Annual Foundation School - II (2015) - IISER, Trivandrum -Speakers and Syllabus

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:

Week - 1
Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
11 May RT1

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12 May DT1 RT2 TDT1 TRT2
13 May FA2 DT2 TFA2 TDT2
14 May RT3 FA3 TRT3 TFA3
15 May DT3 RT4 TDT3 TRT4
16 May FA4 DT4 TFA4 TDT4

 

Week - 2
Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
18 May  RT5

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19 May  DT5  RT6 TDT5 TRT6
20 May  FA6  DT6 TFA6 TDT6
21 May  RT7  FA7 TRT7 TFA7
22 May  DT7  RT8 TDT7 TRT8
23 May  FA8  DT8 TFA8 TDT8

 

Week - 3
Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
25 May RT9

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26 May DT9 RT10 TDT9 TRT10
27 May FA10 DT10 TFA10 TDT10
28 May RT11 FA11 TRT11 TFA11
29 May DT11 RT12 TDT11 TRT12
30 May FA12 DT12 TFA12 TDT12

 

Week - 4
Date 9.30-11.00 11.00-11.30 11.30-1.00 1.00-2.30 2.30-3.30 3.30-4.00 4.00-5.00
1 June RT13

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2 June DT13 RT14 TDT13 TRT14
3 June FA14 DT14 TFA14 TDT14
4 June RT15 FA15 TRT15 TFA15
5 June DT15 RT16 TDT15 TRT16
6 June FA16 DT16 TFA16 TDT16
  • 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