Kerala School of Mathematics (KSOM) will conduct a series of orientation/refresher courses for college teachers in Kerala from August 2014at KSOM. Mathematicians from various research institutes in India will be conducting the classes. Each course of 4 day duration will be on specific subjects such as Functional Analysis, Topology, Number Theory, Real & Complex Analysis, Algebra, and Measure Theory.
All Mathematics teachers from Colleges in Kerala are eligible to apply. Travel expense upto 3 AC will be met by KSOM and accommodation will be arranged at the KSOM guest house. Teachers are encouraged to apply for more than one course.
Tentative schedule of the programme is as follows.
PROGRAMMEI
1417 August, 2014 : Advanced Functional Analysis
Resource persons:
Prof. E. K. Narayanan, IISC, Bangalore
Prof. P. K. Sanjay, NIT, Calicut
Click here for Topics covered
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.
Time Schedule
List of Participants.
PROGRAMMEII
1114 September, 2014 : Topology
Resource persons:
Prof. A.J. Parameswaran, TIFR Bombay
Prof. Parameswaran Sankaran, IMSc Chennai
Click here for Topics
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.
Time Schedule
List of Participants
PROGRAMMEIII
0912 October, 2014 : Number Theory and Cryptography
Resource persons:
Prof. Surya Ramana, HRI Allahabad
Mr. Kasi Viswanadham, HRI Allahabad
Mr. Jay G Mehta, HRI Allahabad
Click here for Topics
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
Time Schedule
List of Participants
PROGRAMMEIV
0609 November, 2014 : Real and Complex Analysis
Resource persons:
Dr. K. Sandeep, TIFRCAM, Bangalore
Dr. Venky Krishnan, TIFRCAM, Bangalore
Click here for Topics
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.
Time Schedule
List of Participants
PROGRAMMEV
0811 January, 2015 : Algebra
Resource persons:
Prof. U. K. Anandavardhanan, IIT Bombay
Prof. A V Jayanthan, IIT Madras
Click here for Topics
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.
Time Schedule
List of Participants
PROGRAMMEVI 1922 February, 2015 : Measure Theory
Registration Closed.
Resource persons:
Prof. B. V. Rao, CMI, Chennai
Dr. A. K. Vijayarajan, KSOM
Prof. Joseph Mathew, KSOM
Click here for Topics
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 Material
Prof. B V Rao
Time Schedule
List of Participants
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 
