The idea of a concept like (differentiable ) manifold germinated from the discovery of Non-Euclidean geometry (like hyperbolic geometry and elliptic geometry), which considers spaces where Euclid's parallel postulate fails, studied first by Saccheri (1733) followed by Lobachevsky, Bolyai, uncovered to the world of mathematics that there could be something more and distinct from the classical Euclidean space.
The theorema egregium of Gauss was the first instance of a (geometric) invariant (the Gaussian curvature) of an abstract spaces (surface, later manifold) is an intrinsic invariant not depending on its situation in any ambient space. Euler characteristic is another such (algebraic topological) invariant. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature.
Henri Poincaré's 1895 paper Analysis Situs studied manifolds(which he called "varieties"), in the context of homology (Betti numbers), and qualitative theory of differential equations, or dynamical systems. Lagrangian mechanics and Hamiltonian mechanics can only be studied satisfactorily in the setting of manifolds.
Hermann Weyl gave an intrinsic definition for differentiable manifolds in 1912. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century
became precise, and developed through differential geometry and Lie group theory. The Whitney embedding theorem, that manifolds intrinsically defined by charts could always be embedded in Euclidean space, as in the extrinsic definition,
showed that the two concepts of manifold were equivalent.
Ideas from differential calculus carry over to differential manifolds. There are, however, important differences in the calculus of vector fields. In brief, the The directional derivative of a vector field (and tensor fields in general) is not welldefined, or at least not defined in a straightforward manner. The Lie derivative is an alternative, which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation. An affine connection, which is not uniquely defined, generalizes in a more complete manner the features of ordinary directional differentiation.
Ideas from integral calculus also carry over to differential manifolds. These are naturally expressed in the language of differential forms. The fundamental theorems of integral calculus in several variables — namely Green's theorem, the divergence theorem, and Stokes' theorem — generalize to a theorem (also called Stokes' theorem) relating the exterior derivative and integration over submanifolds.
Differential geometry does also deals with global properties but its invariants mostly originate locally, like curvature at a point, to name one. In this branch one has to equip the differentiable manifolds with extra structures like a Riemannian
metric. On a Riemannian manifold one has notions of length, volume, and angle. Any differentiable manifold can be given a Riemannian metric.
A symplectic manifold is a manifold equipped with a closed, nondegenerate 2-form. Cotangent bundles, which arise as phase spaces in Hamiltonian mechanics, provided the motivating example of symplectic manifolds. The subject is growing
on its own right ever after.
A Lie group is C^{∞ }of manifolds that also carries a group structure whose product and inversion operations are smooth as maps of manifolds. These objects arise naturally in describing symmetries.
The purpose of the AIS would be to expose some of these topics in a condensed and yet lucid fashion, help students in digesting the topics through tutorials, to direct the development towards some interesting outstanding problems in the subject and also towards applications which are under the process of intensive current development.
Syllabus (Some addition/subtraction may be made in the topics on the advice of resource persons)
Lectures | Detailed Syllabus | |
Topic 1 (ATS/PASS/KVS) |
6 (90 minute) | Quick review of : Differntiable manifolds, tangent bundles, immersion, submersion,embeddings, normal bundles, Whitney’s embedding theorem, Thom’s transversality theorem. |
