ATMW Geometric Invariant Theory

Venue: KSOM, Kozhikode

14-19 May, 2018

Organizers:

Prof.  M.  Manickam,

Kerala School of Mathematics
Kunnamangalam, P.O, Kozhikode – 673571,  Kerala

Prof. Parameswaran A.J.

School of Mathematics, Tata Institute of Fundamental Research.
Homi Bhabha Road, Navy Nagar, Colaba, Mumbai – 400005

Dr. A K Vijayarajan
Associate Professor
Kerala School of  Mathematics, Kunnamangalam, P.O,  Kozhikode – 673571,  Kerala

 Resource Persons Prof. A J Parameswaran Prof. D S Nagaraj Prof. S Kannan Prof. V Balaji Tutorial Assistants Suratno Basu Rohith Varma

Participants

Time Schedule

Workshop on Pedagogical Knowledge of Mathematics

Venue: KSOM, Kozhikode

24-26 March, 2018

This was a workshop to college teachers / M.Phil., and PhD Scholars. The workshop focused around the concepts of teaching Mathematics and the pedagogical knowledge of teaching Mathematicss.  The aim of this workshop was to share and interact on the pedagogical challenges in teaching Algebra, Linear Algebra.

Organizing Committee ( In alphabetical order)

• Arindama Singh, Indian Institute of Technology, Madras
• Manikkam M, Kerala School of Mathematics, Kerala
• Ramesh VP, Central University of Tamil Nadu, Thiruvarur

Participants

Time Schedule

School on Modular forms

Venue: KSOM, Kozhikode

12-24 February, 2018

Organizes
Prof. M Manickam
Director
KSOM
Prof. B Ramakrishnan
Dear & Professor

Topics Covered: Modular Forms of both integral and half integral weight and complete Hecke theory and inter relation etc.

Participants

Resource Persons

Annual Foundation School (AFS) - I (2017)
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

04-30 December, 2017

Organizes
Prof. M Manickam
Director
KSOM
1. Prof.  Partha Sarathi Chakraborty,

Professor
IMSc, CIT Campus, Taramani. Chennai,

IMSc, CIT Campus, Taramani. Chennai,

The Annual Foundations Schools (AFS) are targetted toward fresh Ph. D. Students in the  Universities and Research Institutions. Here in this programme the 29 participants of the program consisted of 7 MSc students, 2 Assistant Professors, 20 research scholars. The program as scheduled started on the morning of 6th June and ended on the evening of 1st July.

Topics covered in the workshop

 Name of the Speakers with their affiliation, who will cover each module of 6 lectures. Detailed Syllabus * Vijay Ravikumar  (CMI, Chennai) (VR) Algebra Anirban Mukhopadhyay (IMSc Chennai) (AM) Algebra M Manickam (KSOM, Kozhikode)  (MM) Algebra Partha Sarathi Chakraborty (IMSc Chennai) (PSC) Analysis Sutanu Roy (NISER) (SR) Analysis Sushmita Venugopal (IMSc Chennai) (SV) Topology Soumen Sarkar (IITM) (SS) Topology

Participants

Time Schedule

Annual Foundation School (AFS) - II (2017)
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

08 May - 03 June 2017

Organizes
Prof. M Manickam
Director
KSOM

Prof. Venkata Balaji T E

Dr. A K Vijayarajan

KSOM Kozhikode

The Annual Foundations Schools (AFS) are targetted toward fresh Ph. D. Students in the  Universities and Research Institutions. Here in this programme the 26 participants of the program consisted of 1 MSc students, 1 MPhil student, 24 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

Algebra

Krishna Hanumanthu

Rings, homomorphisms, ideals, quotient rings by ideals, adjunction of elements, integral domains, prime and maximal ideals, Hilbert's Nullstellensatz (two versions below and their equivalence)
(i) maximal ideals in C[x_1,...,x_n] are in bijective correspondence with C^n; and
(ii) for an ideal I in C[x_1,...,x_n], I(V(I)) is equal to the radical of I.
Finally, I gave a brief introduction to algebraic geometry by defining algebraic varieties in C^n, Zariski topology, and giving some examples of curves in C^2.

