Annual Foundation School – II (2016)


Kerala School of Mathematics


06 June – 01 July, 2016


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name Prof. M Manickam Prof. A J Parameswaran Dr. A K Vijayarajan
Mailing Address Director
Kerala School of  Mathematics,
Kunnamangalam, P.O
Kozhikode – 673571,  Kerala
Tata Institute of Fundamental Research , Mumbai KSOM, Calicut


Speakers and Syllabus 

 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.


Day Date Lecture 1
Lecturer 2
Tutorial 1
Tutorial 2
    name of the speaker   name of the speaker   name of the speaker/tutor    
Mon 06-06-2016 AV   AKV   Alg-Tut -1   Alg-Tut -2
Tues 07-06-2016 PD   AV   Ana-Tut-1   Ana-Tut-2
Wed 08-06-2016 AV   PD   Top-Tut-1   Top-Tut-2
Thu 09-06-2016 AV   AKV   Alg-Tut -3   Alg-Tut -4
Fri 10-06-2016 PD   AKV   Ana-Tut-3   Ana-Tut-4
Sat 11-06-2016 AKV   PD   Top-Tut-3   Top-Tut-4
Mon 13-06-2016 SV   KS   Alg-Tut -5   Alg-Tut -6
Tues 14-06-2016 AS   SV   Ana-Tut-5   Ana-Tut-6
Wed 15-06-2016 KS   AS   Top-Tut-5   Top-Tut-6
Thu 16-06-2016 SV   KS   Alg-Tut -7   Alg-Tut -8
Fri 17-06-2016 AS   SV   Ana-Tut-7   Ana-Tut-8
Sat 18-06-2016 KS   AS   Top-Tut-7   Top-Tut-8
Day Date Lecture 1
Lecturer 2
Tutorial 1
Tutorial 2
Mon 20-06-2016 MK   GR   Alg-Tut -9   Alg-Tut -10
Tues 21-06-2016 AJJ   MK   Ana-Tut-9   Ana-Tut-10
Wed 22-06-2016 GR   AJJ   Top-Tut-9   Top-Tut-10
Thu 23-06-2016 MK   GR   Alg-Tut -11   Alg-Tut -12
Fri 24-06-2016 AJJ   MK   Ana-Tut-11   Ana-Tut-12
Sat 25-06-2016 GR   AJJ   Top-Tut-11   Top-Tut-12
Mon 27-06-2016 SN   BVR   Alg-Tut -13   Alg-Tut -14
Tues 28-06-2016 BVR   SN   BVR   Ana-Tut-14
Wed 29-06-2016 GR   AKV   Top-Tut-13   Top-Tut-14
Thu 30-06-2016 SN   AJP   Alg-Tut -15   Alg-Tut -16
Fri 01-07-2016 AJP   SN   Ana-Tut-15   Ana-Tut-16

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)
Actual Participants 


Serial SID Full Name Gender Affiliation State

Position in College/ University

1 6262 Mr. Athul P Male NIT , Calicut Kerala PhD
2 6327 Ms Thenmozhi Shunmugam Female Bharathidasan University Tamil Nadu PhD
3 6461 Ms M Sabari Female NIT, Karnataka Karnataka PhD
4 6466 Ms. Indumathi A Female Bon Secours College for women Tamil Nadu Assistant professor
5 6567/6500 Mr. B Janaki Raman Male Ramanujan Institute for Advanced Study Tamil Nadu M.Sc Student
6  6626 Mr. Chaitanya G K Male NIT, Karnataka Karnataka PhD Student
7 6765 Mr Repana Devendra Male University of Hyderabad Andhra Pradesh PhD
8 6769 Mr. Ankit Pal Male Savitribai University Of Pune Maharashtra MSc Student
9 7038 Ms. B Prasuna Female University of Hyderabad Andhra Pradesh MSc Student
10 7049 Mr. Nidhish Unnikrishnan Male University of Hyderabad Andhra Pradesh Ph.D. Student
11 6407 Ms R.Eswari - Rajendran Female Bharathidasan University Tamil Nadu Ph.D
12 7274 Mr. Ashok Kumar K Male Department of Mathematics Tamil Nadu PhD
13 6170 Ms.Janani Jayalakshmi Govindarajan Female Bharathidasan University Tamil Nadu PhD
14 6246 Mrs Dhivya Pari Rajmohan Female Bharathidasan University Tamil Nadu PhD Student
15 6364 Ms. Prerona Dutta Female Pondicherry University Pondicherry M.Sc. Student
16 6365 Ms Renuka Kannan Female Bharathidasan University Tamil Nadu MPhil
17 6478 Mr. Sreedeep C D Male NIT, Karnataka Karnataka PhD Student
18 6995 Ms. Reewa Malik Female Banasthali University Rajasthan MSc Student
19 7062 Mr James T Kurian Male Pondicherry University Pondicherry MSc student
20 7137 Mr. Sumit Roy Male Ramakrishna Mission Vivekananda University West Bengal MSc Student
21 7146 Ms. Anushree Jaiswal Female University of Hyderabad Andhra Pradesh MSc Student
22 6392 Mr Geno Kadwin J Male Bharathidasan University Tamil Nadu Phd
23 - Mr.Sandeep E M Male KSOM, Kozhikode, Kerala Research Scholar
24 - Nidhish Unnikrishnan Male University of Hyderabad, Telangana Ph.D. Student
25 - Mohammed Hussain KK Male NIT, Calicut   Research Scholar



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).