Topic 2 (ATS/KVS/PASS) |
6 (90 minute) | Vector fields and flows, Differential forms, exterior and interior products, Lie derivative, Lie bracket, Poisson brackets. Affine connections, Riemannian connections, parallel transport, Geodesics, the exponential map, minimizing properties of geodesics, the Riemannian distance. The curvature tensor, Jacobi fields, The geodesic flow. |
Topic 3 (ARS) | 6 (90 minute) | Elements of Morse Theory: Nondegenerate critical points, The gradient flow, The topology of level sets, Manifolds represented as CW complexes, Morse inequalities, Morse homology. |
Topic 4 (MD) | 6 (90 minute) | Smooth dynamical systems, Discrete dynamical systems, hyperbolic fixed points and periodic orbits. Hyperbolic Systems: Hyperbolic sets, Hyperbolicity criteria, Geodesic flows. Hamiltonian mechanics, The symplectic topology of Euclidean space, Linear symplectic geometry, Symplectic vector spaces, The symplectic linear group, Lagrangian subspaces, The affine nonsqueezing theorem, Complex structures, Symplectic vector bundles, Symplectic manifolds, Basic concepts, Isotopies and Darboux's theorem, Submanifolds of symplectic manifolds, Contact structures |
Topic 5 (HKM/SB) | 6 (90 minute) | Area-preserving diffeomorphisms, Periodic orbits, The Poincare-Birkhoff theorem, Generating functions, Hamiltonian symplectomorphisms , Arnold’s conjecture, Conley index theory, Morse-Witten complexes, Conley-Zehnder’s proof of Arnold’s conjecture on torus, Floer homology, and proof of Arnold’s conjecture in general. |
Topic 6 (AR) | 6 (60 minute) | Symplectic capacities , Nonsqueezing and capacities, Rigidity, The Hofer metric, The Hofer-Zehnder capacity, A variational argument, Weienstein’s conjecture that A Reeb vector field on a closed manifold M 2n−1 admits a periodic orbit. Open problems. |
- AR - Akhil Ranjan, IIT Bombay,Mumbai
- ARS - Anant R. Shastri, IIT Bombay, Mumbai
- ATS - Angom Tiken Singh, NEHU, Shillong
- GM - Goutam Mukherjee, ISI, Kolkata
- HKM - Himadri Kumar Mukerjee, NEHU, Shillong
- MD - Mahua Dutta, ISI, Kolkata
- SB - Somik Basu, Ramkrishna Mission Vivekananda University, Kolkata
- PASS - P.A.S. Sri Krishna, IIT Guwahati
- KVS - K.V. Srikanth, IIT Guwahati
Tea | Lecture 2 | Lunch | Tutorials | Tea & Snacks |
Special Lectures |
|||
Day | Date | 9.30-11.00 | 11.30 -13.00 | 14.30 - 16.30 | ||||
Monday | 15-06-2015 | Topic 1 | Topic 2 | Tutorial on 1 & 2 | ||||
Tuesday | 16-06-2015 | Topic 1 | Topic 2 | Tutorial on 1 & 2 | ||||
Wednesday | 17-06-2015 | Topic 1 | Topic 2 | Tutorial on 1 & 2 | ||||
Thursday | 18-06-2015 | Topic 1 | Topic 2 | Tutorial on 1 & 2 | ||||
Friday | 19-06-2015 | Topic 1 | Topic 2 | Tutorial on 1 & 2 | Special lecture | |||
Saturday | 20-06-2015 | Topic 1 | Topic 2 | Tutorial on 1 & 2 | ||||
Sunday | 22-06-2015 | Break | Break | Break | ||||
Monday | 23-06-2015 | Topic 3 | Topic 4 | Tutorial on 3 & 4 | ||||
Tuesday | 24-06-2015 | Topic 3 | Topic 4 | Tutorial on 3 & 4 | ||||
Wednesday | 25-06-2015 | Topic 3 | Topic 4 | Tutorial on 3 & 4 | ||||
Thursday | 26-06-2015 | Topic 3 | Topic 4 | Tutorial on 3 & 4 | ||||
Friday | 27-06-2015 | Topic 3 | Topic 4 | Tutorial on 3 & 4 | Special lecture | |||
Saturday | 28-06-2015 | Topic 3 | Topic 4 | Tutorial on 3 & 4 | ||||
Sunday | 29-06-2015 | Break | Break | Break | ||||
Monday | 30-06-2015 | Topic 5 | Topic 6 | Tutorial on 5 & 6 | ||||
Tuesday | 01-07-2015 | Topic 5 | Topic 6 | Tutorial on 5 & 6 | ||||
Wednesday | 02-07-2015 | Topic 5 | Topic 6 | Tutorial on 5 & 6 | ||||
Thursday | 03-07-2015 | Topic 5 | Topic 6 | Tutorial on 5 & 6 | ||||
Friday | 04-07-2015 | Topic 5 | Topic 6 | Tutorial on 5 & 6 | Special lecture | |||
Saturday | 05-07-2015 | Topic 5 | Topic 6 | Tutorial on 5 & 6 |
Specific References:
1. McDuff, D., Salamon, D., Introduction to Symplectic Topology, 2nd Ed., Clarendon Press Oxford, 1998.
2. ZEHNDER, E., Lectures on Dynamical systems, Hamiltonian vector fields and symplectic capacities, European Mathematical Society, 2010.