Parvati Shastri

Motivation from number theory, Fundamental Theorem of Arithmetic (Division algorithm) in the ring of integers and polynomial rings over a field and the Gaussian integers, pointing out, division algorithm by a monic polynomial holds for polynomials over any commutative ring with identity. Euclidean domains are PID’s and PID’s are UFD’s. Gauss Lemma. A necessary condition for the existence of a Euclidean function in terms of universal side divisors, the Dedekind Hasse function, existence of Dedekind-Hasse function as a necessary and sufficient condition for an integral domain to be a PID. Historical introduction to algebraic number theory, the theorem of Diophantus and Fermat’s last theorem. Definition of algebraic numbers and integers, the ring of integers of a quadratic field. Unique factorization of ideals in the ring of integers of a number field, the definition of Dedekind domains, statement that for a Dedekind domain, UFD is equivalent to PID. Proof of this last statement was given for the ring of integers of an imaginary quadratic field. Definition of Ideal Class group and its significance, Class number one is equivalent to PID and hence equivalent to UFD. An outline of the proof through guided exercises, that the norm function is Dedekind-Hasse function on the ring of integers of Q( sqrt ( − 19)) and hence it is a PID. But there does not exist any Euclidean function on this ring, thus providing an example that PID does not imply Euclidean. List of nine values of d for which the ring of integers of Q( √ d) is PID and out of which the first five are Euclidean and there can be no Euclidean function on the remaining four. Units in imaginary quadratic fields contrast with real quadratic fields, by examples; relation and history of Brahma Gupta- Pell equation. Eisenstein’s irreducibility criterion and application to cyclotomic polynomials were also done.

M. Manickam
During my five lectures and tutorials, Chapter 14 of M.Artin second edition has been covered. The concept of modules, sub-modules, free modules finitely generated modules and noetherian rings were introduced and developed the theory in order to solve a linear equation over the ring of integers and then derived structure theorem of abelian groups and the same for modules over polynomial rings. Then as an application the linear space of finite dimension over F is equivalent to the F[t] module has been showed and finally rational canonical form associated with one such linear operator has been derived.

Differential Geometry

Sudarshan Gurjar

I gave the initial 5 lectures and conducted 5 tutorials in the geometry section of the school. The focus was on differential geometry of plane curves; more specifically the first 3 chapters from Pressley's book on differential geometry. In the first lecture, reviewed some of the basic aspects of multivariable calculus. Good part of the focus was on the Implicit and the Inverse function theorems. In the next lecture, I began discussing  parametrized curves. I then discussed length of a parametrized curve and reparametrization of a parametrized curve, especially the unit speed reparametrizations of regular parametrized curves.   I then introduced the notion of curvature of plane curves and derived an expression for the same. A related notion of signed curvature was also introduced. I then proved the fundamental theorem on existence and uniqueness of plane curves with a given signed curvature.    In the later lectures I began talking about spaces curves. I introduced
torsion of space curves and derived an expression for the same. I then discussed the Serret-Frenet equations. I then stated (without proof) the analogues theorem for existence and uniqueness of space curves with
given curvature and torsion.   In the last lecture I proved two global theorems on plane curves; namely the isoperimetric inequality and the 4 vertex theorem.    The tutorials were devoted to discussing how to prove Inverse function theorem from Implicit function theorem and vice versa, comparison between level curves and parametrized curves, exercises discussing properties of curvature and fixing some loose ends from the lectures.
The students were alert and enthusiastic. The teaching experience was rewarding. The facilities and arrangements at KSOM  were good.

A R Shastri:

I have engaged totally 6 lectures of 1.5 hours each and 6 tutorials of 1 hour each.
I arrived at the venue one day in advance of my lectures and got the feed-back from Sudarshan Gurjar who was teaching Differential Geometry during the first week.

The partcipants were taught through first three chapters from Pressley's book during the first week which is about curves.  So I began with the question: What is curve? I got some very interesting answers which helped me to take the course that I should for these audience.

