Annual Foundation School - Part II (2013)

School Convener(s)

Speakers and Syllabus 

1. Alok Maloo (IIT, Kanpur)
2. Sameer Chavan (IIT, Kanpur)
3. Vishnu Namboodiri (Central University of Hyd.)

1. S. Dutta (IIT, Kanpur)
2. P. Shunmugaraj (IIT, Kanpur)
3. P. Mohanty (IIT, Kanpur)

1. Mahuya Datta (ISI, Kolkata)
2. Nandini Nilakantan (IIT, Kanpur)
3. Rekha Santhanam (IIT, Bombay)

1. G. M. Misra (IISC, Bangaluru)
2. B. V. Rajaram Bhat (ISI, Bangaluru)

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 chainconditions and Artin’s theorem, Wedderburn’s main theorem.
4. Module 4: (6 lectures) Semisimplicity and tensor products, Skolem Noether theorem, Wedderburn’s theorem about finite division rings, Frobenious theorem about division algebras over the reals.

References:

1. 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.

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 mapping 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) Hilbert spaces, Riesz representation, Lax - Milgram lemma and application to variational inequalities, Orthonormal bases, Applications to Fourier series and examples of special functions like Legendre and Hermite polynomials.

References:

1. Rudin, Functional Analysis.
2. J. B. Conway,  A course in Functional Analysis.
3. P. Lax, Functional Analysis.

1. Module 1: (6 lectures) Statement of basic problems in Algebraic Topology: extension problems and lifting problems; homotopy, relative homotopy, deformation, contraction, retracts etc., Typical constructions: Adjunction spaces, Mapping cones, Mapping cylinder, Smash - product, reduced cones reduced suspension etc. Categories and Functors, Definition and examples. Equivalence of functors, adjoint functors, examples, Computation of fundamental group of the circle and applications.
2. Module 2: (6 lectures) Spaces obtained by attaching n - cells. Basic topological properties, Tubular neighbourhood theorem . Examples, CW -  complexes and basic topological properties, Products of CW complexes, Homotopy theoretic properties of CW complexes, Abstract Simplicial complexes and geometric realization, Barycentric subdivision and Simplicial Approximation theorem, Cellular Approximation theorem and applications.
3. Module 3: (6 lectures) Covering Spaces, Lifting properties, Relation with fundamental group, Classification (proof of existence may be skipped), Computation of Fundamental groups:simpler cases of Van - Kampen theorem. Effect of attaching n - cells.
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 excision axiom), Simplicial and Singular simplicial chain complexes. Statement of equivalence of all these homologies, Computations, Separation theorems, Invariance of Domain, Euler characteristic.

References:

1. (1) A. Hatcher, Algebraic Topology, Cambridge University Press.
2. C.R.F. Maunder, Algebraic Topology, Van Nostrand Reinhold Company, London.
3. E.H. Spanier, Algebraic Topology, Tata McGraw - Hill.
4. A.R. Shastri, Basic Alebraic Topology, http/www.math.iitb.ac.in/ars (under construction).

Monday

06/05/13

Tuesday

07/05/13

Wednesday

08/05/13

Thursday

09/05/13

Friday

10/05/13

Saturday

11/05/13

02.30 pm - 03.30 pm
and
03.45 pm - 04.45 pm

Monday

13/05/13

Tuesday

14/05/13

Wednesday

15/05/13

Thursday

16/05/13

Friday

17/05/13

Saturday

18/05/13

02.30 pm - 03.30 pm
and
03.45 pm - 04.45 pm

Monday

20/05/13

Tuesday

21/05/13

Wednesday

22/05/13

Thursday

23/05/13

Friday

24/05/13

Saturday

25/05/13

02.30 pm - 03.30 pm
and
03.45 pm - 04.45 pm

Monday

27/05/13

Tuesday

28/05/13

Wednesday

29/05/13

Thursday

30/05/13

Friday

31/05/13

Saturday

01/06/13

02.30 pm - 03.30 pm
and
03.45 pm - 04.45 pm