Annual Foundation School (AFS) - II (2016)
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

06 June - 01 July, 2016


Prof. M Manickam
Prof. A J Parameswaran
TIFR Centre for Applicable Mathematics

The Annual Foundations Schools (AFS) are targetted toward fresh Ph. D. Students in the  Universities and Research Institutions. Here in this programme the 25 participants of the program consisted of 8 MSc students, 1 MPhil student, 15 research scholars and 1 college teacher. The program as scheduled started on the morning of 6th June and ended on the evening of 1st July.

Topics covered in the workshop

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.

Suresh Naik (ISI, Bangalore)

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.  Classified all reduced finite-dimensional algebras over complex numbers. Introduced divisional algebras. Described (left, right, 2-sided) ideals in matrix algebras over a division ring.  Simple and semi-simple modules. Described finitely generated (left) modules over matrix algebras.  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.
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).

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, Borsuk- Ulam 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.

Extra topics covered included adjoint operators, Banach algebra, Second dual and reflexivity.
Lecture 5: During this extra session, considering the topics already covered, weak topologies on the space of bounded linear operators on a Hilbert space was discussed. Double commutants, density theorem,  and von Neumann's double commutant theorem were discussed and defined von zNeumann algebra.

For the most part,   the books by S. Kesavan and B. V. Limaye on Functional Analysis were used  as references.

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 Schedule


Workshop on Automorphic forms
Partially supported by HRI, Allahabad

Venue: KSOM, Kozhikode

10-16 February, 2016


Prof. M Manickam
Prof. B Ramakrishnan
Dear & Professor 
HRI, Allahabad

Aim: To bring Mathematicians working in the area of Automorphic Forms especially on Modular Forms, Jacobi Forms and Siegel Modular Forms to share their current research work among the participants. 



Time Schedule

ATM Workshop on PDE & Mechanics (2016)
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

01-06 February, 2016


Prof. M Manickam
Prof. S D Veerappa Gowda
TIFR Centre for Applicable Mathematics

Topics covered in the workshop

Non linear hyperbolic conservation laws play a central role in science and engineering and  form the basis for the mathematical modeling of many physical systems. Their  theoretical and numerical analysis thus plays an important role in applied mathematics and applications. Hyperbolic conservation laws present unique challenges for both theory and numerics as smoothness of their  solution breaks down and produce discontinuities. The main aim of this workshop is to introduce this area  to the young researchers starting from the basics to the advanced level from theoretical as well as computational point of view so that they can  take up this area for their further research.


Veerappa Gowda, TIFR-CAM,Bangalore

Scalar conservation laws and Hamilton-Jacobi equations: Hamilton-Jacobi equations,Legendre transform,Hopf-Lax formula, viscosity solutions. Lax-Oleinik formula for the solution of convex conservation laws and its long time behaviour.

Adimurthi, TIFR-CAM,Bangalore

Conservation laws: weak solutions, entropy conditions, the viscous problem, Existence of an Entropy solution for scalar conservation laws.

Uniqueness result.

K T Joseph, TIFR-CAM,Bangalore

Systems of Conservation laws: Introduction to Riemann problem, Shocks and rarefaction, Entropy condition,General existence and uniqueness result for the Riemann problem for systems with the characteristics fields which are either linearly degenerate or genuinely nonlinear. Example: the p-system. Some results on different regularizations of the system, admissibility of discontinuous solutions and entropy conditions.

Harish Kumar, IIT-Delhi

Numerical approximation of scalar conservation laws: Consistency,stability and Lax-Wendroff theorem. Monotone schemes, Godunov, Enquist-Osher and Lax-Friedrichs schemes. Convergence of monotone schemes to entropy solutions. TVD schemes.

C Praveen, TIFR-CAM,Bangalore

Discontinuous Galerkin method for scalar and system of conservation laws: basis functions, energy and entropy stability, TVD property and limiters, maximum principle satisfying schemes, time integration, numerical implementation.

1. Partial Differerential equations by L C Evans
2. Hyperbolic system of conservation laws by E Godlewski and P A Raviart, Vol I & II
3. Numerical methods for  conservation laws by Leveque
4.  Shock Waves and Reaction Diffusion Equations by J. Smoller




ISL on Number Theory
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

05-17 October, 2015


Prof. M Manickam
Prof. S D Adhikari
HRI, Allahabad

Topics covered in the workshop

Dr. Thangadurai, HRI
1. b-ary Expansion 

Existence of b-ary expansion of real numbers, non-uniqueness of the representation for rationals, classification of rationals, Motivation to Gauss conjecture about the periods and generalization of Gauss Conjecture by E. Artin.

2. Continued fractions.

finite, simple continued fractions, properties of the kth convergent, existence of infinite continued fractions for irrational numbers, classification of quadratic irrationals, computation of infinite continued fraction expression for e using Pade approximation method.

