Initiation into Mathematics 2017  mini MTTS Programme
Venue: KSOM, Kozhikode
2430 April, 2017
A one week long Mini Mathematics Training and Talent Search programme organized during April 2430, 2017 at Kerala School of Mathematics, Kozhikode, Kerala with the funding of National Board of Higher Mathematics, Govt. of India.
The program was targeted to students in the second year/fourth semester B.Sc/Integrated M.Sc programmes having Mathematics as one of the compulsory papers. The topics which discussed in the program were Number Theory, Foundations of Set Theory, Real Analysis, Group Theory, and Linear Algebra. The program was conducted on the solid academic support of the prestigious Mathematics Training and Talent Search (MTTS) programme and it took up and implemented the meticulous teaching methodology of MTTS.
Resource Persons
Prof. A J Jayanthan : Foundations of Mathematics / Real Analysis
Dr. Vishnu Namboothiri K : Linear Algebra
Prof. S Kumaresan : Real Analysis
Local Coordinator  Academic Coordinator 

Prof. M Manickam Director Kerala School of Mathematics 
Dr. Vishnu Namboothiri K Department of Mathematics, Government College for Women, Thiruvananthapuram 
Summer Workshop for M.Sc. students and College Teachers
Venue: KSOM, Kozhikode
0121 April, 2017
A summer workshop on basic subjects of MSc syllabus meant for MSc students conductred at KSOM during 0121, April 2017. A small group of young college teachers also attended the program along with students as tutors and facilitators in this residential program. Subject experts from various nationallevel research institutes handled classes, lead discussions and conducted tutorial sessions.
Resource Persons
Sl. No. 
Name and Address of Resource person 

1 
Prof. Joseph Mathew 

2 
Prof. M Manickam 

3  Prof. Parameswaran Sankaran Professor, I.M.Sc, Chennai 

4  Prof. K N Raghavan Professor, I.M.Sc, Chennai 

5  Prof. B Ramakrishnan Dean & Professor, HarishChandra Research Institute, Allahabad 

6 
Prof. K N Ranganathan 

7 
Dr. V. Selvan 

8 
Dr. R. Thangadurai 

9  Prof. L Venkataraman Retired Professor, RKM Vivekananda College, Chennai 

Coordinator 

Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201617
PROGRAMME  V
Venue: KSOM, Kozhikode
0912 March, 2017
Topics of discussion:Differential Equations
Resourse Persons:
Prof. K Sandeep, TIFRCAM Bangalore
Dr. A. K. Nandakumaran, Professor, Indian Institute of Science, Bangalore
Topics covered:
1 Existence and Uniqueness of solution for ODE
2 Smooth dependence on the data
3 Stability of linear systems
4 First order partial differential equations Method of characteristics
Visit the link at http://nptel.ac.in/courses/111108081/
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201617
PROGRAMME  IV
Venue: KSOM, Kozhikode
0912 February, 2017
Topics of discussion:Algebraic Number Theory
Resourse Persons:
Prof. Sanoli Gun, IMSc Chennai
Prof. Purushotham Rath, CMI Chennai
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201617
PROGRAMME  III
Venue: KSOM, Kozhikode
1922 January, 2017
Topics of discussion:Semisimple Algebras and Group Representations
Resourse Persons:
Prof. K N Raghavan, IMSc Chennai
Prof. Vijay Kodiyalam, IMSc Chennai
Course Content
The resource persons for the program recommend the notes, exercises, and lectures
in the following links:
Notes and exercises:
> http://www.imsc.res.in/~knr/past/afsII_iisertvm_2015/Notes/knr_2.pdf
Videos of lectures:
> https://www.youtube.com/playlist?list=PLhkiT_RYTEU0CcvzU49C5AYmX7mLHUvOb
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
KSOM Workshop for B.Sc. students in Idukki district, Kerala
28th, 29th, 30th November, 1st and 2nd December 2016
KSOM conducted a Workshop at Pavanatma College, Idukki district, Kerala during 28 November  02 December 2016. The program was meant for BSc Mathematics students studying in colleges in Idukki, Kerala.
Resource Persons:
Prof. Sanoli Gun, IMSc Chennai
Prof. Purushotham Rath, CMI Chennai
Prof. B Ramakrishnan, HRI Allahabad
Special Lectures by:
Prof. Gautam Menon.IMSc Chennai
Prof. Murugan V, RKM Vevekananda College, Chennai
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics Tel: 0495 2809001 Email: murugumanick@ksom.res.in 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics Tel: 0495 2809004 Email: vijay@ksom.res.in 
Local Coordinators  

