This volume consists of articles based on lectures delivered at an International Colloquium held at the Tata Institute of Fundamental Research, Bombay. Experts from all over the world spoke at the Conference on different aspects of geometry and analysis. Topics include: impact of geometry on the boundary solutions of a semilinear Neumann problem with critical nonlinearity; algebraic representations of reductive groups over local fields; scalar conservation laws with boundary condition; and the Borel-Weil theorem and the Feynman path integral. 1. Fundamental group of the Affine Line in Positive Characteristic, S.S. Abhyankar 2. Impact of geometry on the boundary on the positive solutions of a semilinear Neumann problem with Critical nonlinearity, Adimurthi 3. Sur a cohomologie de certains espaces de modules de fibres fectoriels, A. Beauville 4. Some quantum analogues of solvable Lie groups, C. De Concini et al 5. Compact complex manifolds whose tangent bundles satisfy numerical effectivity properties, J.-P. Demailly 6. Algebraic Representations of Reductive Groups over Local Fields, W.J. Haboush 7. Poncelet Polygons and the Painleve Equations, N.J. Hitchin 8. Scalar conservation laws with boundary condition, K.T. Joseph 9. Bases for Quantum Demazure modules-I, V. Lakshmibai 10. An Appendix to Basesfor Quantum Demazurew modules-I 11. Moduli Spaces of Abelian Surfaces with Isogeny, Ch. Birkenhake and H. Lange 12. Instantons and Parabolic Sheaves, M. Maruyama 13. Numerically effective line bundles which are not ample, V.B. Mehta and S. Subramanian 14. Moduli of logarithmic connections, N. Nitsure 15. The Borel-Weil theorem and the Feyman path integral, K. Okamoto 16. Geometric super-rigidity, Y.-T. Siu