I began with recalling inverse and implicit function theorems smooth version of invariance of domain,  and lead them the concept of manifolds with boundary in euclidean spaces. Most of the examples were taken from the book Pressley and therefore were surfaces. In particular, the of all  the classes of quadratics was discussed.  I also gave them some glimpses of what is Topology, Algebraic Topology, Differential Topology and Differential Geometry.   Then I proved the regular inverse image theorem, Jacobian Criterion for a subset of euclidean space to be a manifold. Then tangent spaces derivative of smooth maps on tangent spaces were introduced.

In lecture three, I introduced the first fundamental form of any manifold in a euclidean space and and indicated how this leads to the concept of Riemannian manifolds in general. We then discussed local isometry, conformal  mappings and equi-areal mappings. As a consequence, we obtained Archimedes theorem and computed the area of a triangle on the sphere. The last lecture was devoted to the study of the geometry of the sphere, and the unit 2-disc with the Poincare metric as examples of non euclidean geometry.

The participants were on the whole quite enthusiastic though with diverse background.  The distinction between those who had attended AFS-I earlier and those who had not was very clear.
Often the discussion hours went beyond the stipulated time.  I also engaged a session of making Platonic solids, which the participants enjoyed.

I also gave two special talks in which I presented two proofs of Fundamental Theorem of Algebra, one using Linear algebra  and the other using only real analysis.

We had only one tutor who was also not of much use.

The arrangements at KSOM were excellent.

Incidentally, I came to know on the very first day that there is a II edition of Pressely book and got a soft copy. This edition is far better than the first one and I recommend that we follow this in future.

Venkata Balaji T E

The lectures given by the third speaker Venkata Balaji were aimed at covering the three most important results of the course, namely the Gauss-Bonnet Theorem, the Gauss Theorema Egregium and the the Gauss Curvature Theorem.

After beginning with the definition of geodesics as curves on surfaces that obey Newton's First Law, with geodesic curvature as the tangential component of curvature, and Gaussian curvature as the product of minimum and maximum normal curvatures of curves passing through the given point of the surface, the
Gauss-Bonnet Theorem was introduced as a statement of the form total curvature equals a constant. This was first stated for simple smooth closed curves, then for curvilinear polygons and then for compact orientable surfaces. The constant was motivated in the first two cases to be either a plane angle or a solid angle by considering the cases of a closed disc or a sphere. The proof for the compact surface case was deduced from the curvilinear polygon case after assuming the existence of triangulations and it turns out that the constant is the Euler characteristic, showing that the theorem amazingly relates the differential geometric curvature with a topological invariant. The implications of such a statement were explained.

The second fundamental form was introduced as arising naturally from an effort to compute the normal curvature of a curve on the surface. The Weingarten map was defined and the existence of real eigenvalues were shown. The eigenvalues were called principal curvatures and the corresponding eigenvectors principal eigenvectors. It was established that these vectors are orthogonal (when the eigenvalues are distinct) and Euler's Theorem was established relating the normal curvature of an arbitrary curve with these principal
curvatures. As a corollary to this, it followed that these principal curvatures were none other than the maximal and minimal normal curvatures. A formula for Gaussian curvature was deduced using the fact that it is the determinant of the Weingarten map, showing that it is a smooth function.

The above analysis allows approximation of the surface by an equation of degree two and the classification of its points as elliptic, hyperbolic, parabolic or planar was illustrated.

It was shown that the Wiengarten map is also the same as the Gauss map up to a sign,where the Gauss map records the variation of the unit normal to the surface. Gauss's Theorem on Gaussian Curvature as the limiting value of area spanned by the normal vector on the unit sphere per unit area of the surface was proved, generalising the statement that the curvature of a curve is the limiting value of change in the normal direction (or tangent direction) per unit arc length. The total curvature of the torus was shown to be zero easily by considering the Gauss map and the regions of elliptic and hyperbolic points on it.

