AIS in Differential Geometry and Lie groups (2006)
Venue:  BP and Pune U 
Dates:  430 Dec, 2006 
Convener(s)  Speakers, Syllabus and Time table  Applicants/Participants 
Name  R. S. Kulkarni and S. Kumaresan 
AISDGLG being organised in Pune in December 2006 is the Advanced Instructional School on Differential Geometry and Lie Groups being organised on behalf of NBHM.
National Coordinating Committee
 Director: Ravi S. Kulkarni
 Secretary: J. K. Verma
 Members : S. D. Adhikari, Satya Deo, Shobha Madan, I. B. S. Passi, Ravi A. Rao, S. A. Katre
Local Coordinator
 S. A. Katre, University of Pune
Members of the Local Organising Committee
 Bhaskaracharya Pratishthana: C. S. Inamdar (Custodian), R. V. Gurjar (Res. Director, Hon.), R. R. Simha
 University of Pune: B .N. Waphare (HOD, Maths.), S. A. Katre
Speakers and Syllabus

Differential Geometry, in its presentday avatar, is known as the study of connections on fiber bundles. These ``connections" are implicit in any branch of mathematics/applied mathematics/theoretical physics dealing with continuum models, and their discrete analogues are developed even in algebra and arithmetic. In building these ``connections", the Lie groups play a foundational role.
One of the aims of this AIS is to communicate this viewpoint starting from the very concrete study of curves and surfaces in the Euclidean nspace, an introduction to euclidean, spherical, and hyperbolic geometries, and generalities on differentiable manifolds and fiber bundles. The special topics include: A special feature of the ATM programme is the inclusion of ``Unity of Mathematics" Lectures by some distinguished scientists. In this AIS, Prof. JAYANT NARLIKAR, IUCAA, Pune, will deliver three lectures on ``Cosmological Models and Spacetime symmetries", and Prof. M. S. RAGHUNATHAN, TIFR, Mumbai, will deliver 2 lectures. 

A. Mangasuli
email: anandateertha at gmail.com
1) A) Multivariate Calculus, Inverse Function Theorem, Existence and Uniqueness theorem on ODE (3 lectures)
B) Differentiable Manifolds, Tangent/cotangent bundles, Associated bundles, Differential forms, Lie derivative, Stokes's theorem, Frobenius's theorem (5 lectures)

Ravi Kulkarni
email: punekulk at yahoo.com
2) FrenetSerret theory, Euler's theory on normal curvatures, Gaussian curvature and theorema egregium (3 lectures)

G. Santhanam and Ravi Kulkarni
email: santhana at iitk.ac.in
3) 2dimensional Euclidean, Spherical, and Hyperbolic geometries (4 lectures).
The groups of isometries, curves of constant geodesic curvature, the dynamical types of isometries. The alternate representations of isometry groups using complex numbers, when the above geometries are considered as 1dimensional complex Euclidean, elliptic, or hyperbolic geometries. The disk and upper half plane models of hyperbolic geometry. 2dimensional affine geometry.
4) dimensional Euclidean, Spherical, and Hyperbolic geometries, and their special cases when .
Description of dynamical types of isometries.
Hyperboloid model of hyperbolic geometry.
Description of the groups of isometries for using complex numbers and quaternions.
Introduction to some special Lie groups. (3 lectures)

Akhil Ranjan and Harish Seshadri
email: aranjan at math.iitb.ac.in email:harish at math.iisc.ernet.in
5) dim. Riemannian Geometry
Riemannian metrics (examples), LeviCivita connection,distance, geodesics, firstvariation formula, HopfRinow, curvature, secondvariation formula, Jacobi fields, Rauch comparison theorem, BonnetMyers theorem, CartanHadamard theorem.
Ricci Flows (16 lectures)
Examples, Heattype equations and maximum principles, Existence and uniqueness, Ricci flow on surfaces.

S. Kumaresan
email: kumaresa at math.mu.ac.in
6) Compact Lie Groups (8 lectures) ``Handson" introduction to and their representations, and the statements of general theorems on compact Lie groups without proof.