3. Well-ordering Principle, weak and strong Induction equivalence. 


Prof. B Ramakrishnan, HRI

Lecture 1: Arithmetical functions; several examples, multiplicative, additive functions, Mobius identity, Dirichlet convolution of arithmetic functions, some properties of $\mu(n)$, $\varphi(n)$, $\Lambda(n)$. 

Lecture 2: Properties of Dirichlet convolution and its applications, viz., proving certain identities, evaluating the convolution and proving multiplicative property; Asymptotic estimates (Arithmetic means; Summatory functions); certain applications of these estimates; big O and little o notation; the logarithmic integral.

Lecture 3: Euler summation formula and applications; partial sums of $\log n$ and Stirling's formula for $n!$ as an application; integral representation of the Riemann zeta function; Abel summation formula.

Lecture 4: Relation between asymptotic mean and logarithmic mean; Dirichlet series and summatory functions (Mellin transform representation of a Dirichlet series); finding average orders of certain arithmetical functions using convolution method, especially $\varphi(n)$, $\mu^2(n)$, $d(n)$.


Prof. S. D. Adhikari, , HRI

Lecture 1:  Congruences modulo an integer, Some results on finite fields, basic congruences modulo a prime.
Lecture 2: Lagrange theorem for polynomials over Z/pZ,  quadratic congruences, there are infinitely many primes of the form 4n+1, 4n-1 - statement of Dirichlet's theorem on primes over an A.P. , solution of some diophantine equations.
Lecture 3: Chinese Remainder Theorem, some related problems.
Lecture 4: Quadratic reciprocity law - an elementary proof, related problems.


Prof. S. A. Katre,  Savitribai Phule Pune University
Lecture 1: No. of primesis infinite. No. of  primes  of the form 4n- 1 is infinite.
Statement of Dirichlet's Theorem on primes in A.P.. Large primes and their application in RSA Cryptography. Information about  polynomial time algorithm for primality testing by Manindra Agrawal, Statement of prime number theorem and  Bertrand's postulate, Introduction to Chebyshev's Lambda, psi and theta functions.
Lecture 2: Relation between psi and theta functions. Application of Abel's identity to get relations between theta function and  \pi(x) function. Equivalence of the asymptotic results for \pi(x), \psi(x) and \theta(x).
Lecture 3: Proof for the upper bound for theta function and application this upper bound to the Chebyshev bounds: n/6 log n < \pi(n) < 6n/log n.
Lecture 4: Proof for the upper and lower  bounds for psi function and their application to the proof of Bertrand's postulate. Discussion about the relation of PNT with the nonvanishing of the Riemann zeta function on x=1. (Some part also covered in the tutorial time.)
Application of Mobius Inversion Formula for getting a formula for the n-th cyclotomic polynomial was discussed in tutorial time.

References: 1) Introduction to Analytic Number Theory by Tom M. Apostol, UTM, Springer, 1976 (Narosa, Indian Edn.) (Chapter 4)
2) Introduction to the Theory of Numbers, I. Niven, H. S. Zuckerman and L. Montgomery, John Wiley & Sons, 1991. (Chapter 8, Section 8.1)
3) An Introduction to the Theory of Numbers, G. H. Hardy and E. M. Wright, sixth edition, Oxford University Press, 2008.


Prof. M Manickam, KSOM
Existence of finite Fourier series for periodic arithmetic function. The construction of such function like Ramanujan function, the function $s_k(n)$.Gauss sum associated with quadratic character and
derive the reciprocity law for quadratic symbol. Quadratic Gauss sum and the reciprocity of the quadratic Gauss sum using Residue theorem.Primitive roots and their existence.


A. Mukhopadyaya,  IMSc, Chennai

Dirichlet Character and Dirichlet Prime Number Theorem 



Time Schedule

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Workshop on Jacobi forms and Modular forms of half-integral weight
Supported by HIR, Allahabad

Venue: KSOM, Kozhikode

02-12 February, 2015


Prof. M Manickam
Prof. B Ramakrishnan
HRI, Allahabad

Topics covered in the workshop

B Ramakrishnan

1. Review of modular forms of integral weight, for Γo(N) (N>1)

Atkin-Lehner Newform theory.

2. Modular forms of half-integral weight:

i. Transformation formula for the classical theta function.

ii. Definition of a modular form of half-integral weight.

iii. Hecke operators; Action of Hecke operator on the Furier expansion.

iv. Kohnen's plus space, new form theory in the plus space.

v. Shimura & Shintani liftings for the Kohnen's plus space.

vi. Extension of Kohnen's work to the full space.

vii. Recent result on the theorem of newform of half-integral weight; extension of Kohnen's plus space to 'even' levels.

3. i. Rankin convolution.

ii. Rankin's method & its generalizations; Review of the results by Zagier in the LNM627 on Rankin's method.