Mr. Sijo P. George Assistant Professor Pavanatma College, Idukki Email: sijopgorg@gmail.com Mob: 9746105166 
Mr. Bobin George Pavanatma College, Idukki Email: bobinkgeorge@gmail.com Mob: 9747905115 
Mr. Tijo James Assistant Professor Dept. of Mathematics Pavanatma College, Idukki Email: tijojames@gmail.com Mob: 9946884650 
Refresher Course for College Teachers in Kerala State:201617
PROGRAMME  II
Venue: KSOM, Kozhikode
0306 November, 2016
Topics of discussion:Groups and their representations
Resourse Persons:
Prof. A J Parameswaran, TIFR Mumba
Prof. V Balaji, CMI Chennai
Course Content
Semisimple representations.Closed and "algebraic" subgroups of GL(n,C).
Nilpotent and semisimple elements: simultaneous upper triangulability of ommuting
matrices (Jordan decomposition). Nilpotent and unipotent groups, Torus, Classification of connected abelian groups, Solvable groups, LieKolchin Theorem.
Reductive groups: If a connected algebraic group has a faithful irreducible representation then it is a reductive group. Classification of finite subgroups of GL(n,C): Theorem of Jordan.
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201617
PROGRAMME  I
Venue: KSOM, Kozhikode
2023 October, 2016
Topics of discussion:Linear Algebra in Control Theory  An Introduction
Resourse Persons:
Prof. Subramanian Moosath, IIST Thiruvananthapuram
Dr. Priyadarshnam, IIST Thiruvananthapuram
Course Content
Linear algebra, Linear systems, and Control concepts
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201516
PROGRAMME  V
Venue: KSOM, Kozhikode
1821 February, 2016
Topics of discussion:Complex Analysis
Resourse Persons:
Prof. Parameswaran Shankaran IMSc, Chennai
Prof. L Venkataraman Rtd Prof, Vivekananda College, Chennai
Course Content
Prof. Parameswaran Sankaran: Weierstrass theorem on infinite product representation, Hadamard's theorem, MittagLeffler
theorem, Gamma and Zeta functions (basic properties). Weierstrass function.
Ref: Complex Analysis by Ahlfors (Chapter 5)
Prof. L Venkataraman : Cauchy theory (basically complex integration and its applications). CauchyGoursat Theorem, Cauchy Integration Formulas, Singularities, Residue Theorem, Evaluation of Ontegrals using Residues, Argument Principle, Maximum Modulus Principle.
1. Conformal Mappings
2. Analytic Continuation
3. Riemann Mapping Theorem
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
KSOM Workshop for B.Sc. students in Wayanad district, Kerala
0913 December, 2015
KSOM conducted a Workshop at Wayanad during 0913 December, 2015. The program was meant for BSc Mathematics students studying in colleges in Wayanad.
Resource Persons
Dr. S Kumaresan (Department of Mathematics,, Hyderabad University)
Dr. V. Muruganandam (Department of Mathematics, NISER, Bhuvaneswar)
Dr. M. Manickam (KSoM)
Dr. A. K. Vijayarajan (KSoM)
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics Tel: 0495 2809001 Email: murugumanick@ksom.res.in 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics Tel: 0495 2809004 Email: vijay@ksom.res.in 
Local Coordinators  

