Annual Foundation School - Part II (2014 @ KSOM)

1. Ring Theory.

1. Modules over Principal Ideal Domains Modules, direct sums, free modules, finitely geneover a PID, structure of finitely generated abelian groups, rational and Jordan canonical form.
2. Basics Commutative rings, nil radical, Jacobson radical, localization of rings and modulrings, primary decomposition of ideals and modules.
3. Integral extensions of rings, Going up and going down theorems, finiteness of intdiscrete valuation rings, Krull's normality criterion, Noether normalization lemma, Hilbert's Nullstellensatz
4. Semisimple rings Wedderburn's Theorem, Rings with chain conditions and ArWedderburn’s main theorem.

Speakers:M Manickam (KSOM, Kozhikode),Sanoli Gun (IMSc, Chennai),R. Thangadurai (HRI, Allahabad)
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. Functional Analysis.

1. Normed linear spaces, Continuous linear transformations, application to differential equBanach theorems-analytic and geometric versions, vector valued integration.
2. Bounded Linear maps on Banach Spaces Baire's theorem and applications: Uniformprinciple and application to Fourier series, Open mapping and closed graph theoremscomplemented subspaces, unbounded operators and adjoints.
3. Bounded linear functionals Weak and weak* topologies, Applications to reflexive sepauniformly convex spaces, Application to calculus of variations.
4. Hilbert spaces, Riesz representation theorem, Lax-Milgram lemma and applicationinequalities, Orthonormal bases, Applications to Fourier series and examples of special funcLegendre and Hermite polynomials.

Speakers:B. Ramakrishnan (HRI, Allahabad),D. S. Ramana (HRI, Allahabad),Purusottam Rath (CMI, Chennai)
References
(1) J. B. Conway, A Course in Functional Analysis, II edition, Springer, Berlin 1990.
(2) C. Goffman, G. Pedrick, First Course in Functional Analysis, Prentice-Hall, 1974.
(3) S. Kesavan, Functional Analysis Volume 52 of Texts and readings in mathematics, HAgency (India), 2009.
(4) B. B. Limaye, Functional Analysis, II edition New Age International, 1996.
(5) A. Tayor and D. Lay, Introduction to Functional Analysis, Wiley, New York, 1980.

3. Differential Topology.

1. Review of differential calculus of several variables: Inverse and implicit function theoremsmooth functions; smooth partition of unity, Submanifolds of Euclidean spaces (withboundary) Tangent space, embeddings, immersions and submersions, Regular valutheorem, Transversality and Stability. [The above material should be supported amply by exand examples from matrix groups.]
2. Abstract topological and smooth manifolds, partition of unity, Fundamental gluing lemmfor Hausdorffness of the quotient, classification of 1-manifolds. Definition of a vector bundbundle as an example. Sard's theorem. Easy Whitney embedding theorems.
3. Vector fields and isotopies Normal bundle and Tubular neighborhood theorem. Omanifolds and on normal bundles. Vector fields. Isotopy extension theorem. Disc Theorem. neighborhood theorem.
4. Intersection Theory: Transverse homotopy theorem and oriented intersection numbmaps both oriented and non oriented cases, winding number, Jordan Brouwer separation theorem, Borsuk- Ulam theorem.

Speakers:Krishna Hanumanthu* (CMI, Chennai), Sukhendu Mehrotra* (CMI, Chennai)
References:
(1) V. Gullemin and A. Pollack, Differential Topology, Englewood Cliff, N.J. Prentice Hall (19(2) W. Hirsch, Differential Topology, Springer-Verlag.
(3) J. W. Milnor, Topology from the Differential Viewpoint, Univ. Press, Verginia.
(4) Anant R. Shastri, Elements of Differential Topology, CRC Press, 2011.

 * to be confirmed.

 Time Schedule
Week-1

Week-2

Week-3

Week-4

A : Algebra (Ring Theory)
FA : Functional Analysis
DT : Differential Topology
T : Tutorial

Please read the following instructions carefully.

1.  All further inquiries/communications, regarding the school should be addressed to the respective local organizers. 
2. Selected candidates should communicate the confirmation of their participation on or before 05th March 2014, along with a copy of their onward journey ticket where-ever applicable, failing which their selection will be cancelled automatically.
3. If you are a selected candidate and decide NOT to attend the school for any reason what-so-ever, at any stage even after after confirmation, please communicate this immediately to the organizers so that some wait-listed candidates may benefit.
4. Organizers are requested to fill-up the vacancies as and when they occur by informing the candidates from the wait-list. 
5. Wait-listed candidates may therefore book their tickets so as not miss such a chance merely because they would not get a train-ticket.

How To Reach
Kerala School of Mathematics (KSOM) is located at a distance of about 15 Kms from the city centre (train/bus stations) situated off the Kozhikode-Medical College-Kunnamangalam bus route.Prepaid auto counter is available at Railway Station itself  (Near 1st number Platform).