M Manickam

1. Review of the modular forms for the full modular group

2. Jacobi forms:

i. Introduction of Jacobi group, its actions on H X c and analytic function of H X c, definition of Jacobi forms.

ii. Proving finite dimensionality of the space of Jacobi forms by computing the no. of zeroes in the fundamental parallelgram and obtaining Taylor maps around Z=0.

iii. Discussing JK,1 explicitly

iv. Construction of Ek,1 – Eisenstein series of weight K, index 1.

v. Operators: Um, Vm, Tm and discus their commuting properties with themselves and with Taylor maps and their action on Eisenstein Series, Poincare Series.

vi. Fundamental decomposition of φ into theta functions and obtain Eichler-Zagier map. Also proved that these maps naturally acts on Poincare series.

vii. Siegel modular forms, Fourier-Jacobi expansion and the Saito-Kurokava lift, Mass Space, and proved their 1-1 correspondence through Hecke Eigen variant lift between mass space and integral weight cusp forms.

viii. Theta functions, Waldspurger's formula, derive adjoint of D*2ν, index changing operator Im

R Thangadurai

Irrationality of Zeta (3) proved by Beukers.

S Boecherer

Introduction of Siegel modular forms; modular forms mod p.

J Meher

Product of Hecke eigenform.



Time Schedule

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Advanced Instructional School(AIS) (2014)
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

01-19 December, 2014


Prof. D S Nagaraj
IMSc., Chennai
Prof. V Balaji
CMI, Chennai

A Brief Description of the Subject
The development of Algebraic Geometry in the 20th century went through a few salient phases; the first one was dominated by the Italian geometers. This phase was when algebro-geometric techniques were closely linked with topology and analytic geometry and were used in developing the theory of algebraic surfaces. But the shortcoming of this phase was that the intuitive aspects of the geometry was given more importance and there was a neglecting of the key aspects of proofs and rigour as well as analysis of special examples which were needed to support the intuition.  The period 1930-1960 was dominated by the work of Zariski, Weil intended to set right these shortcomings; this culminated in the grand synthesis in the hands of Grothendieck. An immense programme was launched by Grothendieck which introduced tools from commutative algebra into algebraic geometry which afforded a uniform language to handle geometry over characteristic p and characteristic zero. The goal was to create a geometry which synthesized the formal arithmetic aspccts of geometry as well as classical projective geometry. Some of the spectacular successes achieved by algebraic geometry both in its geometric as well as its arithmetic aspects can be directly linked to this grand synthesis initiated by the work of Grothendieck. Just to name a few, Deligne's proof of Weil conjectures, Faltings proof of the finiteness of rational points, Wiles's proof of Fermat's last theorem rely indispensably on this super structure.  
The AIS will be directed towards making young researchers in India familiar with the developments in Algebraic Geometry, with special emphasis on Grothendieck's programme. The three week programme is primarily aimed at early researchers in this field to become familiar and users of the tools and techniques in this subject. To acquire a good knowledge of modern algebraic geometry it is essential to see the “local” aspects coming  from Commutative algebra in its interplay in geometry as well as the immense machinery of cohomology. These two themes will be the main ones for the workshop. To give a flavour of the manner in which these tools have lead to fundamental theorems in algebraic geometry, the  final week will stress on some research themes.  A team of active researchers in the field has been invited for this purpose.  The basic reference for this course will be the book by Hartshorne supplemented by books by Mumford, Ueno and others. 
Schemes: Sheaves, schemes,  elementary properties, morphisms,  invertible sheaves and bundles, differentials, valuative criterion, Bertini’s theorems,  Lefschetz theorem. Derived functors, cohomology of sheaves, Cech cohomology, Cohomology of projective space, Serre duality, semicontinuity  theoremes , Zariski’s main theorem and connected theorem,  Fulton Lazarsfeld connectedness theorem.

1. R. Hartshorne,  Algebraic Geometry
2. K. Ueno, Algebraic Geometry I, II, III
3. Lazarsfeld,  Positivity in Algebraic Geometry 
4. Griffiths - Harris, Principles of Algebraic Geometry


Dr. Krishna Chaitanya, CMI, Chennai

Prof. K N Raghavan, IMSc, Chennai

Prof. D S Nagaraj, IMSc, Chennai

Dr. Manoj Kummini, CMI, Chennai

Prof. A J Parameswaran, TIFR, Mumbai

Prof. V Balaji, IMSc, Chennai

Tutorial Assistants

Narasimha Chari, CMI, Chennai
Pabitra Barik,
IMSc, Chennai
Krishanu  Dan,
IMSc, Chennai
Rohith Varma,
CMI, Chennai
Suratno Basu
, CMI, Chennai


Time Schedule

Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

01-28 May, 2014


M Manickam
Professor & Director(i/c)
Kerala School of Mathematics, Kozhikode
B Ramakrishnan
HRI, Ahmedabad

Description: The main objectives of AFS are to bring up students with diverse background to a common level and help them acquire basic knowledge in algebra, analysis and topology. This programme is organised by National Centre for Mathematics (NCM).