Dr. Pamy Sebastian Head, Dept. of Mathematics Mary Matha Arts & Science College, Mananthavady, Wayanad Email: pamyseb@gmail.com Mob: +919446970984 
Mr. Latheesh Kumar A R Sulthan Bathery, Wayanad Email: latheesby@gmail.com Mob: +919947049112 
Mr. Santhosh Kumar C G Assistant Professor Dept. of Mathematics W M O Arts & Science College, Muttil, Wayanad Email: skumarcg@gmail.com Mob: +919447780543 
Refresher Course for College Teachers in Kerala State:201516
PROGRAMME  IV
Venue: KSOM, Kozhikode
1215 November, 2015
Topics of discussion:Topologies, Filters and Uniformities
Resourse Persons:
Prof. K. N. Raghavan, IMSc Chennai
Course Content
Basic reference: Bourbaki, General Topology, Part I, Chapters I and II
Other references:
J L Kelley, General Topology
Gilman & Jerison, Rings of Continuous Functions
Abstract:
Following Bourbaki (Chapter I), we will study filters and their limits. Filters, like nets, generalize sequences. They enable us to restore intuition to the notion of continuity: a map between general topological spaces is continuous if and only if, for every filter in the domain that converges to a point, the direct image filter converges to the image of that point. Moreover, Hausdorfness may be defined as the property that every filter has at most one limit point, (quasi)compactness as the property that every filter admits at least one cluster point.
Following Gilman & Jerison, we will illustrate the use of filters in the study of rings of continuous functions (such as C[0,1]).
Following Bourbaki again (Chapter II), we will study uniformities. Cauchy sequences and uniform continuity are not topological notions (they are not preserved under homeomorphisms; their usual definitions presume a metric) but they are closely related to topological notions. Uniform spaces generalize metric spaces. Like a metric, a uniformity too induces a topology. Uniform continuity, Cauchy filters, and completions may be defined in the context of uniform spaces.
Some notes following Bourbaki may be found at http://www.imsc.res.in/~knr/past/top15 (these notes are being currently added to).
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201516
PROGRAMME  III
Venue: KSOM, Kozhikode
1821 October, 2015
Topics of discussion: Differential Geometry
Resourse Persons:
Prof. S Kumaresan, University of Hyderabad
Prof. G Santhanam, IIT, Kanpur
Course Content
1. Review of Multivariable Calculus (3 lectures)
2. Curves: Definition, vector field, Frenet – n frame, existence of curves with prescribed curvatures, curvature and torsion of a space curve, FrenetSerret formula. (3 L)
3. Surfaces: Definition, vector field, quadratic form, first fundamental form, second fundamental form, curvature of curves on a surface, Principal curvature, Gaussian and mean curvatures. (6 L)
References:
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201516
PROGRAMME  II
Venue: KSOM, Kozhikode
1013 September, 2015
Topics of discussion: Matrix Analysis & Applications
Resourse Persons:
Prof. A. Chandrashekaran, Central University of Tamilnadu
Prof. Sachindranath Jayaraman, IISER, Thiruvananthapuram
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201516
PROGRAMME  I
Venue: KSOM, Kozhikode
2326 August, 2015
Topics of discussion: Topology
Resourse Persons:
Prof. A J Parameswaran, TIFR Mumbai
Prof. A J Jayanthan, Goa University
Differentiable maps, Tangent space, Implicit function theorem, Critical Points, Structure Theorem for nondegenerate critical points, Topological and Differentiable manifolds, Differentiability of functions on a manifold, Sard’s theorem, Partitions of unity, para compactness., Whitney’s Embedding Theorem, Morse functions and their Existence, CW complexes, Change of Topology as one passes a non degenerate critical point.
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Initiation into Mathematics 2015  mini MTTS Programme
Venue: KSOM, Kozhikode
0409 May, 2015
A one week long Mini Mathematics Training and Talent Search programme organized during May 49, 2015 at Kerala School of Mathematics, Kozhikode, Kerala with the funding of National Board of Higher Mathematics, Govt. of India.
The program was targeted to students in the second year/fourth semester B.Sc/Integrated M.Sc programmes having Mathematics as one of the compulsory papers. The topics which discussed in the program were Number Theory, Foundations of Set Theory, Real Analysis, Group Theory, and Linear Algebra. The program was conducted on the solid academic support of the prestigious Mathematics Training and Talent Search (MTTS) programme and it took up and implemented the meticulous teaching methodology of MTTS.
Resource Persons
Dr. P.S. Srinivasan : Foundations of Mathematics / Real Analysis
Dr. Subramanyan Moosath : Linear Algebra
Dr. Vishnu Namboothiri K : Group theory
Local Coordinator  Academic Coordinator 

Prof. M Manickam Director Kerala School of Mathematics 
Dr. Vishnu Namboothiri K Department of Mathematics, BJM Govt. College Chavara, Kerala 
OUTREACH PROGRAMME: Refresher Course for M.Sc. students in Kerala State
Venue: KSOM, Kozhikode
1726 April, 2015
KSOM conducted a refresher course during April 2015 for final year M.Sc. Mathematics students in Kerala. The course covered some basic topics of Algebra, Analysis, and Number Theory.
Resource Persons
Prof. M Manickam, KSOM : Algebra
Dr. A K Vijayarajan, KSOM : Linear Algebra
Prof. B Ramakrishnan, HRI Allahabad : Number Theory
Prof. Joseph Mathew, Formerly KSOM : Analysis
Prof. A J Parameswaran, TIFR Mumbai : Topology Lecture Notes
Coordinator 

Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201415
PROGRAMME  VI
Venue: KSOM, Kozhikode
1922 February, 2015
Topics of discussion: Measure Theory
Resourse Persons:
Prof. B. V. Rao, CMI, Chennai
Dr. A. K. Vijayarajan, KSOM
Prof. Joseph Mathew, KSOM
Prof. B V Rao & Prof. Joseph Mathew
1 Quick review of basic measure theory including Monotone Class theorem, Caratheodory extension; MCT, DCT, Fatou;
2 RadonNikodym theorem; Lebesgue decomposition; various notions of convergence;
3. Product spaces and Fubini;
4. L_p spaces; Specializing to p=2 simple introduction to Fourier series.
Dr. A K Vijayarajan
Spectral Measures
Course Materials
Prof. B V Rao
Prof. Joseph Mathew
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201415
PROGRAMME  V
Venue: KSOM, Kozhikode
0811 January, 2015
Topics of discussion: Algebra
Resourse Persons:
Prof. U. K. Anandavardhanan, IIT Bombay
Prof. A V Jayanthan, IIT Madras
Prof. U. K. Anandavardhanan : Representation Theory
Generalities on linear representations, characters, Schur's lemma, orthogonality relations, induced representations, Mackey's criterion.
Prof. A. V. Jayanthan : Commutative Algebra
Modules over Commutative rings, ascending and descending chain conditions, Noetherian rings and modules, Hilbert basis theorem.
References:
1. Abstract Algebra by Dummit and Foote
2. Algebra by Artin.
3. Algebra by Lang.
4. Introduction to Commutative Algebra by Atiyah and Macdonald.5. Linear Representations of Finite Groups by Serre.
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201415
PROGRAMME  IV
Venue: KSOM, Kozhikode
0609 November, 2014
Topics of discussion: Real & Complex Analysis
Resourse Persons:
Dr. K. Sandeep, TIFRCAM, Bangalore
Dr. Venky Krishnan, TIFRCAM, Bangalore
Dr. K Sandeep
R^n calculus, continuity, various notions of differentiability, inverse function and implicit function theorems, extension of these notions to Banach spaces, finally some applications.
Dr. Venky Krishnan
From Cauchy Riemann equations to some advanced topics of Analytic continuation.
Course Materials
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201415
PROGRAMME  III
Venue: KSOM, Kozhikode
0912 October, 2014
Topics of discussion: Number Theory & Cryptography
Resourse Persons:
Prof. Surya Ramana, HRI Allahabad
Mr. Kasi Viswanadham, HRI Allahabad
Mr. Jay G Mehta, HRI Allahabad
D. Surya Ramana
Arithmetical functions
Proof of prime number theorem
G. Kasi Viswanadham
Some topics in Elementary Number Theory  Time estimates, divisibility and Euclidean algorithm, congruences, applications to factoring
Finite Fields and Quadratic Residues  finite fields, Quadratic residues and reciprocity
Jay Mehta
Classical Cryptography  some simple cryptosystems, different types of Ciphers, Cryptanalysis of Ciphers
Public Key cryptography  idea of public key cryptography, RSA and
discrete log
Reference books
Introduction to analytic and probabilistic number theory G. Tenenbaum
Introduction to analytic number theory T. Apostol
Number theory and Cryptography N. Koblitz
Cryptography Theory and Practice Douglas R. Stinson
Course Materials
Cryptography Theory and Practice
Introduction to Cryptography
Course in Number Theory And Cryptography
Prime Number Theory
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201415
PROGRAMME  II
Venue: KSOM, Kozhikode
1114 September, 2014
Topics of discussion: Topology
Resourse Persons:
Prof. A.J. Parameswaran, TIFR Bombay
Prof. Parameswaran Sankaran, IMSc Chennai
Homological Algebra: Complexes and homology, Exact sequences, Long exact sequence assciated to a short exact sequence of complexes.
Topology: Simplicial complexes, CW Complexes.
Homology Thoeries: Simplicial Homology, Cellular Homology, Singular Homology.
Main Theorems (Only statements with examples): Excision Theorem, Homology sequence of a pair/tripple, Functoriality. Invariance of domain.
Further if time permits: Brouwer fixed point theorem, Lefschetz fixed point theorem and BorsukUlam theorem.
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Refresher Course for College Teachers in Kerala State:201415
PROGRAMME  I
Venue: KSOM, Kozhikode
14th, 16th & 17th August, 2014
Topics of discussion: Advanced Functional Analysis
Resourse Persons:
Prof. E. K. Narayanan, IISC, Bangalore
Prof. P. K. Sanjay, NIT, Calicut
Diagonalization of matrices
Singular value decomposition
Self adjoint operators
C* algebras
Gefland theory
GelfandNaimark theorem
Spectral theorem
Spectral measures
Compact operators etc.
The two series of lectures will complement each other; First one, demystifying the finite dimensional spectral theorem (diagonalization of self adjoint matrices) following a paper of Adam Koranyi by doing the singular decomposition first (see A. Koranyi: Around the finite dimensional spectral theorem, Amer. Math. Monthly 108 (2001) no.2 120125) and the second, introducing the participants to the infinite dimensional version.
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
OUTREACH PROGRAMME
Refresher Course for B.Sc. Mathematics Students in Kerala State
Venue: KSOM, Kozhikode
22  30 April, 2014
KSOM conducted refresher course during April 2014 for B.Sc. Mathematics IV Semester students studying in Kerala state. The course covered some basic topics of Algebra, Analysis, Linear Algebra and Number Theory.
The course with relaxed schedule and interactive sessions discussed abstract topics in Number Theory, Analysis, Algebra and Linear Algebra. The sessions were handled by Joseph Mathew, A. K. Vijayarajan, V Srivatsa, P Jisna, P Shankar(KSOM) and Kirankumar V B.
Topics covered:
Analysis:
Real number system, Rational numbers, Irrational numbers, Completeness axiom
Sequences, Cauchy sequences, Convergence
Series, Convergence
Topology of the real line
Algebra:
GroupssubgroupsCyclic groupsGroup homomorphisms and IsomorphismsNormal subgroups
Introduction to Rings and Fields
Linear Algebra:
Vector spaces, Sums and Direct Sums
Linear Independence, Bases and Dimension
Linear maps, Matrices
Number Theory:
Divisibility
Congruences and some basic theorems
Quadratic reciprocity law
Some arithmetic functions
Primitive roots
Coordinators  

