The texts we will be following 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.
Office hours : Wed from 45 pm. (Feel free to come during other times after emailing me first.)
Classroom and timings : Tuesday and Thursday from 1:503:20 in LH2 (basement).
The Grading policy : 25% for Homeworks, Midterm25%, and
50% for the Final. Under NO circumstances will makeup exams be held
for the midterm. If you have a valid and provable excuse, (Schedule
conflicts with other courses do NOT constitute as valid excuses. You
are supposed to resolve them before registering for the courses.)
then your performance on the other exams shall determine your grade on your midterms.
Moodle : The grades for your HW, Midterms, and Final will be posted on this webpage
Exams :
The Midterm shall be held on from 1:503:20 in the class on 27th Feb 2018 (Tuesday).
The final for this course will be held on April 25 (Wednesday) from 2 PM5PM in TBA.
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, goals of the course, the Poisson ODE (Tuesday notes), the Poisson PDE, weak derivatives ((Thursday notes) 
2 
7th Jan13th Jan 
Mollification, Sobolev spaces, Sobolev embedding (Tuesday notes), Compactness, Definition of constant coefficient elliptic operators (Thursday notes) 
3 
14th Jan20th Jan 
Parametrix for constant coefficient elliptic operators on a torus and Fredholm maps (Tuesday notes), Fredholm alternative, elliptic regularity, Definition and examples of Riemannian manifolds (Thursday notes) 
4 
21st Jan27th Jan 
Geodesic normal coordinates, Gauss lemma (Tuesday notes), Geodesically convex balls (Thursday notes) 
5 
28st Jan 3rd Feb 
HopfRinow theorem (Tuesday notes), Connections (Thursday notes) 
6 
4th Feb  10th Feb 
Curvature (Tuesday notes), Induced connections on direct sums, tensor products, etc. The LeviCivita connection (Thursday notes) 
7 
11th Feb  17th Feb 
Curvature(s) of the LeviCivita connection (Tuesday notes), Sectional curvature and theorems about it (Thursday notes) 
8 
18th Feb  24th Feb 
Midterm week (no classes) 
9 
24th Feb  3rd Mar 
Midterm : Everything we covered so far (Tuesday), Divergence theorem and the Hodge star (Thursday notes) 
10 
4th Mar  10th Mar 
Hodge and Rough Laplacians (Tuesday notes), The Hodge theorem and applications (Thursday notes) 
11 
11th Mar  17th Mar 
Sobolev spaces (Tuesday notes), Sobolev embedding, Ellipticity (Thursday notes) 
12 
18th Mar  24th Mar 
Elliptic regularity for L^2 distributional solutions (Tuesday notes), Elliptic operators are Fredholm (Thursday notes) 
13 
25th Mar  31st Mar 
No class 
14 
1 Apr  7 Apr 
The statements of L^p regularity and Schauder estimates (Tuesday notes), The Riemannian uniformisation theorem (Thursday notes) 
Wk 
To be handed to me on 
Homework (subject to changes; please check regularly) 
1 
4th Jan 
No HW due 
3 
18th Jan 

6 
8th Feb 

7 
15th Feb 

12 
24th Mar 

14 
5th April 