Hopf's Umlaufsatz was motivated on the plane using the case of a circle and its proof for the surface case was explained using the ideas of isotopy.  A calculation relative to a moving frame on the tangent plane and
forming a right handed orthonormal system with the unit normal shows that the difference between the rate of angular change between the tangent and the first vector of the frame, and the geodesic curvature, is measured
on the bounding curve by the dot product of the first vector with the rate of change of the second (rates measured relative to arclength). This is essentially an infinitesimal version of the Gaussian curvature
on the inner surface recorded by the bounding curve and integrating this on the curve leads to the proof of Gauss-Bonnet for the case of a smooth curve, from which the polygon case can be deduced. Green's Theorem and all basic formulas involving the second and first fundamental forms are put to use here.

It was pointed out that the same moving frames calculation applied to the Gram-Schmidt orthonormalisation of the frame of a patch (chart) leads to the proof of Gauss's Theorem Egregium, which showed that the notion of
Gaussian curvature is intrinsic, is preserved by isometries, and showed that the determinant of the second fundamental form depends on the first fundamental form. It was thus explained why any map of the earth distorts distances, must distort either angles or areas and not preserve both.

Geodesics, their appearance in normal sections, and their determination on surfaces of revolution (Clairaut's Theorem) were explained. There was not enough time to cover the Geodesic Equations, Gauss Equations or Geodesic Coordinates. These the students were requested to study later on their own.

Several hours of extra classes were taken in order to complete the above proofs and most of the students attended these sessions from 5.30PM to 7.30PM with interest, enthusiasm and appreciable patience. The lecturer greatly enjoyed lecturing on these beautiful topics and wishes to do the same in future given the opportunity.

Measure Theory

A. K. Vijayarajan

Lecture I:
Review of Riemann integration - The The Riemann characterisation of integrability  -Fundamental theorem of integral calculus.

Lecture II:
(Abstract) Measurable spaces - sigma algebras-existence of sigma algebras - Measurable functions-Borel sigma algebras.

Lecture III:

(Abstract )Measure spaces - Measures- examples(Lebeque measure on R^n - no construction or detailed discussion here) - Sequences of measurable functions-properties and theorems

Lecture IV:

Complete measure spaces(this was done with complete detailed construction and important results) - the notion of almost everywhere and related results.

Lecture V:

Integration - simple functions-positive measurable functions - monotone convergence theorem-measurable functions - complex measurable functions-properties and examples.

Lecture VI:

Fatou's lemma-Dominated convergence theorem - absolute continuity of Lebesgue integral(an abstract measure was referred to as Lebesgue measure) - absolutely continuous measure and  Radon Nykodm theorem were mentioned - Revisit of Riemann integration - the Lebesgue criterion of Riemann integrability.

In all the 6 tutorial hours, problems were worked out and left over verifications were done.

As prescribed all topics in the first chapter of Rudin's book on Real and Complex Analysis were covered though I did not strictly follow the material there. Several other textbooks such as Real Analysis by Folland and Royden were made use of.

In the next two lecture hours and tutorials hours Spectral measure, spectral integrals and spectral theorems were done in detail. This was done as participants showed enthusiasm to see where measure theory methods can be used and applied.

Printed class notes were given after each class so that students could concentrate on attending and interacting rather than taking down notes.

The whole audience was interested in the subject and a few of them were really good. Quite a few of them need a push and some assistance to start on a problem.

I have given them notes for all the classes. Some corrections are needed in the notes and there are typos as well. I will get back and give them the corrected notes in full.

Indrava Roy

Lecture 1:

Review of topological concepts- Locally Compact Hausdorff spaces, Urysohn's Lemma, Partition of Unity

Lecture 2:

Riesz Representation Theorem: Statement and preliminary steps in proof- Construction of sigma-algebra from a positive linear functional, Inner and Outer regularity, Uniqueness of measure thus obtained

Lecture 3:

Riesz Representation Theorem (continued): Approximation Lemma, Countable sub-additivity and additivity on pairwise disjoint coverings, inner regularity on Borel sets with finite measure.