Dr. A K Vijayarajan Associate Professor email: vijay@ksom.res.in Phone: 0495 2809004 Fax: 0495 2809010 
Prof. Joseph Mathew Visiting Professor email: jmathew@ksom.res.in Phone: 0495 2809005 Fax: 0495 2809010 
Discussion Meeting on ANALYTIC & ALGEBRAIC GEOMETRY
Venue: KSOM, Kozhikode
24  28 March, 2014
A. J. Parameswaran & Indranil Biswas
TIFR
Speakers  Participants 

Nitin Nitsure Jens Kristian Egsgaard Arijith Dey 
Nitin Nitsure Usha Bhosle Amit Hogadi Chanchal Kumar. Sukhendu Mehrotra D.S. Nagaraj S. S. Kannan. Arijit Dey V. Balaji Sergey Arkhipov JeanBaptiste Meilhan Jens Kristian Egsgaard Florian Schätz. Daniel Tubbenhauer Joergen Ellegaard Andersen Henning Haahr Andersen Niels Leth Gammelgaard Søren Fuglede Jørgensen Jens Jakob Kratmann Nissen Arash Momeni Kenneth Rasmussen Simon Stoltze Anders D. Jakobsen Troels B. Andersen Dennis H. Pedersen Paolo Masulli 
Orientation/Refresher Course for College Teachers in Kerala State201314
PROGRAMME  VI
Venue: KSOM, Kozhikode
0710 March, 2014
Topics of discussion: Measure Theory
Resourse Persons:
Dr. K. Sandeep,
TIFR Centre for Applicable Mathematics, Bangalore
Dr. Joseph Mathew, KSOM
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Orientation/Refresher Course for College Teachers in Kerala State201314
PROGRAMME  V
Venue: KSOM, Kozhikode
27 February  02 March, 2014
Topics of discussion: Topology
Resourse Persons:
Prof. A J Parameswaran, TIFR Mumbai
Algebraic Topology: (A J Parameswaran & Sagnik )
1: $\pi_1$: definition, Homotopy, Exapmles.
2: VanKampen Theorem; Computation of the Fundamental groups of various spaces.
3: Covering Spaces: Path lifting, Homotopy lifting, Codition for arbitrary map lifting....
4: Invariance of Domain, Separation Theorem, Jordan curve Theorem ...
5: Universal covering (Existence  statement only) Relation between $\pi_1$ and fibres of covering maps.
Prof. Parameswaran Sankaran, IMSc, Chennai
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Orientation/Refresher Course for College Teachers in Kerala State201314
PROGRAMME  IV
Venue: KSOM, Kozhikode
1013 January, 2014
Topics of discussion: Functional Analysis
Resourse Persons:
Prof. Dr. T. Sengadir, Central University, Tamilnadu
Topic: Distribution Theory, Sobolev Spaces, Semigroup Theory , Basic PDE, Nonlinear Analysis, Fourier Transforms
Dr. D. Surya Ramana, HRI, Allahabad
Topic: The basic theorems of functional analysis  Hahn Banach, Completeness and Banach Spaces, Operators, Baire Category Theorem, Uniform Boundedness, Closed Graph Open Mapping together with some wellknown applications.
Dr. A K Vijayarajan, Kerala School of Mathematics, Kozhikode
Topic: Topologies of B(H)
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Orientation/Refresher Course for College Teachers in Kerala State201314
PROGRAMME  III
Venue: KSOM, Kozhikode
