Name of the Speaker with affiliation

No. of Hours

Topics

Dr. Narendra Singh Sijwali (NSS), Govt PG, College , Dwarahat &

Dr .Gopal Datt (GD), BBAU, Lucknow

(3 hrs & 3 hrs resp. )

Complex Function Theory I: Arithmetic of complex numbers, Exponential functions, Complex Differentiation, Holomorphic functions, The Exponential and complex trigonometric functions, Polynomials, Logarithmic function, Power series. Complex line integral. Cauchy’s theorem. General contour integration. Application of Cauchy’s formula: Zeros of a holomorphic function. Entire function, Liouville’s theorem.

Dr .Gopal Datt (GD), BBAU, Lucknow

6 hrs

Complex Function Theory II: Homotopy version of Cauchy’s theorem, Jordan curve theorem. Branches of logarithm on simply connected domains. Singularities, Laurant Series, Residue theorem, Argument principle, Open mapping theorem, Maximum modulus principle, Riouche’s theorem, Mittag-Leffer theorem, Little and Big Picard theorems.

Prof E K Narayanan (EKN)

IISc.

9 hrs

Several Complex Variables:Holomorphic functions of several variables, basic properties such as open mapping theorem, the Schwarz lemma. Hartogs’ separate analyticity theorem. Invariant metrics such as the Bergman, Caratheodory and Kobayashi metric.

Dr. K. Gongopadhyay (KG), IISER Mohali

&

Dr. Abhishek Mukherjee (AM), Kalna College, Burdwan

9 hrs

(6 & 3 resp.)

Conformal Maps: Euclidean similarity geometry using complex numbers. Riemann sphere.Cross ratios. Ideas of conformal transformation. Introduction to hyperbolic plane, Several models of the hyperbolic plane. Groups of Mobius transformations. Action of Mobius transformations. Geodesics. Area of a triangle. Gauss-Bonnet theorem. Schwarz’s Lemmas; Schwarz-Pick Lemma;

Professor Harish Chandra (HC) , BHU

6 hrs

Harmonic Functions, Riemann Mapping theorem; Grotzsche’s problem.