This Annual Foundation School is aimed at first and second year Ph.D. students. The main objective is to bring up students with diverse backgrounds to a desirable level and help them acquire basic knowledge in Algebra, Analysis and Topology.   Knowledge of the material covered in AFS-I Schools in Group Theory, Real Analysis and Point-set Topology will be assumed. Scholars who have previously attended AFS-I are encouraged to apply and will be given top priority in selection. 

Participants will attend morning lectures in the topics of Ring Theory, Functional Analysis and Differential Topology. The afternoons will feature tutorials on these topics. A key goal of the program is the development of problem-solving ability and a mastery of examples and counter examples.

The School will draw expertise from reputed institutions such as IISER, Trivandrum, IISc Bangalore,TIFR Mumbai, IMSc, Chennai, IISER Pune, and RKMVU

1. Ring Theory
   1. Modules over Principal Ideal Domains Modules, direct sums, free modules, finitely generated modules over a PID, structure of a finitely generated abelian groups,rational and Jordan canonical form.
   2. Basics Commutative rings, nil radical, Jacobson radical, localization of rings and modules, Noetherian rings, primary decomposition of ideals and modules.
   3. Integral extensions of rings Going up and going down theorems, finiteness of integral closure, discrete valuation rings, Krulls normality criterion, Noether normalization lemma,Hilberts Nullstellensatz.
   4. Semisimple rings Wedderburns theorem, rings with chain conditionsand Artins theorem, Wedderburns main theorem.
   1. Speakers: Viji Thomas(IISER TVM) and K. N. Raghavan(IMSc)
   2. Course Associates: TBA
   5. References
   1. M.F. Atiyah and I.G. MacDonald:Introduction to Commutative Algebra, Westview Press.
   2. S. Lang: Algebra, Revised 3rd Edition, Springer.
   3. D.S. Dummit and R.M.Foote: Abstract Algebra, 2nd Edition, John-Wiley.
   4. N. Jacobson: Basic Algebra, Vol.1 and Vol.2, Dover,2011.
   5. A.W. Knapp: Advanced Algebra, Birkhauser,2011.
2. Functional Analysis
1. Normed linear spaces Continuous linear transformation, application to differential equations, Hahn-Banach theorems - analytic and geometric versions, vector valued integration.
2. Bounded linear maps on Banach spaces Baires theorem and applications, uniform boundedness principle and applications to Fourier series, open mapping and closed graph theorems, annihilators, complemented subspaces, unbounded operators and adjoints.
3. Bounded linear functionals Weak and weak* topologies, applications to reflexive separable spaces, uniformly convex spaces, application to calculus of variations.
4. 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.
1. Speakers: Siddhartha Bhattacharya (TIFR Mumbai), Manjunath Krishnapur(IISc), Partha Sarathi Chakraborty(IMSc)
2. Course Associates: TBA
5. References
1. J.B. Conway: A course in Functional Analysis, 2nd Edition, Springer, Berlin, 1990.
2. C. Goffman and G. Pedrick: First Course in Functional Analysis, Prentice Hall, 1974.
3. S. Kesavan: Functional Analysis, Volume 52, TRIM, Hindustan Book Agency (India), 2009.
4. B.V. Limaye: Functional Analysis, 2nd Edition, New Age International, 1996.
5. A.Taylor 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 theorems, richness of smooth functions, smooth partition of unity, submanifolds of Euclidean spaces ( without and with boundary), tangent space, embeddings, immersions and submersions, regular values, pre-image theorem, transversality and stability.[The above material should be supported amply by exercises and examples from matrix groups.]
2. Abstract topological and smooth manifolds Partition of Unity, fundamental gluing lemma with criterion for Hausdorffness of the quotient, classification of 1-manifolds, definition of a vector bundle (and tangent bundle as an example), Sards theorem, easy Whitney embedding theorems.
3. Vector fields and isotopies Normal bundle and tubular neighborhood theorem, orientation on manifolds and on normal bundles, Vector fields, isotopy extension theorem, disc theorem, collar neighborhood theorem.
4. Intersection Theory Transverse homotopy theorem and oriented intersection number, degree of maps (both oriented and non-oriented cases), winding number, Jordan Brouwer separation theorem, Borsuk - Ulam theorem.
1. Speakers: Tejas Kalelkar (IISER Pune), Samik Basu (RKMVU)
2. Course Associates: TBA
5. References
1. V. Guillemin and A. Pollack: Differential Topology, AMS Chelsea Publishing, AMS, Providence,1974.
2. Morris W. Hirsch: Differential Topology, Springer-Verlag, New York Inc., 1976.
3. John W. Milnor: Topology from the Differentiable Viewpoint, Princeton Univ. Press, Princeton, New Jersey,1965.
4. Anant R. Shastri: Elements of Differential Topology, CRC Press, 2011.

Time Table:

• RTn : nth lecture in ring theory
• TRTn: nth tutorial in ring theory
• FAn : nth lecture in functional analysis
• TFAn: nth tutorial in functional analysis
• DTn : nth lecture in differential topology
• TDTn: nth tutorial in differential topology