2225 November, 2013
Topics of discussion: Complex Analysis
Time Schedule  
Date  9:0010:15  10:15 10:30  10:3011:45  11:4512:00  12:001:15  1:152:30  2:303:30  3:303:45  3:454:45 
22Nov13  AJP  Hot Break  EKN  Cool Break  AJP  Lunch Break  Tutorial EKN  Hot Break  Tutorial AJP 
23Nov13  EKN  AJP  EKN  Tutorial AJP  Tutorial EKN  
24Nov13  AJP  EKN  AJP  Tutorial EKN  Tutorial AJP  
25Nov13  EKN  AJP  EKN  Tutorial AJP  Tutorial EKN  
Resource Persons:
Prof. A J Prameswaran, TIFR Mumbai
Topic: Quick review of topological preliminaries, simply connectedness, homotopy etc. Definition of log and n^th roots, lifting of maps, covering maps. Covering spaces of C*, C** and D*. Applications: Characterization of nowhere vanishing functions, Picard Theorems, Fundamental Theorem of Algebra.
Prof. E K Narayanan, IISc, Bangalore
Topic: Quick review of holomorphic (analytic) functions, CauchyRiemann equations, Cauchy's theorem, power series. Properties of holomorphic functions, zeros, maximum modulus principle, Liouville's theorem, isolated singularities, arguments principle, open mapping theorem. If time permits some major theorems like, Montel's theorem, Riemann mapping theorem and Pciard's theorem (Ahlfors's proof).
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Orientation/Refresher Course for College Teachers in Kerala State201314
PROGRAMME  II
Venue: KSOM, Kozhikode
0811 November, 2013
Topics of discussion: Real Analysis
Time Schedule  
Date  9:0010:15  10:15 10:30  10:3011:45  11.4512:00  12:001:15  1:152:30  2:303:45  3:454:00  4:005:00 
08Nov13  EKN  Hot Break  SS  Cool Break  EKN  Lunch Break  EKN  Hot Break  Tutorial SS 
09Nov13  SS  EKN  SS  EKN  Tutorial JM  
10Nov13  JM  SS  EKN  EKN  Tutorial EKN  
11Nov13  EKN  SS  SS  JM  Discussion/ Feedback  
Resource Persons:
EKN: E K Narayanan, IISc, Bangalore
Topic: Borel \sigmaalgebra, outer measure, measurable sets, construction of Lebesgue measure on \mathbb R, measurable functions, introduction to Lebesgue integration and limit theorems.
SS: S. Somasundaram, Manonmaniam Sundaranar University,Tirunelveli
Topic: Nets and Ordinal Topology
JM: Joseph Mathew, Kerala School of Mathematics
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Orientation/Refresher Course for College Teachers in Kerala State201314
PROGRAMME  I
Venue: KSOM, Kozhikode
1114 October, 2013
Topics of discussion: Galois Theory, Field Extension and Commutative Algebra
Time Schedule  
Date  9:3011:00  11:0011:30  11:301:00  1:002:30  2:303:30  3:304:00  4:005:00 
11Oct13  PS  Tea Break  PS  Lunch Break  PS  Tea Break  PS 
12Oct13  PS  PS  PS  PS  
13Oct13  RT  RT  RT  RT  
14Oct13  RT  RT  RT  RT  
Resource Persons:
PS : Parameswaran Sankaran, IMSC, Chennai : Galois Theory
RT: R Thnagadurai, HRI, Allahabad : Commutative Algebra
Coordinators  