Syllabus :

Ring Theory.
(1) Modules over Principal Ideal Domains Modules, direct sums, free modules, nitely generated modules over a PID, structure of nitely 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, niteness of integral closure, discrete valuation rings, Krull's normality criterion, Noether normalization lemma, Hilbert's Nullstellensatz
(4) Semisimple rings, Wedderburn's Theorem, Rings with chain conditions and Artin's theorem, Wedderburn's main theorem,

Functional Analysis.
(1) Normed linear spaces, Continuous linear transformations, application to di erential equations, Hahn-Banach theorems-analytic and geometric versions, vector valued integration.
(2) Bounded Linear maps on Banach Spaces 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) Bounded linear functionals Weak and weak* topologies, Applications to re exive 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.

Differential Topology.
(1) Review of di erential 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 Hausdor ness of the quotient, classi cation of 1-manifolds. De nition of a vector bundle and tangent bundle as an example. Sard's theorem. Easy Whitney embedding theorems.
(3) Vector elds and isotopies Normal bundle and Tubular neighbourhood theorem. Orientation on manifolds and on normal bundles. Vector elds. Isotopy extension theorem. Disc Theorem. Collar neighbourhood 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.

Resource Persons

Prof. B Ramakrishnan, HRI, Allahabad

Prof. M. Manickam, KSOM, Kozhikode

Dr. Kavita Sutar, CMI, Chennai

Dr. Priyavrat Deshpande, CMI, Chennai

Prof. Purusottam Rath, CMI, Chennai

Prof. Sanoli Gun, IMSc, Chennai

Dr. Manoj Kummini, CMI, Chennai

Dr. Krishna Hanumanthu,CMI, Chennai

Prof. R. Thangadurai,HRI, Allahabad

Prof. D.Surya Ramana ,HRI, Allahabad

Prof. Satya Deo ,HRI, Allahabad



Time Schedule

ATM Workshop on Graph Theory(2014)
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

17-22 March, 2014


M Manickam
Professor & Director(i/c)
Kerala School of Mathematics, Kozhikode
S S Sane
Department of Mathematics, IIT Bombay

Description and salient features of the syllabus: This workshop covered some topics in Graph Theory, including both algorithmic and non-algorithmic that are not normally covered in a first course on Graph Theory. The topics include spectral graph theory, some topics in algebraic graph theory leading to Tutte polynomial of a graph, extremal graph theory including the Szemeredi regularity lemma and some topics in algorithmic graph theory including reducibility of 3-SAT.

Resource Persons

Professor R. Balakrishnan, Department of Mathematics, Bharathidasan University, Trichy

Professor Ajit Diwan, Department of Computer Science, I.I.T. Bombay, Mumbai

Professor S.A. Choudum, (Retired) Professor, I.I.T. Madras, (at present in) Bangalore


Time Schedule

Annual Foundation School - Part I (2013)
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

02-28 December, 2013


Group Theory:

Speaker: B. Ramakrishnan
Group actions, Sylow Theory, direct and semi-direct products, simplicity of the alternating groups, solvable groups, p-groups, nilpotent groups, Jordan-Holder theorem.

Speaker: M. Manickam
Free groups, generators and relations, finite subgroups of SO(3), SU(2), simplicity of  PSL(V).

  • Speaker: M. Manickam

Representations and characters of finite groups: Maschke’s Theorem, Schur’s lemma, characters, orthogonality relations, character tables of some groups, Burnside’s theorem.


  • Analysis:

Speaker: R. Thangadurai
Abstract measures, outer measure, completion of a measure, construction of the Lebesgue measure, nonmeasurable sets.
Measurable functions, approximation by simple functions, Cantor function, almost uniform convergence, Ego-roff and Lusin’s theorems, convergence in measure.

Speaker: P. K. Ratnakumar
Integration, monotone and dominated convergence theorems, comparison with the Riemann integral, signed measures and Radon-Nikodym theorem.

Speaker: K. Sandeep
Fubini’s theorem, Lp - spaces.


Differential Topology:

Speakers: A. J. Jayanthan
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

Speakers: P. K. Ratnakumar
Topological and smooth manifolds, partition of unity , Fundamental gluing lemma and classication of 1-manifolds, Vector bundle; Tangent bundle, Morse - Sard theorem, Easy Whitney embedding theorems.

Speakers: P. Sankaran
Orientation on manifolds. , Transverse Homotopy theorem and oriented intersection number , Degree of maps both oriented and non oriented case, Winding number, Jordan Brouwer Separation theorem, Borsuk - Ulam Theorem, Vector eld and isotopies (statement of theorems only) with application to Hopf Degree theorem.

Speakers: G. Santhanam 
Morse functions, Morse Lemma, Connected sum, attaching handles , Handle decompostion theorem, Application to smooth classication of compact smooth surfaces.


Workshop and International Conference on Automorphic Forms and Number Theory

Venue: KSOM, Kozhikode

30th August - 3rd September, 2013

Recent developments in various branches of mathematics and physics show an increasing interest in the explicit construction of Jacobi forms, in particular, Jacobi forms of matrix (or, equivalently, lattice) index. This has to do with their appearance in quantum field theory, module spaces of surfaces, infinite dimensional Lie algebras. Though Jacobi forms of matrix index are known since quite long it is only recently that there was an essential breakthrough in the theory. For example, recent work of Gritsenko, Skoruppa, Zagier shows that Jacobi forms can be constructed in a surprisingly explicit way. These constructions in turn are closely related to classical problems in the arithmetic theory of lattices and in the theory of trigonometric polynomials.

