MAT 339. Geometric Analysis - Spring 2020 - Vamsi Pingali

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 pre-requisites) can be found on this webpage.

Instructor : Vamsi Pritham Pingali,

Office : N23 in the mathematics building.

Classroom and timings : Tuesday and Thursday from 2:00-3:30 in LH-5.

The Grading policy : 25% for Homeworks, Midterm-25%, 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 2:00-3:30 in LH-5 (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.)



 Syllabus to be covered


1st Jan-6th Jan

Logistics, Poisson ODE (Thursday notes)


7th Jan- 13th Jan

Poisson PDE, Weak derivatives (Tuesday notes); Sobolev spaces and Sobolev Embedding (Thursday notes)


14th Jan- 20th Jan

Constant coefficient elliptic operators on the torus (Tuesday notes), Fredholm operators ( Thursday notes)


21st Jan- 27th Jan

Constant coefficient elliptic operators on the torus are Fredholm (Tuesday notes), Review of vector bundles (Thursday notes)


28th Jan- 3rd Feb

Review of geodesics and normal coordinates (Tuesday notes); Connections (Thursday notes)


4th Feb - 10th Feb

Connections, Curvatures. Definition of PDE (Tuesday notes); Levi-Civita connection and its curvatures, Divergence and gradient (Thursday notes)


11th Feb - 17th Feb

Hodge Star, Laplacians (Tuesday notes); The Hodge Theorem and its applications (Thursday notes)


18th Feb - 24th Feb

Midterm week (our Midterm is on Thursday)


25th Feb - 2nd Mar

Sobolev spaces on manifolds (Tuesday notes); Elliptic regularity (Thursday notes)


3rd Mar - 9th Mar

Elliptic regularity (Tuesday notes); Elliptic operators-Fredholmness (Thursday notes)


10th Mar - 16th Mar

Diagonalisability of strongly elliptic formally self-adjoint operators (Tuesday notes); Parabolic equations (Thursday notes)


31 Mar - 6 April

Riemannian uniformisation theorem using the continuity method (Tuesday notes); Using Calculus of Variations (Thursday slides)


6 April-12 April

Riemannian uniformisation theorem using the method of sub and super solutions (Tuesday slides)

HW : There'll be peace when you are done....Lay your weary head to rest....Don't you cry no more...


 To be handed to me on

 Homework (subject to changes; please check regularly)


23rd Jan (Changed the date)

HW 1


13 Feb

HW 2


5 Mar

HW 3


26 Mar (Change of date !)

HW 4 and 5


Some time before end of April

HW 6