Lecture 4:

Riesz Representation Theorem (continued): Completion of proof of RRT, Definition of sigma-compactness and sigma-finiteness, Regularity theorem for Riesz measures on sigma-compact, Locally Compact Hausdorff spaces

Lecture 5:

Equivalent formulation of Riemann integration on $\R^n$, Proof that continuous compactly supported functions are Riemann integrable, Definition of Lebesgue measure on $\R^n$ via the Riesz Representation Theorem, Caratheodory's approach using outer measures for constructing measures, equivalence of Lebesgue measure obtained via RRT and Caratheodory constructions.

Lecture 6:

$L^p$-spaces: Convex functions, Inequalities of Jensen, Holder and Minkowski

Lecture 7:

$L^p$-spaces: Definition of a Banach spaces, definition of $L^p$-spaces, Proof that $L^p$-spaces are Banach.

Lecture 8:

Hausdorff measures: definition of metric outer measure, definition, and properties of Hausdorff measures, Definition of Hausdorff dimension, Computation of Hausdorff dimension of the Cantor set

Haar measures: Definition of Topological groups, Definition of Radon and Haar measures, examples of Haar measures via Riesz Representation Theorem

In all tutorial classes, problems pertaining to relevant topics were given including some leftover exercises from Lecture classes.

Participants

Time Schedule

AIS on Harmonic Analysis
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

02-24 December, 2016

Organizes
Prof. M Manickam
Director
KSOM

Prof. E K Narayanan,

IISc Bangalore

An AIS in Harmonic Analysis was held at the Kerala School of Mathematics, Kozhikode, Kerala from 5th of Dec 2016 to 24th of Dec 20L6. A total of 25 participants attended the school. The resource persons were Sanjay P.K. from NIT, Calicut, E. K. Narayanan from IISc, Bangalore, Rudra Sarkar and Swagato Ray from ISI Kolkata, S. Thangavelu from IISc Bangalore and Parasar Mohanty from IIT Kanpur.

Topics covered in the workshop

Prof. Sanjay P. K (NIT, Calicut)
Lecture 1: Review of Functional Analysis (without proofs): Banach spaces, Hahn-Banach Theorem, dual of L^p spaces, Uniform boundedness principle, Open mapping theorem, Closed graph theorem, Hilbert spaces, Orthonormal basis, Hermite functions.

Lecture 2: The spaces C_c and C_0, density of simple functions, continuity of translation in L^p, Minkowski's integral inequality, Convolution, Young's inequality, Convolution and differentiability, Approximate identity- L^p convergence, Density of C_c^\infty functions

Tutorial 2 and Lecture 3: Weak and weak* convergence, Banach-Alaoglu Theorem, Compact operators, Finite rank operators, Integral operators, Hilbert-Schmidt operators, Compact operators on Hilbert spaces, Spectral theorem for self-adjoint compact operators.

Lecture 4: Phragmén–Lindelöf principle, Hadamard three-lines theorem, Riesz-Thorin interpolation theorem.

Lecture 5: Distribution function, Weak L^p spaces, weak-type (p,q), Marcinkiewicz interpolation theorem (diagonal case), Generalised Young's Inequality.

Prof E. K. Narayanan (IISc, Bangalore)
Lecture 1: Banach algebras, *-algebras, several examples like L^1, C(X) where X is
compact T_2, bounded operators on a Hilbert space etc. Some easy results were established, for example, invertible elements in a commutative unital Banach algebra form an open set etc.
Lecture 2: Spectrum of an element. Resolvent and its properties, in particular R is an analytic function and consequently spectrum is non-empty. Gelfand-Mazur theorem and the spectral radius formula.
Lecture 3: Multiplicative linear functions on a Banach algebra, spectrum of a Banach algebra and its connection to the spectrum of an element. C* algebras, examples and the Gelfand-Naimark theorem
Lecture 4: Review of finite dimensional spectral theorem. Continuous functional calculus for a self adjoint operator as a consequence of the Gelfand-Narimark theorem. Extension to bounded function calculus and the associated projection valued measure.
Lecture 5: Abstract projection valued measure and the construction of corresponding algebra of operators via integration, thus completing the spectral theorem for self adjoint operators.
Remarks: 6th of Dec 2017, there were no lecture due to holiday declared by the Keral Govt after Ms Jayalalitha's death.