For getting interested researchers or graduate students up to date in recent developments we have conducted a 3 days instructional workshop on Jacobi forms of lattice index, followed by a 2 days conference, where participants of the workshop got an opportunity to report on their own research projects for initiating discussions, obtaining feedback or help.


Part 1: Workshop on Jacobi forms of lattice index
Organizers: Prof. M. Manickam, Prof. Ramakrishnan B., Prof. Nils Peter Skoruppa
Lecturers: Hatice Boylan (Istanbul Üniversitesi and Max-Planck-Institute Bonn), Fabien Cléry (University of Siegen), Nils-Peter Skoruppa (University of Siegen and Max-Planck-Institute Bonn)
Overview: The workshop aimed to give a careful and thorough introduction into the theory of Jacobi forms of lattice index with emphasis on explicit constructions, in particular, for low weights and maximal lattices as index. In addition, it showed how Jacobi forms can be used to construct other types of automorphic forms via the so-called additive and multiplicative liftings.
● Basic notions: Lattices, shadows, discriminant modules, definition of Jacobi forms, discussion of the definition, first examples, functorial properties
● Jacobi forms as theta functions: vector valued modular forms, relation between Jacobi forms and vector valued modular forms, dimension formulas
● Explicit constructions of Jacobi forms: Taylor expansion around 0, Theta blocks, invariants of Weil representations
● Maximal lattices as index, explicit description of forms of singular and critical weight and maximal index
● Additive and multiplicative liftings, product expansions

Friday to Sunday: each day 3 lectures of 90 min and 60 minutes for discussions, questions and problem session

Part 2: Conference on Automorphic Forms and Number Theory
List of Speakers

1 Mr. Ali Ajouz Siegen University, Germany
2 Dr. Anandavardhanan .U.K IIT, Mumbai
3 Dr. Brundaban Sahu NISER, Bhubaneshwar
4 Dr. Fabien Cléry France
5 Dr. Hatice Boylan Istanbul University and Max-Planck Institute for Mathematics, Bonn
6 Dr. Jaban Meher IMSc., Chennai
7 Dr. Jagathesan .T RKM vivekananda College, Chennai
8 Dr. Jayakumar R RKM vivekananda College, Chennai
9 Ms. Jisna P KSOM, Kozhikode
10 Dr. Karam Deo Shankhadhar IMSc., Chennai
11 Dr. Kumarasamy .K RKM vivekananda College, Chennai
12 Dr. Manickam .M KSOM, Kozhikode
13 Dr. Nils Peter Skoruppa Siegen University, Germany
14 Mr. Rahothaman .R Sona College of Technology, Salem
15 Dr. Ramakrishnan .B HRI, Allahabad
16 Mr. Shankar P KSOM, Kozhikode
17 Mr. Srivatsa .V KSOM, Kozhikode
18 Dr. Sujay Ashok IMSc. Chennai
19 Ms. Tamil Selvi Alpha College of Engineering, Chennai

The programme was partially suported by HRI, Allahabad


Instructional Workshop on Logic and Set Theory

Venue: KSOM, Kozhikode

18th February - 1st March, 2013

The workshop is meant to benefit research scholars and young teachers of Mathematics, Statistics and Computer Science. 

The course will cover fundamentals of propositional logic, first order logic, completeness theorem, model theory with applications in number theory, algebra and geometry.
Cardinal arithmetic, partially-ordered sets, well-ordered sets, transfinite induction, ordinal numbers, axiom of choice and its equivalent forms such as Zorn's lemma and well-ordering principle and their applications in mathematics, Zermelo-Fraenkel axioms.


Resource Persons

S. M. Srivastava (ISI, Kolkata)  
H. Sarbadhikari (ISI, Kolkata)  
B.V.Rao (CMI, Chennai) Lecture Notes
N. Raja (TIFR, Mumbai)  
R. Ramanujam  (IMSc, Chennai).  


Workshop on Number Theory and Dynamical Systems

Venue: KSOM, Kozhikode

04-08 February, 2013

Topics Covered:

Linear recurrent sequences and iterations of linear maps

Diophantine approximation and dynamical systems

Continued fractions and the geodesic flow

Scientific Committee

Prof. Yann Bugeaud, Strasbourg University, Mathematics, 7, rue Rene Descartes, 67084 STRASBOURG Cede, France
Prof. Pietro Corvaja, Dipartimento di Matematica e Informatica, University of Udine, Italy. 
Prof. S.G. Dani, Indian Institute of Technology, Mumbai, India 
Prof. Michel Waldschmidt, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, Paris 6, France

Local Organising Committee
Prof. M Manickam, Kerala School of Mathematics, Kozhikode, India
Prof. Michel Waldschmidt, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, Paris 6, France