Ravi Kulkarni
7) Noncompact Lie Groups (2 lectures)

Kaushal Verma
email: kverma at math.iisc.ernet.in
8) Harmonic maps (8 lectures)
(A) Quick review of Sobolev spaces, in particular , followed by the solution to the Dirichlet Problem on domains in . Regularity theory for Laplacian.
(B) Harmonic maps between Riemann surfaces: topics to be covered include
(i) definition and examples
(ii) theorem on the existence of harmonic maps
(iii) second variation of a harmonic map and some applications (e.g. the finiteness of the conformal automorphism group of a compact hyperbolic surface)
(iv) The theorem of SchoenYau which shows that harmonic maps are actually diffeomorphisms when the surfaces under consideration satisfy relevant topological properties.
(v) Some parts of M. Wolf's paper (J. Diff. Geom. no. 29 , 1989) titled `The Teichmüller theory of harmonic maps'.
(C) Definition of a harmonic map in higher dimensions. Examples and statement of the main existence theorem due to EellsSampson.
References:
1. Glen E. Bredon, Topology and Geometry, Springer GTM 139, Indian reprint2006.
2. Do Carmo, Riemannian Geometry, Boston, Birkhauser, 1993.
3. J. Hubbard, Teichmüller Theory, Vol. 1, Matrix Edition, 2006.
4. Jurgen Jost, Compact Riemann Surfaces, J. Jost, Springer, Universitext, 1997.
5. S. Kumaresan, A Course in Differential Geometry & Lie Groups, Hindustan Book Agency, 2002.
6. S. Kumaresan, A Course in Riemannian Geometry  Lectures notes (To be published).
7. S. Kumaresan and G. Santhanam, An Expedition to Geometry, New Delhi, Hindustan Book Agency, 2005
8. J. Milnor, Morse Theory, Princeton University Press, 1963.
9. Peter Petersen, Riemannian Geometry, Springer GTM 171.
UM Lectures
 J. V. NARLIKAR : Cosmological Models with Spacetime Symmetries
 M. S. RAGHUNATHAN : Arithmetic and Differential Geometry
1. Introduction: the expectations of a mathematical model in cosmology. 2. Spacetime symmetries: Groups of motions, Killing equations, concepts of homogeneity and isotropy, RobertonWalker spacetimes, de Sitter metric. 3. Homogeneous but anisotropic spacetimes: some examples of models with spin, Godel's metric,the HeckmannSchucking metric. 4. Evidence for symmetries in the actual universe. 
References:
1. Riemannian Geometry by Eisenhart (Princeton) 
The uniformisation theory of Riemann surfaces tells us the following. Let S (resp. C, resp. H) denote the sphere (resp. the complex plane, resp.the upper half plane) equipped with the Riemannian metric of constant sectional curvature 1 (resp. 0, resp. 1). Then the universal cover of any connected compact surface can be identified with one and only one of S, C or H in such a way that the decktransformation group acts as a group of isometries of that constant curvature space. One cannot expect such a strong result in higher dimensions. A simply connected manifold of higher dimension does not in general carry a homogeneous Riemannian metric. In fact it is a nontrivial fact that there do exist compact manifolds in higher dimension that carry metrics with the universal covering being a homogeneous Riemannian manifold (of nonnegative sectional curvature). The only methods of construction of such manifolds that will cover all dimensions involve arithmetical considerations. A famous result due to G. A. Margulis asserts in fact that in most cases the construction necessarily comes from arithmetic. In these lecture I will discuss some of the ideas involved in these matters. 
Associate Teachers
1.Vikram Aithal email: vikram at mri.ernet.in
2. Sanjay Pant email: sanjpant at yahoo.co.in
3. Srikanth K. V. email: kvsrikantha at gmail.com (1528 Dec)
1 st Dec., Fri  2 nd, Sat  4 th, Mon  5 th, Tue  6 th, Wed  7 th, Thur  8 th, Fri  9 th, Sat  
9.30  11.00  AM  AM  AM  AM  AM  AM  AM  AM 
11.00  11.30  Tea  Tea  
11.30  1.00  RK  RK  RK  RK  RK  GS  GS  GS 
1.00  2.15  Lunch  Lunch  
2.15  3.15  Tutorials  Tutorials  
3.15  3.45  Library / Tea  Library / Tea  
3.45  5.15  Tutorials  Tutorials 
11 th Dec., Mon  12 th,Tue  13 th, Wed  14 th,Thurs  15 th, Fri  16 th, Sat  
9.30  11.00  GS  GS  GS  GS  GS  RK 
11.00  11.30  Tea  
11.30  1.00  JVN  JVN  JVN  SK  SK  SK 
1.00  2.15  Lunch  
2.15  3.15  Tutorials    
3.15  3.45  Library / Tea    
3.45  5.15  Tutorials  SK  Seminar/Tutorial   
18 th Dec., Mon  19 th,Tue  20 th, Wed  21 th, Thurs  22 th, Fri  23 th, Sat  
9.30  11.00  HS  HS  HS  HS  HS  HS 
11.00  11.30  Tea  
11.30  1.00  SK  SK  SK  SK  RK  RK 
1.00  2.15  Lunch  
2.15  3.15  Tutorials  
3.15  3.45  Library / Tea  
3.45  5.15  RK  RK  KV  KV  KV  KV 
25 th, Dec. Mon  th, Tue  27 th, Wed  28 th, Thur  
9.30  11.00  HS  HS  HS  HS 
11.00  11.30  Tea  
11.30  1.00  KV  KV  KV  MSR 
1.00  2.15  Lunch  
2.15  3.15  Tutorials  FeedBack Session  
3.15  3.45  Library / Tea  Valedictory Function  
3.45  5.15  KV  Seminar/Tutorial  MSR   
AM Anand Mangasuli  RK Ravi Kulkarni,  KV Kaushal Verma 
GS G. Santhanam  HS Harish Seshadri  SK S. Kumaresan 
UMLecture Series :  JVN Jayant Narlikar  MSR M. S. Raghunathan 
Vikram Aithal,  Sanjay Pant,  Srikanth K. V. 