Prof Parasar Mohanty (IIT Kanpur)
Lecture 1: Fourier transform on L^1(R),  various properties of Fourier transform, Schwartz Space
Lecture 2: Fourier Inversion, Plancherel Theorem, Hausdorff -Young inequality
Lecture 3 : Paley Weiner Theorem
Lecture 4 : Hardy Littlewood Maximal function, Pointwise convergence of Poission Integral and Lebesgue differentiation theorem
Lecture 5: Calderon-Zygmund decomposition on R^n.
Lecture 6:  Hilbert transform, weak (1,1) and point wise convergence.

Prof Swagato Ray (ISI, Kolkata)
1. Translation invariance of the measure on T. Convolution and Young's inequality. Approximate identity and denseness of trigonometric polynomials in L^p(T). Fourier coefficients and Riemann Lebesgue lemma. Arbitrary slow decay of the Fourier coefficients of integrable functions. Uniqueness of Fourier coefficients.

2. Dirichlet kernel and convergence of Fourier series for differentiable functions. Riemann's localization theorem. Uniform convergence of Fourier series for twice differentiable functions. Fourier series for square integrable functions and Plancherel theorem for circle.
3. Summability: Summable implies Cesaro summable implies Abel summable (converse is not true), Fejer's theorem and L^p convergence of Cesaro mean and Abel mean of Fourier series.
4. Uniform boundedness of the partial sum operator and introduction to Hilbert transform. Proof of L^p boundedness of Hilbert transform and L^p convergence of the Fourier series. Relation between Hilbert transform and conjugate Poisson kernel.
5. Dirichlet problem for unit disc, Poisson integral and Harmonic functions. Characterzation of Harmonic functions on the open unit disc as Poisson integral of L^p functions on T.
6. Symmetry breaking of the Fourier series and construction of a continuous function whose Fourier series diverges at a point on the circle.

Prof S. Thangavelu (IISc, Bangalore)
Lecture 1.  After defining Linear Lie Groups, we discussed  their natural actions on the underlying vector spaces $\R^n$ and $\C^n$ and the induced actions on certain function spaces with special emphasis on the action of $SO(n)$ on $L^2(\R^n)$ and that of $SU(2)$ on the Fock space.

Lecture 2. We began with the action of $SU(2)$ on the finite dimensional spaces $V_m$ of  holomorphic polynomials in two variables which are homogeneous of degree $m.$ After introducing a suitable Hilbert space structure on $V_m$ we showed that the natural action of $SU(2)$ gives rise to unitary representations $\pi_m$ which are then proved to be irreducible.

Lecture 3. After recalling the definition and properties of characters of compact Lie groups, we explicitly calculated the characters of the representations $\pi_m.$ We then proved that any irreducible unitary
representation of $SU(2)$ is unitarily equivalent to one and only one of $\pi_m.$

Lecture 4.  After defining the exponential of a matrix, we introduced the notion of Lie algebras. We calculated the Lie algebras of $SU(n)$ and $SO(n)$ and introduced the adjoint representations of a Lie group on its Lie algebra.

Lecture 5. Using the adjoint representation of $SU(2)$ on its Lie algebra, we showed that $SO(3)$ is isomorphic to a quotient group of $SU(2).$ This was  then used to construct all the irreducible unitary
representations of $SO(3)$ in terms of $\pi_m.$

Lecture 6. Returning to the action of $SO(n)$ on $L^2(\R^n)$ we introduced the spaces of spherical harmonics and discussed representations of $SO(n)$ which are realised  on such spaces. We then discussed the invariance of certain family of subspaces of $L^2(\R^n)$ under the Fourier transform. We ended the lecture with a statement of the Hecke-Bochner identity.
Prof Rudra Sarkar (ISI, Kolkata)
Lecture -1 Topological groups: Definition, example  and general preliminaries. Group-action, uniform continuity, approximate identity, Existence and uniqueness of Haar measure (statements), some consequence e.g. compactly supported continuous functions are dense in $L^p$. Modular function and integration useful formulae for  left-right translations. Convolution and Young's inequality for unimodular groups.

