**[Cr:4, Lc:3, Tt:1, Lb:0]**

- Presheaf, Sheaf, étale space, Differentiable Manifolds — sheaf theoritic approach.
- First order Differential operators, locally free sheaves and Vector Bundles, theorem of Frobenius.
- Differential operators of higher order.
- Integration on Manifold and adjoints of Differential operators.
- Local analysis of Elliptic operators : Schwartz space and Densities, Fourier transforms, Distributions, Sobolev’s theorem, Interior regularity of Elliptic solutions, Rellich’s theorem.
- Elliptic operators on Differentiable Manifolds, Regularity theorem, finiteness theorem, Elliptic Complexes and Laplacian.

- Pseudo-Differential operators on Manifolds.
- Fredholm operators and the Index of a Fredholm operator.

- S. Ramanan, Global Calculus, American Mathematical Society, Providence, RI (2005).
- Raghavan Narasimhan, Analysis on Real and Complex Manifolds, North-Holland Publishing
Co., Amsterdam (1968).

- R. O. Wells, Differential analysis on complex manifolds, Second edition, GTM 65, Springer-Verlag (1980).
- P. B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Second edition, CRC Press (1995).