Annual Foundation School – II (2016)

 Anandavardhanan (IIT, Bombay)
Lecture 1, June 06: Basics of Rings & Modules, Examples, Free modules.
Lecture 2, June 07: Discussed free modules over a PID in some detail.
Lecture 3, June 08: Discussed structure theorem for a f.g module over a PID.
Lecture 4, June 09: Applications to canonical forms. Also discussed the elementary divisors theorem.
Lectures closely followed relevant sections in Lang's Algebra.

Sandeep Varma (TIFR Mumbai)
The topics  covered were: brief introduction to the following topics: Noetherian rings and modules (proof of Hilbert's basis theorem omitted), localization (many proofs were omitted), nil radical and Jacobson radical, primary decomposition (stated the result, sketched part of the proof of existence).

Manish Kumar (ISI Bangalore)
Except for valuation rings, other materials from the syllabus was discussed. Integral extensions were talked about in details, their behaviour with respect to Quotients and localizations, properties of normal domains, going up and going theorem were proved. Noether normalization was also proved. Finiteness of ingeral closure were discussed (the proofs require background (galois theory) which the participants lacked and hence were only sketched). Various versions of Hilbert nullstellensatz was also discussed though there was not enough time to prove it. In my opinion the syllabus needs to be reworked a little bit. In particular each of the three AFS should have a mix of Groups, Rings and Fields to sort out interdependency. And given the general nature of the audience some materials can be taken of the syllabus.

Suresh Naik (ISI, Bangalore)
1.Basic examples of finite dimensional commutative algebras over a field,classified all 2-dimensional algebras over real and complex numbers and modules over such algebras, role of idempotents and nilpotents.
2.Classified all reduced finite-dimensional algebrasover complex numbers. Introduced divisional algebras. Described (left, right,2-sided) ideals in matrix algebras over a division ring.
3.Simple and semi-simple modules.Described finitely generated (left) modules over matrix algebras.
4.Schur lemma,Artin-Wedderburn decomposition for finite-dimensional semi-simple algebras over a field, central division algebras over complex numbers. Mentioned group rings for finite groups.

 Priyavrath D (CMI, Chennai)
Lecture 1: Review of derivatives including and inverse and implicit function theorems. Unfortunately I was not able to spend time on smooth partitions of unity. I did try and give an intuitive idea though.
Lecture 2: Manifolds in Euclidean spaces with relevant definitions and examples; submanifolds, products of manifolds etc.
Lecture 3: Tangent spaces and derivative of a map between manifolds.
Lecture 4: immersions, submersions and the preimage theorem (with all the standard examples). I could not reach stability and transversality.

Anant Shastri (IIT Bombay)
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. Sard’s theorem. Easy Whitney embedding theorems.
 

Jayanthan A.J. (Uni. Goa)
Vector fields and isotopies Normal bundle and Tubular neigh- bourhood theorem. Orientation on manifolds and on normal bun- dles. Vector fields. Isotopy extension theorem. Disc Theorem. Col- lar neighbourhood theorem.

A J Parameswaran (TIFR, Mumbai)
Intersection Theory: Transverse homotopy theorem and oriented intersection number. Degree of maps both oriented and non oriented cases, winding number, Jordan Brouwer separation theorem, BorsukUlam theorem.

 A K Vijayarajan (KSOM, Kozhikode)
Lect 1: General theory of normed linear spaces with an emphasis on examples and simple properties,Demonstration of algebra-analysis interaction, subspaces, and quotient spaces.
Lect. 2: Continuous linear maps on normed linear spaces, functionals, and operators. Examples of continuous linear maps. Banach's fixed point theorem and application to Picard's theorem.
Lect 3: Computation of norms of linear transformations, Hahn-Banach extension theorem, and consequences, Equivalence of norms and isometric isomorphisms, dual spaces with several examples.
Lect 4: Separating convex sets using linear functionals, Hahn-Banach separation theorem, and vector-valued integration.

K. Sumesh (IMSc, Chennai)
Lecture 1. Baire's category theorem-different versions, applications, examples, counter examples.
Lecture 2. Strong (pointwise) and uniform boundedness, uniform boundedness principle, Banach Steinhaus theorem, strong (pointwise) and uniform convergence, strong and uniform closed subspaces of B(X, Y )
Lecture 3. Open mapping theorem, bounded inverse theorem and two-norm theorem.
Lecture 4. Closed graph theorem, projections and complemented subspaces.

G.Ramesh (IIT Hyderabad)
Lecture 1: Topology, basis, subbasis, topology generated by family of functions (weak topology),examples, properties.
Lecture 2: Weak convergence, weak boundedness, weakly closed sets, weak and norm topologies on finite dimensional normed linear spaces, the weak and the norm topologies are not the same in infinite dimensional normed linear spaces
Lecture 3: Weak star topology, Banach-Alaouglu's theorem, applications
Lecture 4: Reflexive spaces, Kakutani's theorem, other characterizations of reflexivity, Uniformly convex spaces, examples, Milman-Petti's theorem (statement only) and theorem due to M. M. Day (statement only)
Lecture 5: Best approximation in a Hilbert space, projection theorem, existence of orthogonal projections, Riesz-representation theorem, existence of adjoint of a bounded operator, spectrum of an operator, computation of spectrum for few operators.

B V Rajarama Bhat (ISI Bangalore)
Hilbert spaces, Riesz representation theorem, Lax-Milgram lemma and application to variational inequalities, Orthonormal bases, Ap- plications to Fourier series and examples of special functions like Legendre and Hermite polynomials.

Time-Table

Resource Persons:

• AV  Anandavardhanan (IIT, Bombay)
• SV Sandeep Varma (TIFR Mumbai)
• MK Manish Kumar (ISI Bangalore)
• SN Suresh Naik (ISI, Bangalore)
• PD Priyavrath D (CMI, Chennai)
• AS Anant Shastri (IIT Bombay)
• AJJ Jayanthan A.J. (Uni. Goa)
• AJP A J Parameswaran (TIFR, Mumbai)
• AKV A K Vijayarajan (KSOM, Kozhikode)
• KS K. Sumesh (IMSc, Chennai)
• GR G.Ramesh (IIT Hyderabad)
• BVR G.Ramesh (IIT Hyderabad)

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).