Lecture-2 Unitary representations, example, equivalence of representation, sub-representation, irreducible representations, contra-gradient representation, direct-sum and tensor product of representations. Schur's lemma. Gelfand-raikov theorem (statement).

Lecture-3 Compact groups, general preliminaries like $L^p$-spaces are comparable. Irreducible representations are $1$-dim, Weyl's unitary trick, complete reducibility of unitary representations.

Lecture 4 Peter-Weyl theory: Matrix coefficient of representations and the space generated by them, various properties.  Schur's orthogonality relation,

Lecture 5 Peter-Weyl theory contd: Embedding of  elements of unitary dual in left/right regular representations, multiplicities. Proof of the result that matrix coefficients are dense in $L^2$ using Gelfand-Raikov and Stones-Weierstrass theorem and a direct proof of this result without using Gelfand-Raikov and as a consequence proving Gelfand-raikov for compact groups.

Lecture-6 Summarizing Peter-Weyl theory. Faithful representation and proving when a compact group is a matrix group. Definition and use of Characters of a representations, Fourier analysis on compact group using Hilbert space theory.

Participants

Time Schedule

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

Venue: KSOM, Kozhikode

06 June - 01 July, 2016

Organizes
Prof. M Manickam
Director
KSOM
Prof. A J Parameswaran
Professor
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.

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.

Participants

Time Schedule

Workshop on Automorphic forms

Venue: KSOM, Kozhikode

10-16 February, 2016

Organizes
Prof. M Manickam
Director
KSOM
Prof. B Ramakrishnan
Dear & Professor

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.

Participants

Time Schedule

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

Venue: KSOM, Kozhikode

01-06 February, 2016

Organizes
Prof. M Manickam
Director
KSOM
Prof. S D Veerappa Gowda
Professor
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.

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.

References:
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

Participants

ISL on Number Theory
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

05-17 October, 2015

Organizes
Prof. M Manickam
Director
KSOM
Professor

Topics covered in the workshop

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.

Dirichlet Character and Dirichlet Prime Number Theorem

Participants

Time Schedule

Workshop on Jacobi forms and Modular forms of half-integral weight

Venue: KSOM, Kozhikode

02-12 February, 2015

Organizes
Prof. M Manickam
Director
KSOM
Prof. B Ramakrishnan
Professor

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

Irrationality of Zeta (3) proved by Beukers.

S Boecherer

Introduction of Siegel modular forms; modular forms mod p.

J Meher

Product of Hecke eigenform.

Participants

Time Schedule

Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

01-19 December, 2014

Organizes
Prof. D S Nagaraj
Professor
IMSc., Chennai
Prof. V Balaji
Professor
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.

References:
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

Speakers

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

Participants

Time Schedule

ANNUAL FOUNDATION SCHOOL (AFS) - II (2014)
Supported by National Centre for Mathematics

Venue: KSOM, Kozhikode

01-28 May, 2014

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

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. 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

Participants

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

Venue: KSOM, Kozhikode

17-22 March, 2014

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

Participants

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:

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

Objective
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.
Subjects:
● 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

Schedule:
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

Organiser

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.

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. 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

Participants

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)

Participants

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)

Participants

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

Raja Sreedharan , TIFR Bombay

A K Nandakumaran, IISc Bangalore

K Sandeep, TIFR Bangalore

Mahuya Datta, ISI, Kolkatta

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. 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

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

 Name Affiliation Designation Address/e-mail Dr. B. V. Rajarama Bhat Indian Statistical Institute Professor Indian Statistical Institute, R. V. College(PO) Bangalore-560059 bhat@isibang.ac.in Dr. G. Ramesh Indian Statistical Institute Post-doctoral Fellow Indian Statistical Institute, R. V. College(PO) Bangalore-560059 rameshhcu@gmail.com Dr. Vijayarajan A. K. Kerala School of Mathematics Associate Professor Kerala School of Mathematics Kunnamangalam(PO) Kozhikode-673571 vijay.ksom@gmail.com

@2010 Kerala School of Mathematics