IST - Complex Analysis from a geometric viewpoint (2022)
Speakers and Syllabus
|
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.
|
In addition to the above courses, there will be a 3 hrs mini-course by Professor S. Ponnusamy (online) that will give a research survey on harmonic maps in complex analysis.
Tutors:
-
Dr. Chandan Maity (CM), IISER Mohali
-
Dr. Narendra Singh Sijwali, Govt P G College, Dwarahat
-
Dr. Abhishek Mukherjee (AM), Kalna College, WB
-
Tejbir Lohan (IISER Mohali)
-
Ramanpreet Kaur(RK), DU
-
Nidhi Gahlyan (NG), DU
References:
1. E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, 2003.
2. G. A. Jones and D. Singerman, Complex Functions: An algebraic and geometric viewpoint, Cambridge University Press, 1987.
Time Table
First week:( 13 June - 18 June)
|
09.30
|
11.00 |
11.30 |
1.00 |
2.30 |
3.30 |
3.45 |
4.45 |
|
|
Mon |
NSS |
T E A |
KG |
L U N C H |
T1 NSS TL RK
|
T E A |
T2 KG AM CM |
S N A C K S
|
|
Tues |
NSS |
KG |
T3 KG AM CM |
T4 NSS TL RK
|
||||
|
Wed |
GD |
KG |
T5 KG AM CM |
T6 GD RK NG
|
||||
|
Thur |
GD |
KG |
T7 AM CM TL |
T8 GD RK NG |
||||
|
Fri |
AM |
GD |
T9 AM CM TL |
T10 GD RK NG |
||||
|
Sat |
AM |
GD |
T11 AM CM TL |
T12 GD RK NG |
Second Week : (20 June - 25 June)
|
|
09.30
|
11.00 |
11.30 |
1.00 |
2.30 |
3.30 |
3.45 |
4.45 |
5–6:30 |
|
Mon |
EKN |
T E A |
GD |
L U N C H |
T1 GD NG RK |
T E A |
T2 EKN RK TL |
S N A C K S
|
|
|
Tues |
EKN |
GD |
T3 GD NG RK |
T4 EKN RK TL |
|
||||
|
Wed |
HC |
EKN |
T5 HC NSS CM |
T6 EKN RK TL
|
|
||||
|
Thur |
HC |
EKN |
T7 HC NSS CM |
T8 EKN RK TL |
Samy-1 |
||||
|
Fri |
HC |
EKN |
T9 HC NSS CM |
T10 EKN RK TL |
Samy-2 |
||||
|
Sat |
HC |
EKN |
T11 HC NSS CM |
T12 EKN RK TL |
Valedictory* |
* The valedictory meeting would include a special address (online) by the NCM director Prof (Dr.) Jugal Verma & the USERC director Prof (Dr.) Anita Rawat.