ATMW Hilbert Modular forms and varieties (2013)

Venue: KSOM, Kozhikode

21-31 January, 2013


National Centre for Mathematics, a joint centre of IIT Mumbai and TIFR, Mumbai

Brief description of the workshop
The goal of the workshop will be to introduce researchers from scratch to some of the basic concepts in the theory of automorphic forms and varieties attached to GLn over totally real fields. We shall quickly treat some of the basic concepts over the first two or three days, and reserve the latter part of the first week for more advanced topics, some of these may include: p-adic Hilbert modular forms, Herzig's classification of the mod p representations of GLn over a local field and Taylor's recent construction of Galois representations for GLn over totally real fields.During the second week, there will be a short conference to collect together local experts in the area of modular forms and related areas of number theory. We will in particular concentrate on representation theoretic and p-adic aspects of the theory of automorphic forms. The conference will have several one hour research level talks every day.

Click here for details

School on Modular Forms

Venue: KSOM, Kozhikode

03 - 19 October, 2012

In connection with the National Mathematical Year celebrations we had conducted a Workshop on Theory of Prime Numbers and Related Areas from 06-10 May 2012 at KSOM. The workshop,School on Modular Forms conducted from 03rd October, 2012 to 19th october 2012 was also related to Ramanujan's work. 

Topics Covered:
Basic of modular forms for the full modular group and for its congruence subgroups and related areas.

Lectures delivered by:
Prof. M. Manickam, Kerala School of Mathematics, Kozhikode
Prof. B. Ramakrishnan, Harish Chandra Research Institute, Allahabad.
Prof. Nils-Peter Skoruppa, Siegen University, Germany

Special Lectures delivered by:
Prof. Ananda Vardhanan, IIT, Mumbai
Prof. Eknath Ghate, TIFR, Mumbai
Prof. A. Raghuram, IISER, Pune
Prof. J. Sengupta, TIFR, Mumbai
Dr. R. Thangadurai, HRI, Allahabad.
Dr. Brundaban Sahu, NISER, Bhubaneshwar

Theory of Prime Numbers and Related Areas

Venue: KSOM, Kozhikode

06 - 10 May, 2012

The Workshop was meant for research fellows, young teachers, PG and bright UG students of Mathematics. The course program covered: Ramanujan's Proof of Betrand's Postulate, Prime Number Theorem, Highly Composite Numbers and its applications and Ramanujan's way of summation.

Resource Persons:

Prof. Ram Murty
Queens University, Canada

Dr. Anirban Mukhopadyaya

Institute of Mathematical Sciences

Dr. D. S. Ramana
Harish-Chandra Research Institute

Dr. R. Thangadurai
Harish-Chandra Research Institute



Instructional Workshop on Ergodic Theory

Venue: KSOM, Kozhikode

17 October - 04 November 2011

The Workshop was meant to benefit research fellows and young teachers of Mathematics and Statistics. The course program covered: Basic Ergodic Theory and Topological Dynamics, Probabilistic Systems, Spectral Theory of Dynamical Systems, Ergodic Theory with groups and Diophantine Approximations, Ergodic Theory in Geometry.

Resource Persons:

M. G. Nadkarni (IIT, Indore) Lecture Notes
S. G. Dani (TIFR, Mumbai)  
S. C. Bagchi (ISI, Kolkata)  
Siddharta Bhattacharya (TIFR, Mumbai) Lecture Notes
B. V. Rao (CMI, Chennai) Lecture Notes
J. Dani (University of Mumbai)  
C. S. Aravinda (TIFR, Bangalore)  
Shiva Shankar (CMI, Chennai)  
J. Mathew (KSOM, Kozhikode)  



School on Algebra, Analysis & Topology

Venue: KSOM, Kozhikode

01-07 September, 2011

The School was aimed at M.Sc. Mathematics students of Kerala. The course covered some topics of Algebra, Analysis and Topology which are not usually covered in the M.Sc. curriculum

Resource Persons:

Basudeb Dutta (I.I.Sc., Bangalore)
U. K. Anandavardhanan (I.I.T., Mumbai)
A. J. Parameswaran (TIFR, Mumbai)
Jathavedan (CUSAT, Kochi)
Joseph Mathew (KSOM, Kozhikode)
A.K. Vijayarajan (KSOM, Kozhikode)



Workshop on Functional Analysis and Harmonic Analysis

Venue: KSOM, Kozhikode

01 June 2011 - 10 June 2011

The workshop was aimed at senior post-graduate students with aptitude for research and junior research fellows with research interests broadly in Analysis. The workshop also benefit young teachers of Mathematics teaching at post-graduate level.

