Syllabus and Speakers

Session I

(05 April- 17 April 2021)

• Group Theory

Speaker: Dr A A Ambily CUSAT

Groups, Subgroups, Cyclic group, Permutation groups, Dihedral groups Direct product(Fundamental theorem for finitely generated abelian groups), Cosets, Normal Subgroups, Normal series, Class Equations, Cayley’s theorem, Sylow theorems

• Rings and Fields

Speaker: Dr Dipankar Ghosh, IIT Hyderabad

Rings, integral domains, ideals, prime and maximal ideals, quotient rings, Fields, Gaussian Integers, Polynomial Ring, Irreducibility, UFD, PID, ED, Field Extensions, Cyclotomic Extension, Algebraic and Transcendental Extension, Galois Extension, Galois Groups, Separable Extension, Simple Extensions

• Real Analysis

Speaker: Dr I Subramania Pillai, University of Pondicherry

Convergence of Sequences and series, Continuity, compactness and continuity, connectedness and continuity, Differentiability, Mean value theorem, Sequences and series of functions, Power series, Radius of convergence, Tests for determining region of convergence.

• Multivariable Calculus

Speaker: Dr I Subramania Pillai, University of Pondicherry

Definition of differentiability, derivative as a linear transformation, directional derivative, partial derivative, Mixed partial derivatives, inverse and implicit function theorems (Examples and Applications)

Session II

(24 May- 05 June 2021)

• Linear Algebra

Speaker: Prof G P Youvaraj, Formerly of RIASM, University of Madras

Eigenvalues and eigenvectors, Cayley-Hamilton theorem. Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms, Quadratic forms, reduction, and classification of quadratic forms

• Complex Analysis

Speaker: Dr Sairam Kaliraj, IIT Ropar

Power series, Tests for convergence of power series, Analytic functions, Cauchy-Riemann equations. Contour integral (Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Taylor series, Laurent series, calculus of residues), Conformal mappings, Mobius transformations.

• Ordinary Differential Equations

Speaker: Dr Manil T Mohan, IIT Roorkee

Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs, Sturm-Liouville boundary value problem, Green’s function.