The texts we will be referring to are as follows :
1. Do Carmo, Riemannian Geometry.
2. Griffiths and Harris, Principles of Algebraic Geometry.
3. S. Donaldson, Lecture Notes for TCC Course "Geometric Analysis" .
4. J. Kazdan, Applications of Partial Differential Equations To Problems in Geometry.
5. L. Nicolaescu, Lectures on the Geometry of Manifolds .
6. T. Aubin, Some nonlinear problems in geometry.
7. C. Evans, Partial differential equations.
8. Gilbarg and Trudinger, Elliptic partial differential equations of the second order.
9. G. Szekelyhidi, Extremal Kahler metrics.
10. R.O. Wells, Differential Analysis on Complex Manifolds.
11. Kodaira, Complex Manifolds and Deformation of Complex Structures.
The course description (along with prerequisites) can be found on this webpage.
Instructor :
Vamsi Pritham Pingali, vamsipingali@iisc.ac.in.
Office : N23 in the mathematics building.
Classroom and timings : Tuesday and Thursday from 2:003:30 in LH5.
The Grading policy : 25% for Homeworks, Midterm25%, and
50% for a LaTeX report on some analytic topic (merely copying something will cause me to penalise you).
Exams :
The Midterm shall be held on 20th Feb from 2:003:30 in LH5 (our usual classroom) .
Ethics: Read the information on the
IISc student ethics page. In short, cheating is a silly thing. Don't do
it. As for homeworks, write them up on your own. You are allowed to
discuss them amongst yourselves but please write the solutions on
your own. That said I must hasten to add that you learn mathematics
best when you solve the problems entirely by yourself.
Here
is the tentative schedule. (It is subject to changes and hence
visiting this webpage regularly is one of the best ideas in the
history of best ideas.)
Wk 
Dates 
Syllabus to be covered 
1 
1st Jan6th Jan 
Logistics, Poisson ODE (Thursday notes) 
2 
7th Jan 13th Jan 
Poisson PDE, Weak derivatives (Tuesday notes); Sobolev spaces and Sobolev Embedding (Thursday notes) 
3 
14th Jan 20th Jan 
Constant coefficient elliptic operators on the torus (Tuesday notes), Fredholm operators ( Thursday notes) 
4 
21st Jan 27th Jan 
Constant coefficient elliptic operators on the torus are Fredholm (Tuesday notes), Review of vector bundles (Thursday notes) 
5 
28th Jan 3rd Feb 
Review of geodesics and normal coordinates (Tuesday notes); Connections (Thursday notes) 
6 
4th Feb  10th Feb 
Connections, Curvatures. Definition of PDE (Tuesday notes); LeviCivita connection and its curvatures, Divergence and gradient (Thursday notes) 
7 
11th Feb  17th Feb 
Hodge Star, Laplacians (Tuesday notes); The Hodge Theorem and its applications (Thursday notes) 
8 
18th Feb  24th Feb 
Midterm week (our Midterm is on Thursday) 
9 
25th Feb  2nd Mar 
Sobolev spaces on manifolds (Tuesday notes); Elliptic regularity (Thursday notes) 
10 
3rd Mar  9th Mar 
Elliptic regularity (Tuesday notes); Elliptic operatorsFredholmness (Thursday notes) 
11 
10th Mar  16th Mar 
Diagonalisability of strongly elliptic formally selfadjoint operators (Tuesday notes); Parabolic equations (Thursday notes) 
13 
31 Mar  6 April 
Riemannian uniformisation theorem using the continuity method (Tuesday notes); Using Calculus of Variations (Thursday slides) 
13 
6 April12 April 
Riemannian uniformisation theorem using the method of sub and super solutions (Tuesday slides) 
Wk 
To be handed to me on 
Homework (subject to changes; please check regularly) 
4 
23rd Jan (Changed the date) 

7 
13 Feb 

10 
5 Mar 

12 
26 Mar (Change of date !) 

14+ 
Some time before end of April 