Resource Persons

Prof. Gadadhar Misra (IISc, Bangalore)
Prof. M Krishna(IMSC, Chennai)
Prof. Rajarama Bhat ( ISI, Bangalore)
Prof. K. Parathasarathy(Ramanujan Institute, Chennai)
Prof. M N Narayanan Namboodiri (CUSAT, Cochin)
Prof. M Sundari (CMI, Chennai)
Prof. Joseph Mathew (KSOM, Kozhikode)



School on Analysis, Algebra & Topology

Venue: KSOM, Kozhikode

13 Dec 2010 - 07 Jan 2011

The programme was aimed at junior research fellows and final year post-graduate students in Mathematics from Universities/Colleges in Kerala. Those who have completed post-graduate programme recently and intending to pursue mathematics as a career also appllied. The School was conducted by eminent researchers from Universities/Institutions in India and covered some advanced topics in Algebra, Topology and Analysis which are generally considered to be prerequisites for research scholars in all disciplines of Mathematics. The course consist of rigorous class room lectures and problem solving tutorials.

Resource Persons

U. K. Anandavardhanan , IIT Bombay

Manoj Kumar Keshari , IIT Bombay

Balwant Singh , CBS Bombay

Sudhir Ghorpade  , IIT Bombay

Raja Sreedharan , TIFR Bombay

Rudra Pada Sarkar, ISI, Kolkatta

A K Nandakumaran, IISc Bangalore

K Sandeep, TIFR Bangalore

P K Rathnakumar, HRI Allahabad

P K Rathnakumar, HRI Allahabad

Mahuya Datta, ISI, Kolkatta

V Uma , IIT Madras

Amit Hogadi , TIFR Bombay

A J Parameswaran, KSOM

Advanced Instructional School on Schemes & Cohomology

Venue: KSOM, Kozhikode
28 June - 16 July, 2010
Conveners: A. J. Parameswaran & V. Balaji

A Brief Description of ATM Schools
Advanced Training in Mathematics (ATM) Schools are a joint effort of a large number of mathematicians in the country for training mathematics research scholars and teachers with generous support from the National Board for Higher Mathematics. The programmes are conducted in reputed mathematics departments in Summer and Winter each year. In these Schools, the emphasis is on problems solving and on understanding interrelations of basic subjects in mathematics. At the initial stage, ATM Schools consist of two Annual Foundation Schools (AFSI & II) in algebra, analysis, and topology. At a later stage, Advanced Instructional Schools (AIS) and workshops (ATMW) in all major areas are organised. Several advanced instructional schools (ATML) are organized each year exclusively for young lecturers in colleges and universities.

Advanced Instructional School on Schemes and Cohomology
The AIS was directed towards making young researchers in India familiar with the developments in Algebraic Geometry, with special emphasis on Grothendieck’s programme. The three week programme was primarily aimed at early researchers in this field to become familiar and users of the tools and techniques in this subject. To acquire a good knowledge of modern algebraic geometry it is essential to see the “local” aspects coming from Commutative algebra in its interplay in geometry as well as the immense machinery of cohomology. These two themes were the main ones for the workshop. To give a flavour of the manner in which these tools have lead to fundamental theorems in algebraic geometry, the final week stress on some research themes. A team of active researchers in the field had participated for this purpose.

Resource persons
TEV Balaji V. Balaji Vivek Mallik
D. S. Nagaraj Suresh Naik ,
A. J. Parameswaran, K. N. Raghavan

Unity of Mathematics Lectures
  Nitin Nitsure, Kapil Paranjape, S. Ramanan,
 Pramathanath Sastry , C. S. Seshadri, C. S. Rajan
Jugal Verma, Shiva Shankar

Summer School on Probability: Foundations & Applications

Venue: KSOM, Kozhikode

17 May - 04 June, 2010

The Summer School was meant to benefit senior post-graduate students, junior research fellows and young teachers of Mathematics, Statistics and Computer Science. The course covered fundamentals of discrete and continuous probability models, law of large numbers, weak convergence and limit theorems, conditional probability, Markov Chains and Martingales, large deviations and entropy. It also covered some applications to number theory, graph theory, computer science and other related fields.
Resource Persons

M. G. Nadkarni (Univesity of Mumbai) Lecture Notes
B. V. Rao (C. M. I., Chennai) Lecture Notes
B. Rajeev (I. S. I., Bangalore) Lecture Notes
Inder Rana (I. I. T., Mumbai) Lecture Notes
R. L. Karandikar (C. M. I., Chennai) Lecture Notes
S Ramasubramanian (I. S. I. Bangalore) Lecture Notes


Workshop on Statistics

Venue: KSOM, Kozhikode

05-06 March, 2010

The workshop was meant exclusively for the Scientists of KSCSTE and the R&D institutions under KSCSTE.
Resource Persons

T.Krishnan obtained his Master's and Ph.D. degrees from the Indian Statistical Institute, Kolkata.  He has been on the faculty of this Institute since 1965 until his retirement in 1998 as a Professor of Applied Statistics.  After retirement he worked as a full-time consultant for Cranes Software International Limited, Bangalore, on the development of a statistical software SYSTAT for eight years. Currently he is working for Strand Life Sciences, a Bioinformatics company in Bangalore as a Statistician.  His book  (with G.J. McLachlan) on the EM Algorithm is a definitive and standard reference on the subject.

