The Laplacian in Riemannian Geometry (2024) | National Centre for Mathematics

Convener(s)


Name:	Dr. Atreyee Bhattacharya	Dr. Anandateertha Mangasuli
Mailing Address:	Room no. 301, Academic Building 1 (Infinity building), IISER Bhopal, Bhauri, bhopal Bypass Road, Bhopal, Madhya Pradesh: 462066	Room no. 214, Academic Building 1 (Infinity building), IISER Bhopal, Bhauri, bhopal Bypass Road, Bhopal, Madhya Pradesh: 462066
Email:	atreyee at iiserb.ac.in	anand at iiserb.ac.in

The Laplacian and its eigenvalues play a huge role in understanding various geometric (in particular, curvature) properties of a Riemannian manifold. In this advanced instructional school we would like to give an introduction to Riemannian geometry together with an overview of some of the useful aspects of Laplacian and its eigenvalues in relation with the global analysis of the manifold.

Dates:

Monday, December 2, 2024 - 09:00 to Saturday, December 14, 2024 - 21:00

Venue:

IISER Bhopal

Venue Address:

Department Of Mathematics,
Academic Building 1 (Infinity building), IISER Bhopal,
Bhauri, bhopal Bypass Road,
Bhopal, Madhya Pradesh: 462066

Venue State:

Madhya Pradesh

Venue City:

Bhopal

PIN:

462066

Syllabus:

Syllabus:

Name of the Speaker with affiliation	No. of Lecture s	Detailed Syllabus
Outstation speakers
Dr. Soma Maity, IISER Mohali (SM)	4 (4x 1.5 hrs) =6hrs	Eigenvalues and eigenfunctions of the Laplacian in a Riemannian manifold, Sturm Liouville decomposition of the eigenspace of the Laplacian.
Dr. Harish Seshadri, IISc Bangalore (HS)	4 (4x 1.5 hrs) =6hrs	First eigenvalues of Laplacian and its relations to geodesics and curvatures, Laplacian in space forms, comparison theorems for sectional curvature bounded above and Ricci curvature bounded below.
Local speakers (From IISER Bhopal)
Dr. Sombuddha Bhattacharyya (SB)	4 (4x 1.5 hrs) =6hrs	Laplacian on domains in the Euclidean space, The spectral theorem for the Laplace operator with Dirichlet boundary condition and Neumann boundary condition on a bounded domain in \mathbb{R}^n. (will need Lax-Milgram theorem, and Rellich’s lemma)
Dr. Atreyee Bhattacharya (AB)	4 (4x 1.5 hrs) =6hrs	Introduction to Riemannian Geometry: Riemannian metric, and Riemannian connections, parallel transport, geodesics, exponential map, convex neighborhoods, geodesics as locally minimizing curves, completeness of Riemannian metrics, the Hopf-Rinow Theorem.
Dr. Anandateertha Mangasuli (AM)	4 (4x 1.5 hrs) =6hrs	Riemann Curvature Tensor, Bianchi identity, sectional, Ricci and scalar curvatures, properties of curvature, Lie and covariant derivatives of tensors, gradient and Hessian of smooth functions, divergence of vector fields, rough Laplacian on tensors of a Riemannian manifold.
Dr. Manas Kar, IISER Bhopal (MK)	4 (4x 1.5 hrs) =6hrs	First eigenvalues of Laplacian : the co-area formula, the Faber Krahn inequality, the Cheeger, Sobolev and isoperimetric constants, the Sobolev constants and eigenvalue, eigenfunction estimates.

References:

Evans, Partial Differential equations, AMS, (2010)
S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer (2004).
I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press,Inc. (1984)

Tutorial Assistants:

S. No.	Name	Affiliation
1	Ms. Sayoojya Prakash (SP)	PhD student, IISER Bhopal
2	Dr. Aditya Tiwari (AT)	Postdoctoral fellow, IISER Mohali
3	Ms. Jahnabi Chakraborty (JC)	PhD student, IISER Bhopal
4.	Mr. Tuhin Mondal (TM)	PhD student, IISER Bhopal
5.	Mr. Pranav Kumar (PK)	PhD student, IISER Bhopal

Time Table:

Day	Date	Lecture 1 (9.30–11. 00)	Tea (11.05 –11.25)	Lecture 2 (11.30–1.0 0)	Lunch (1.05–2. 25)	Tutorial (2.30–3.3 0)	Tea (3.35-3. 55)	Tutorial (4.00-5.0 0)	Snacks 5.05-5. 30
		(name of the speaker)		(name of the speaker)		(name of the speaker + tutors)		(name of the speaker + tutors)
		(name of the speaker)		(name of the speaker)		(name of the speaker + tutors)		(name of the speaker + tutors)
Mon	02/12/2024	SB		AB		SB, TM		AB, SP
Tues	03/12/2024	SB		AB		SB, TM		AB, SP
Wed	04/12/2024	SB		AM		SB, TM		AM, JC
Thu	05/12/2024	AB		AM		AB, SP		AM, JC
Fri	06/12/2023	AB		AM		AB, SP		AM, JC
Sat	07/12/2024	SB		AM		SB, TM		AM, JC
				SUNDAY	:OFF
Mon	09/12/2024	MK		HS		MK,PK		HS, JC,SP
Tues	10/12/2024	HS		HS		SM,AT		HS, JC,SP
Wed	11/12/2024	SM		HS		SM,AT		HS, JC, SP
Thu	12/12/2024	SM		SM		SM,AT		SM,AT,
Fri	13/12/2024	MK		SM		SM,AT		MK,PK
Sat	14/12/2024	MK		MK		MK,PK		MK,PK

Selected Applicants:

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. Degree Date
1	58962	Mr. Aniruddha Vishwas Deshmukh	Male	Indian Institute of Technology Indore	PhD	Sardar Vallabhbhai National Institute of Technology Surat	2020
2	59021	Mr. Sayan Shit	Male	Indian Statistical Institute	PhD	Indian Statistical Institute	2024
3	59246	Mr. Sourasish Karmakar	Male	IISER Kolkata	PhD	IISER Kolkata	2024
4	59300	Mr. Arkadeepta Roy	Male	Harish-Chandra Research Institute	Senior Research Fellow	Ramakrishna Mission Vidyamandira	2019
5	59322	Mr Sagar Basak	Male	INDIAN INSTITUE OF TECHNOLOGY	PhD	Savitribai Phule Pune University	2021
6	59348	Mr. Abir Rangan Datta	Male	IISER PUNE	PhD	Indian Statistical Institute ,Kolkata	2024
7	59350	Mr. Shubham Yadav	Male	Harish-Chandra Research Institute	PhD	Sardar Vallabhbhai National Institute of Technology, Surat	2020
8	59417	Mr. Subash Chandra Behera	Male	IIT Goa	PhD scholar	Pondicherry University	2019
9	59489	Mr. Niteesh Kumar	Male	IISER Tirupati	Integrated PhD	IISER Tirupati	2020	10/09/2024
10	59555	Mr. Manan Jain	Male	Indian Institute of Science Education and Research-Mohali	MSc Student	IISER Mohali, M.Sc.-PhD
11	59584	Mr. Abhijit Manna	Male	Indian Institute of Technology (IIT) Gandhinagar	PhD Student	Indian Institute of Technology (IIT) Gandhinagar	2024
12	59611	Ms. Jyoti Yadav	Female	Central University of Punjab	Research Scholar	University of Rajasthan	2018
13	59710	Ms Ashmita Singh	Female	IIT BHU (VARANASI).	Ph.D	Banaras Hindu University	2020
14	59749	Ms. Jaya Rani	Female	Banaras Hindu University	PhD	MJPRU, Bareilly, UP	2019
15	60374	Ms. Adeeba Zaidi	Female	Central University of Punjab, Bathinda	PhD Scholar	Hemvati Nandan Bahuguna Garhwal University, Srinagar, Uttarakhand	2017
16	60533	Ms. Harmandeep Kaur	Female	Central University of Punjab	PhD (Research Scholar)	Panjab University	2019
17	60568	Mr Bhola Nath Saha	Male	Indian Institute of Technology Kanpur	PhD	University of North Bengal	2017
18	60582	Mr Imran Hussain	Male	IIT Bombay	MSc&PhD Dual Degree	IIT Bombay	2024
19	60683	Mr. Vikram Yallapa Naik	Male	BITS Pilani	PhD Student	University of Mumbai M.Sc	2019
20	60963	Ms Nidhi Yadav	Female	Dr. Harisingh Gour Vishwavidyalaya	PhD	Dr. Harisingh Gour Vishwavidyalaya Sagar, Madhya Pradesh	2020

How to Reach:

TBA

School Short Name:

tlrg

Last Date Application:

Sunday, May 14, 2023

School Type:

AIS