the University is about 16 km.

When one enters the university there are 2 roads, take the right road to go to Mathematics Dept. or to University Guest House/ Univ. SET Guest House. Dept. of Math. comes first on this road on the left side, and it is a building  with Green Colour Fence. To go to SET Guest House, one goes ahead and at the end turns to the
left, to go near main Guest House. The main Guest House is very near the main building (with Flag) and is next to VC Bungalow and from the main Guest House, a road goes to IUCAA. The SET Guest House is located on this road about 150 meters from main Guest House on the right side.