Prof. M Manickam Director Kerala School of Mathematics 
Dr. A K Vijayarajan Associate Professor Kerala School of Mathematics 
Outreach Programme 
Refresher Course for B.Sc. Mathematics Students in Kerala State
Venue: KSOM, Kozhikode
1422 September, 2013
KSOM conducted refresher course during September 2013 for B.Sc. Mathematics Final year students in Kerala state. The course covered some basic topics of Algebra and Analysis.
The course with relaxed schedule and interactive sessions discussed abstract topics in Number Theory, Analysis, Algebra, Linear Algebra and Graph Theory. The sessions were handled by V. Krishna Kumar(NICER, Bhuvaneswar), P. K. Sanjay(NIT, Calicut), Ananthavardhanan (IIT, Bombay), Jijo Sukumaran (Govt. College, Kasargod), M. Manickam, Joseph Mathew, A. K. Vijayarajan and Guram Donadze (KSOM).
Time Schedule  
Date  9:3011:00  11:0011:30  11:301:00  1:002:30  2:303:30  3:304:00  4:005:00 
14Sep13  MM  Tea Break  SPK  Lunch Break  JM  Tea Break  AKV 
15Sep13  GD  AKV  SPK  GD  
16Sep13  MM  GD  JM  AKV  
17Sep13  MM  JM  No sessions  No sessions  
18Sep13  MM  JM  VK  VK  
19Sep13  JM  AKV  GD  VK  
20Sep13  JM  VK  JS  GD  
21Sep13  GD  AKV  JM  Res. Sch.  
22Sep13  AKV  GD  Discussion  Discussion  
MM: M. Manickam  Algebra
JM: Joseph Mathew  Topics in Analysis
VK:V Krishna Kumar  Construction of Number system
SPK: Sanjay P K  Topics in Analysis
GD: Guram Donadze  Graph Theory
AKV: A. K. Vijayarajan  Linear Algebra
JS : Jijo Sukumaran  Linear Algebra
Res. Sch.: Research Scholar, KSOM  Tutorial
Coordinator
Dr. A K Vijayarajan
Associate Professor, KSOM
Short Term Discussion Meeting
31 December 2012  13 January 2013
Visitors
1. Bipin Balaram, Assistant Professor, Dept. of Mechanical Engineering, Amrita School of Engineering, Amrita Viswa Vidyapeetham, coimbatore.
2. Vinod V., Research Scholar, Dept. of Mechanical Engineering, NIT Calicut.
Venue: KSOM, Kozhikode
PLAN OF WORK
The basic objective of this work is to develop a robust numerical method for obtaining the periodic solutions of nonlinear ordinary differential equations and to device a strategy to quantitatively estimate the change in this solution when the system parameters are slowly varied. Historically, homotopy based methods like parametric continuation coupled with shooting algorithms has been used to perform this task. With the help of an arclength reparameterisation technique at singular points in the parameter space, this can be used for practical bifurcation analysis too. But this scheme has two serious shortcomings.
First, it makes the assumption that the frequency of the solution is the same as that of the forcing
function and hence, runs into trouble when dealing with selfexcited systems like Van der Pol
oscillators. This means that systems like coupled selfexcited oscillators, which are of great
practical importance in areas like physics, engineering and biology and which show such intresting
behaviour as synchronisation and clustering, cannot be studied using this simple continuation
strategy. So, an effort will be made to develop a continuation strategy based on generalised
shooting method which treats the frequency of the periodic solution also as an unknown. This
means that the new scheme should converge to a point in the enhanced state space of dimension
2n+1; 2n being the dimension of the initial condition vector. Preliminary efforts show that the use of
homotopy methods will reduce this problem to an algebraic one with 2n equations in 2n+1
variables. This poses a challenge which we hope to address. This scheme shall be used to study the
synchronisation in (dissipatively and reactively) coupled Van der Pol oscillators in a ring by treating
the coupling constant as the continuing parameter with an emphasis on tracking down possible
bifurcation phenomenon.
Second, existing continuation strategies are equipped to carry out the solution continuation in the
euclidean state space alone. But in many cases involving n coupled oscillators, the dynamics is