3. Hofer, H. and Zehnder, E, Symplectic invariants and Hamiltonian dynamics, Birkhauser Verlag, 1994.
4. Moser, J. and Zehnder, E., Lectures on dynamical systems, Courant Inst. Lec Notes no. 12, Amer. Math. Soc., 2005.
5. Biran, P., Cornea O., and Lalonde, F., Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, Springer, 2006.(especially the papers (i) ―Homotopical dynamics in symplectic topology― by
BARRAUD, J.F. and Cornea O. , (ii) The LS-Index: A Survey, by M. Izydorek, (iii) Symplectic topology and Hamilton-Jacobi equations, by C. Viterbo)
6. Hofer, H. Dynamics, Topology and holomorphic curves, Documenta Mathematica, ICM 1998, DMV
7. Morse theory, the Conley index and Floer homology, by D. Salamon, Bull. L.M.S., 22 (1990).
8. Gang Liu & Gang Tian, Floer homology and Arnold conjecture, J. differential geometry, 49 (1998) 1-74.
General References (some titles are common):
9. Audin, M. and Lafontaine, (Editors), J-Holomorphic curves in symplectic Geometry, Springer 1994
10. Biran, P, Cornea, O. and Lalonde, F. (Editors), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, Spinger 2004
11. BREDON G., Topology and Geometry, Springer 1993. This book introduces many of the tools needed for manifold surgery.
12. Burns, K. and Gidea, M., Differential Geometry and Topology With a View to Dynamical Systems, Chapman and Hall/CRC
13. GAULD David B., Differential Topology: An Introduction, Dover Publications, 2006.
14. GUILLEMIN Victor and POLLACK Alan, Differential Topology, AMS, 2010.
15. HIRSCH M.W., Differential Topology, Springer-Verlag, 1997.
16. Hofer, H. and Zehnder, E, Symplectic invariants and Hamiltonian dynamics, Birkhauser Verlag, 1994.
17. HUSEMULLER Dale, Fibre bundles, GTM-20, Springer, 1994.
18. Jost Jürgen, Riemannian Geometry and Geometric Analysis, Springer, 2011
19. KOSINSKI, A.A., Differential manifolds, Academic Press, 1993 and Dover publications, 2007.
20. Laudenbach, F., Symplectic Geometry and Floer homology, SOCIEDADE BRASILEIRA DE MATEMATICA
21. Lecture notes - Introduction to symplectic topology, unknown author
22. McDuff, D., Salamon, D., J-holomorphic curves and quantum cohomology, Univ. Lec. Ser. No. 6, American Mathematical Society, 1994.
23. McDuff, D., Salamon, D., J-holomorphic curves and symplectic topology, American Mathematical Society, colloq. Publ. Vol. 52, 2004
24. MILNOR J., ``Differential Topology'', Volume 3 of his collected papers, Amer. Math. Soc. 2007. Surgery theory was initiated by Milnor in the period 1956-1960 for the purpose of classifying smooth manifolds that are homotopy equivalent to a sphere of dimension \geq 5. In addition to published research articles, this volume contains four sets of notes for lectures he gave in this period; these lectures had remained in mimeographed form until their formal publication in this volume. They still provide excellent instruction in differential topology.
25. MILNOR J., Morse theory, (based on lecture notes by M. Spivak and R. Wells) Annals of Mathematical Studies No. 51, Princeton University Press, 1963.
26. MILNOR J., Topology from the Differentiable Viewpoint, Univ. Pross of Virginia at Charlotesville, 1965.
27. MILNOR J. and STASHEFF J., Characteristic classes, Annals of Mathematics Studies No. 76, Princeton University Press, 1974. This monograph is based on more succinct mimeographed notes by Milnor and Stasheff
for lectures of Milnor given in 1959 and still available at <http://www.maths.ed.ac.uk/~aar/papers/milnorcc.pdf>.
28. Moser, J. and Zehnder, E., Lectures on dynamical systems, Courant Inst. Lec Notes no. 12, Amer. Math. Soc., 2005.
29. Mukherjee, Amiya, Topics in Differential Topology, Hindistan Book Agency, 2005.
30. Shastri, A.R., Elements of Differential Topology, CRC Press, 2011.
31. STEENROD Norman J. , Topology of Fibre bundles, Princeton, 1951; Landmarks in Mathematics and Physics; Princeton, 1999.
32. WALLACE A.H., Differential Topology: First Steps, Dover Books on Mathematics, 2006.
33. ZEHNDER, E., Lectures on Dynamical systems, Hamiltonian vector fields and symplectic capacities, European Mathematical Society, 2010.
There are many more new books and resources in the internet.