AIS in Differential Geometry and Lie groups (2006)
|Venue:||BP and Pune U|
|Dates:||4-30 Dec, 2006
|Convener(s)||Speakers, Syllabus and Time table||Applicants/Participants|
|Name||R. S. Kulkarni and S. Kumaresan
AIS-DGLG 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
- 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 present-day 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 n-space, 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.
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)
email: punekulk at yahoo.com
2) Frenet-Serret 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) 2-dimensional 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 1-dimensional complex Euclidean, elliptic, or hyperbolic geometries. The disk- and upper half plane models of hyperbolic geometry. 2-dimensional 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 e-mail:harish at math.iisc.ernet.in
5) -dim. Riemannian Geometry
Riemannian metrics (examples), Levi-Civita connection,distance, geodesics, first-variation formula, Hopf-Rinow, curvature, second-variation formula, Jacobi fields, Rauch comparison theorem, Bonnet-Myers theorem, Cartan-Hadamard theorem.
Ricci Flows (16 lectures)
Examples, Heat-type equations and maximum principles, Existence and uniqueness, Ricci flow on surfaces.
email: kumaresa at math.mu.ac.in
6) Compact Lie Groups (8 lectures) ``Hands-on" introduction to and their representations, and the statements of general theorems on compact Lie groups without proof.
7) Noncompact Lie Groups (2 lectures)
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 Schoen-Yau 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 Eells-Sampson.
1. Glen E. Bredon, Topology and Geometry, Springer GTM 139, Indian reprint-2006.
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.
- 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, Roberton-Walker spacetimes, de Sitter metric.
3. Homogeneous but anisotropic spacetimes: some examples of models with spin, Godel's metric,the Heckmann-Schucking metric.
4. Evidence for symmetries in the actual universe.
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 deck-transformation 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 non-negative 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.|
1.Vikram Aithal email: vikram at mri.ernet.in
2. Sanjay Pant email: sanjpant at yahoo.co.in
3. Srikanth K. V. e-mail: kvsrikantha at gmail.com (15-28 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|
|UM-Lecture 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, R-5-> in BP main building, flat-> near by flat|
|1.||Gautam Borisagar, IIT Bombay, Mumbai||Flat|
|2.||Mr. Sanjit Das, IIT Kharagpur||H-4|
|3.||Mr. Priyabrat Gochhayat, Berhampur Uni., Bhajabihar||R-5|
|4.||Mr. Sanjeev Kumar, G. B. Pant Univ., Uttaranchal||R-5|
|5.||Mr. Jaydeep Sengupta, Univ. of North Bengal, Darjeeling||H-4|
|6.||Mr. Ajay Singh Thakur, IMSc, Chennai||H-7|
|7.||Mr. Umesh V. Dubey, IMSc, Chennai||H-7|
|8.||Mr. Tarakanta Nayak, IIT-Guwahati||K-2|
|9.||Mr. Sushil Gorai, IISc, Bangalore||H-4|
|10.||Ms. Prachi Mittal, IISc, Bangalore||H-1|
|11.||Dr. T. Venkatesh, Karnatak Univ.||H-7|
|12.||Mr. Pavinder Singh, Univ. of Jammu, Jammu||H-4|
|13.||Mr. Nandakumar M. Univ. of Calicut, Kerala||R-5|
|14.||Mr. Jagmohan Tanti, BP, Pune||K-2 (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||H-5|
|18.||Mr. Diganta Borah, I.I.Sc. Bangalore||R-5|
|19.||Ms. Shilpa Gondhali, TIFR||H-1|
|20.||Mr. Chandrasheel Bhagwat, TIFR, Mumbai||Self Accom.|
|21.||Mr. Hemant Pawar, NDA, Pune||Local|
|22.||Mr. Vadiraja Bhatta, NITK, Surathkal||R-5|
|23.||Mr. Debasis Sen, ISI, Kolkata||H-5|
|24.||Mr. Dheeraj Kulkarni, HRI||Self Accom.|
|25.||Mr. Rabiul Islam, Univ. of Calicath, Kolkata||H-5|
|26.||Mr. Abhijit Pal, ISI, Kolkata||H-5|
|27.||Mr. Prem Pandey, IMSc, Chennai||H-7|
|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||H-1|
|33.||Mr. Rajeev B., Alappuzh, Kerala||Flat|
|34.||Mr. Subrata Bhowmik, Univ. of Tripura||H-4|
|35.||Mr. Nanasaheb Phatangare, Ahmednagar, Maharashtra||Flat|
|36.||Vikas Jadhav, Wadia College, Pune||Local|
|37.||Manjusha Joshi, BP, Pune||Local|