restricted to a manifold embedded in the 2n dimensional state space. For example, the dynamics of
a double pendulum can be studied on a 4 dimensional euclidean state space, but the modulo 2π
nature of dynamics means that it is restricted to a torus. There can be significant computational
gains if this continuation can be restricted to the manifold. Keeping this in mind, the possibility of
adapting the developed continuation scheme to work on the appropriate manifold will be explored.
This will involve the use of techniques from computational differential geometry.
Outreach Programme 
Refresher Course for B.Sc. Mathematics Students in Kerala State
Venue: KSOM, Kozhikode
2329 December, 2012
This outreach programme was meant for B.Sc. Mathematics third year students, who were studying in Kerala state. The course covered some basic topics of Algebra, Analysis and Number Theory. Some first year students of M.Sc. Mathematics also participated.
Outreach Programme for Higher Secondary School Students
Venue: KSOM, Kozhikode
02 October, 2012
Second phase of outreach programme for Higher Secondary School Students in the nearby region was held at KSOM in continuation with the Outreach Programme for Higher secondary School Students conducted in May 2012. This programme connected students with mathematical role models and shared their knowledge of and passion for mathematics and its various subspecialities and related fields.
Speakers
1  Dr. M Manickam  Director, KSOM 
2  Prof. E. Krishnan  Formerly Professor, University College, TVM 
3  Prof.U. K. Anandavardhanan  Professor, IITB 
4  Prof. A K Vijayarajan  Associate Professor, KSOM 
5  Prof. Joseph Mathew  Visiting Professor, KSOM 
Outreach Programme for Higher Secondary School Teachers
Venue: KSOM, Kozhikode
30 September 01 October, 2012
A two days discussion meeting with MathematicsTeachers of Higher Secondary Schools in the nearby region was held at KSOM in continuation with the Outreach Programme for Higher secondary School Teachers conducted in May 2012.
Speakers
1  Dr. M Manickam  Director, KSOM 
2  Prof. E. Krishnan  Formerly Professor, University College, TVM 
3  Prof.U. K. Anandavardhanan  Professor, IITB 
4  Prof. A K Vijayarajan  Associate Professor, KSOM 
5  Prof. Joseph Mathew  Visiting Professor, KSOM 
Outreach Programme for Higher Secondary School Students
Venue: KSOM, Kozhikode
5th May, 2012
In continuation with the discussion meeting with Higher Secondary School Mathematics Teachers held at KSOM on 7th April, 2007, an outreach programme for HSS students conducted at KSOM. KSCSTE Member Secretary Dr. K. K. Ramachandran inauguratede the programme.
Panorama Lecture
Venue : KSOM, Kozhikode
01  05 May, 2012
Prof. Ram Murty from Queens University, Canada delivered a series of "Panorama Lectures" on Ramanujan & Lfunctions from 01 to 04, May 2012.
Discussion meeting with Higher Secondary School Mathematics Teachers
Venue: KSOM, Kozhikode
7th April, 2012
Mathematics in Schools
Mathematics underpins so many aspects of our lives and provides the foundation for science, engineering, economics, finance and many other professional fields. For all these reasons, it is critical to help students early on to appreciate the beauty and power of mathematics and to begin to understand how it plays out in so many areas. Of course, teachers are already experts at encouraging and guiding students, but having the added support of a mathematical research institute provides obvious benefits not only for students but for the teachers and mathematicians as well.
Discussion Meeting
As a prelude to the outreach program for Higher Secondary School students, a one day discussion meeting held at KSOM with mathematics teachers of Higher Secondary Schools who are teaching for +1/+2 students in the near by region to discuss and find out the modalities of conducting the outreach program.
Prof. Michael A. Dritschel of University of Newcastle, UK visited KSOM
08  12 March, 2012
The visit was for a discussion meeting on `Operator Spaces and Operator Theory' with KSOM Faculty and researchers working in the area.