Transformations & Isometries (2022)

Venue:	IISER, Kolkata, West Bengal
Date:	6th, Jun 2022 to 18th, Jun 2022

School Convener(s)
Name	Dr. Shibananda Biswas	Dr. Somnath Basu	Dr. Sushil Gorai
Mailing Address	Department of Mathematics & Statistics, Indian Institute of Science Education & Research Kolkata	Department of Mathematics & Statistics, Indian Institute of Science Education & Research Kolkata	Department of Mathematics & Statistics, Indian Institute of Science Education & Research Kolkata

List of the actual speakers:

Name of the speaker	Position	Affiliation and address
Dr. Shibananda Biswas	Associate Professor	IISER Kolkata
Dr. Somnath Basu	Associate Professor	IISER Kolkata
Dr. Sushil Gorai	Associate Professor	IISER Kolkata

List of the actual tutors:

Name of the tutor	Position	Affiliation and address
Sanjoy Chatterjewe	PhD student	IISER Kolkata
Sandip Samanta	PhD student	IISER Kolkata
Amar Deep Sarkar	Post-doc fellow	IISER Kolkata

Syllabus:

Actual syllabus covered by each speaker:

Name of the Speaker	:	Detailed Syllabus
Dr. Shibananda Biswas		We covered distance and angle preserving maps in $\mathbb{R}^2$. Isometries in $\mathbb{R}^2$ and complex formula of plane isometries, orientations, finite group of plane isometries as well as isometries in $\mathbb{R}^n$ were discussed. A detailed discussion about matrix groups,permutation groups and classification of isometries was made along with (orthogonal) projection maps and its properties. Finally we discussed finite group of isometries in $\mathbb{R}^3$ and classification of conics under isometries. References: Geometry - Ancient & Modern by J. R. Silvester.
Dr. Somnath Basu		We discussed the topic of isometries of surfaces embedded in $\mathbb{R}^3$, starting with the basics of curves and surfaces in $\mathbb{R}^3$, revisiting osculating circles (and planes), Serret-Frenet formula. We discussed curvature (Gaussian and mean) of surfaces in variousforms.The first and second fundamental form was discussed, followed by various implications of the sign of the Gaussian curvature. We talked about Gauss-Bonnet theorem and the local-global phenomenon it illustrated. We also discussed isometries of a surface, geodesics and some relevant examples (for the three basic prototypes). References: Differential Geometry of Curves & Surfaces by M. Do Carmo,Differential Geometry: A First Course in Curves and Surfaces by T. Shifrin.
Dr. Sushil Gorai		We discussed conformal isometries, dealing with hyperbolicity of planar domains. We started with classical Schwarz lemma, Schwarz-Pick lemma and computation of the automorphism group of the unit disk and then discussed Poincar\'{e} metric on $\mathbb{D}$ in detail. Schwarz lemma was revisited from the viewpoint of the Poincar\'{e} metric and its curvature.Some complex analysis results like Liouville's Theorem and Montel's Theorem were presented from a geometric viewpoint. A proof of the big Picard Theorem was given Hyperbolicity of the domain was emphasized throughout. References: Complex Analysis: The Geometric Viewpoint by S. G. Krantz.

Time Table

Actual time-table that was followed during the programme:

Day	Date	Lecture 1 9:30 - 11:00	Tea (time)	Lecture 2 11:30 - 13:00	Lunch (time)	Tutorial 1 14:30 - 15:30	Tea (time)	Tutorial 2 16:00 - 17:00	Snacks (time)
		name of the speaker (abbreviation)		name of the speaker (abbreviation)		name of the tutors (abbreviation)		name of the tutors (abbreviation)
Mon	06/06	SB		SG		SB+SC		SG+AS
Tues	07/06	SB2		SB		SB2+SS		SB+SC
Wed	08/06	SB		SB2		SB+SC		SB2+SS
Thu	09/06	SG		SB		SG+AS		SB+SC
Fri	10/06	SB		SG		SB+SC		SG+AS
Sat	11/06	SG		SB2		SG+AS		SB2+SS
Mon	13/06	SB		SB2		SB2+SS		SB+SC
Tues	14/06	SB2		SB		SB2+SS		SB+SC
Wed	15/06	SB		SG		SG+AS		SB+SC
Thu	16/06	SB2		SG		SB2+SS		SG+AS
Fri	17/06	SB2		SG		SB2+SS		SG+AS
Sat	18/06	SB2		SG		SB2+SS		SG+AS

Full forms for the abbreviations of speakers and tutors:
SG: Sushil Gorai; SB: Shibananda Biswas; SB2: Somnath Basu;SS: Sandip Samanta; SC: Sanjoy Chatterjee; AS: Amar Deep Sarkar

List of actual Participants

Sr.n	SID	Full Name	Gender	Affiliation	Position in College/ University	University/ Institute M.Sc./ M.A.	Year of Passing M.Sc./ M.A	Ph.D. Deg.
1	32815	Dr. N Paramaguru	Male	Government Arts College for Women	Assistant Professor	Annamalai University	2003	11/09/2014
2	33705	Prof Rajasekar S P	Male	Government Arts College for Women	Assistant Professor	Pondicherry University	2004
3	33975	Dr. Chandru Hegde	Male	Mangalore University	Asst. prof.	Mangalore University	2006	14/07/2018
4	38409	Mr. Sapan Gupta	Male	Sri Sathya Sai Institute of Higher Learning	Assistant Professor	IIT Madras	2018
5	38688	Mr Bindumadhaba Mohanty	Male	Radhakrishna Institute of Technology and Engineering	Assistant Professor	Sambalpur University	2014
6	38975	Dr. Surya Prakash	Male	C.M.P. Degree College, University of Allahabad	Assistant Professor	University of Allahabad	2015	20/07/2021
7	38976	Mr. Ajim Uddin Ansari	Male	C.M.P. Degree College, University of Allahabad, Prayagraj	Assistant Professor	University of Allahabad	2015
Local participant (without accommodation)
8	37431	Mr. Tamoghna Kar	Male	Ramakrishna Mission Vivekananda Educational and Research Institute	MSc Student	Ramakrishna Mission Vivekananda Educational and Research Institute	2022
9	39014	Ms Anuvertika Pandey	Female	NISER, Bhubaneswar	PhD student	Banaras Hindu University	2020
10	39015	Ms Saswati Mukherjee	Female	National Institute of science education and research	Phd student	University of Burdwan	2019
11		Ms. Rachna Aggarwal			PhD student	University of Delhi
12		Mr. Rakhal B Chandran			BS-MS student	IISER Kolkata
13		Mr. Harshith Sairaj			BS-MS student	IISER Kolkata

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