R.V.Ramamoorthi  obtained his Master's degree from Utkal University and his Ph.D. degree from the Indian Statistical Institute, Kolkata.  Since 1982 he has been on the faculty of the Michigan State University, East Lansing, Michigan, U.S.A.  where he is currently a Professor of Statistics.  He is currently on  leave from this position and is a visiting professor at the Indian Institute of Science, Bangalore.  . His earlier research was on sufficiency and decision theory. For the last few years he has been working on Bayesian Nonparametric Inference and his book (with J.K.Ghosh) on Bayesian Nonparametrics is an authoritative book on the subject.


Time Schedule

Friday, 05 March 2010

09:30--10:00 Inauguration Dr. K. V. Jayakumar  
10:00--11:15 Introduction to Data Analysis TK (T. Krishnan) Lecture Notes
11:30--12:45 Basic Probability RVR (R. V. Ramamoorthi) Lecture Notes
12:45--14:00 Lunch    
14:15--15;30 Descriptive and Graphical Statistics TK

Lecture Notes

Lecture Notes

15;45--17:00 Sampling Distributions and Confidence Intervals RVR Lecture Notes

Saturday, 06 March 2010

10:00--11:15 Hypothesis Testing  RVR Lecture Notes
11:30--12:45 Analysis of Variance and Regression TK Lecture Notes
12:45--14:00 Lunch    
14:15--15;30 Experimental Designs and Survey Sampling TK

Lecture Notes

Lecture notes

15;45--17:00 Computer-Intensive Statistical Methods RVR, TK

Lecture Notes

Lecture Notes

Workshop on Functional Analysis and Harmonic Analysis

Venue: KSOM, Kozhikode

1-10, February 2010

The workshop was meant to benefit senior post-graduate students, junior research fellows and young teachers of Mathematics and aims to cover the necessary background material for understanding, pursuing and teaching advanced level topics in the two main areas of the workshop. The workshop is meant for senior MSc. students, Junior Research Scholars and young teachers from university departments and col leges from the country.

Topics covered:
Functional analysis: Banach algebras, maximal ideals, Gelfand-Naimark theorem, spectral theorem
Harmonic analysis: Fourier series, Fourier transforms and Paley-Wiener theorems

Resource persons

Prof. Gadadhar Mishra, Indian Institute of Science, Bangalore
Prof. E. K. Narayanan, Indian Institute of Science, Bangalore
Prof. Alladi Sitaram, Indian Institute of Science, Bangalore
Prof. V. S. Sunder, Institute for Mathematical Sciences, Chennai
Prof. B. V. Rajarama Bhat, Indian Statistical Institute, Bangalore

Instructional Workshop on Spectral Theorem

Venue: KSOM, Kozhikode

15th and 16th June 2009

Co-Sponsored by KSCSTE

Day 1, June 15, 2009, Monday:

Registration  : 8:30 a.m.      
Inauguration : 10:00 a.m. – 11:00 a.m.
Tea : 11:00 a.m. - 11:30 a.m.  
Lecture 1, 11:30 a. m-12.30 p.m.: Spectral theorem for normal matrices – Dr. Rajarama Bhat
Lecture 2, 12:30 p.m-1:30 p.m: Spectral theorem for compact self-adjoint operators - Dr. G. Ramesh
Lunch break: 1:30 p.m-2:30 p.m
Lecture 3, 2:30 p.m-3:30 p.m:  Topologies of B (H) –Dr. A K Vijayarajan
Tea: 3:30 p.m- 4:00 p.m.
Lecture 4, 4:00 p.m.-5:00 p.m.: General spectral theorem for bounded operators-I- Dr. Rajarama Bhat                                      

Day 2, June 16, 2009, Tuesday:

Lecture 1, 9:30 a.m-10:30 a.m.: General spectral theorem for bounded operators –II- Dr. Rajarama Bhat
Tea break: 10:30 a.m-11:00 a.m.

Lecture 2, 11:00 a.m-12:00 p.m: Double Commutant Theorem - Dr. A K Vijayarajan

Discussions: 12:00 p.m-12:30 p.m
Lunch break: 12:30 p.m-2:00 p.m
Lecture 3, 2:00 p.m-3:00 p.m: Trace Class Operators and Duality- Dr. G. Ramesh
Tea break: 3:00 p.m-3:30 p.m
Lecture 4, 3:30 p.m-4:30 p.m: Some Applications of spectral Theorem-                                            

Discussion: 4:30 p.m.-5:00 p.m. Dr. Rajarama Bhat

Resource persons & Speakers





Dr. B. V. Rajarama Bhat

Indian Statistical Institute


Indian Statistical Institute, R. V. College(PO) Bangalore-560059

Dr. G. Ramesh

Indian Statistical Institute

Post-doctoral Fellow

Indian Statistical Institute, R. V. College(PO)

Dr. Vijayarajan A. K.

Kerala School of Mathematics

Associate Professor

Kerala School of Mathematics

@2010 Kerala School of Mathematics