Sr. No.  Name of Participant  Accommodation H,K> in Hostel of BP, R5> in BP main building, flat> near by flat 
1.  Gautam Borisagar, IIT Bombay, Mumbai  Flat 
2.  Mr. Sanjit Das, IIT Kharagpur  H4 
3.  Mr. Priyabrat Gochhayat, Berhampur Uni., Bhajabihar  R5 
4.  Mr. Sanjeev Kumar, G. B. Pant Univ., Uttaranchal  R5 
5.  Mr. Jaydeep Sengupta, Univ. of North Bengal, Darjeeling  H4 
6.  Mr. Ajay Singh Thakur, IMSc, Chennai  H7 
7.  Mr. Umesh V. Dubey, IMSc, Chennai  H7 
8.  Mr. Tarakanta Nayak, IITGuwahati  K2 
9.  Mr. Sushil Gorai, IISc, Bangalore  H4 
10.  Ms. Prachi Mittal, IISc, Bangalore  H1 
11.  Dr. T. Venkatesh, Karnatak Univ.  H7 
12.  Mr. Pavinder Singh, Univ. of Jammu, Jammu  H4 
13.  Mr. Nandakumar M. Univ. of Calicut, Kerala  R5 
14.  Mr. Jagmohan Tanti, BP, Pune  K2 (Local) 
15.  Dr. Bhimashankar Waghe, Kashibai Nawale Engg. College, Pune  Local 
16.  Mr. Devendra Shirolkar, Dept. of Maths, Univ. Pune, Pune  Local 
17.  Mr. Soumen Sarkar, ISI, Kolkata  H5 
18.  Mr. Diganta Borah, I.I.Sc. Bangalore  R5 
19.  Ms. Shilpa Gondhali, TIFR  H1 
20.  Mr. Chandrasheel Bhagwat, TIFR, Mumbai  Self Accom. 
21.  Mr. Hemant Pawar, NDA, Pune  Local 
22.  Mr. Vadiraja Bhatta, NITK, Surathkal  R5 
23.  Mr. Debasis Sen, ISI, Kolkata  H5 
24.  Mr. Dheeraj Kulkarni, HRI  Self Accom. 
25.  Mr. Rabiul Islam, Univ. of Calicath, Kolkata  H5 
26.  Mr. Abhijit Pal, ISI, Kolkata  H5 
27.  Mr. Prem Pandey, IMSc, Chennai  H7 
28.  Mr. Niraj Prasad, Mumbai Univ.  Flat 
29.  Mr. Angom Tiken Singh, Nehu, Shillong, Meghalaya  Flat 
30.  Mr. Patel Jayminkumar M., Sardar Patel Univ., Gujrat  Flat 
31.  Mr. Manoj Kumar Pandey, A.P.S. Univ., Rewa  Flat 
32.  Ms. Tummala Vinutha, Andhra Univ., A.P  H1 
33.  Mr. Rajeev B., Alappuzh, Kerala  Flat 
34.  Mr. Subrata Bhowmik, Univ. of Tripura  H4 
35.  Mr. Nanasaheb Phatangare, Ahmednagar, Maharashtra  Flat 
36.  Vikas Jadhav, Wadia College, Pune  Local 
37.  Manjusha Joshi, BP, Pune  Local 