MAT 339. Geometric Analysis - Spring 2018 - Vamsi Pingali

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 pre-requisites) can be found on this webpage.

Instructor : Vamsi Pritham Pingali,

Office : N23 in the mathematics building.

Office hours : Wed from 4-5 pm. (Feel free to come during other times after emailing me first.)

Classroom and timings : Tuesday and Thursday from 1:50-3:20 in LH-2 (basement).

The Grading policy : 25% for Homeworks, Midterm-25%, 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 1:50-3:20 in the class on 27th Feb 2018 (Tuesday).

The final for this course will be held on April 25 (Wednesday) from 2 PM-5PM 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.)



 Syllabus to be covered


1st Jan-6th Jan

Logistics, goals of the course, the Poisson ODE (Tuesday notes), the Poisson PDE, weak derivatives ((Thursday notes)


7th Jan-13th Jan

Mollification, Sobolev spaces, Sobolev embedding (Tuesday notes), Compactness, Definition of constant coefficient elliptic operators (Thursday notes)


14th Jan-20th 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)


21st Jan-27th Jan

Geodesic normal coordinates, Gauss lemma (Tuesday notes), Geodesically convex balls (Thursday notes)


28st Jan- 3rd Feb

Hopf-Rinow theorem (Tuesday notes), Connections (Thursday notes)


4th Feb - 10th Feb

Curvature (Tuesday notes), Induced connections on direct sums, tensor products, etc. The Levi-Civita connection (Thursday notes)


11th Feb - 17th Feb

Curvature(s) of the Levi-Civita connection (Tuesday notes), Sectional curvature and theorems about it (Thursday notes)


18th Feb - 24th Feb

Midterm week (no classes)


24th Feb - 3rd Mar

Midterm : Everything we covered so far (Tuesday), Divergence theorem and the Hodge star (Thursday notes)


4th Mar - 10th Mar

Hodge and Rough Laplacians (Tuesday notes), The Hodge theorem and applications (Thursday notes)


11th Mar - 17th Mar

Sobolev spaces (Tuesday notes), Sobolev embedding, Ellipticity (Thursday notes)


18th Mar - 24th Mar

Elliptic regularity for L^2 distributional solutions (Tuesday notes), Elliptic operators are Fredholm (Thursday notes)


25th Mar - 31st Mar

No class


1 Apr - 7 Apr

The statements of L^p regularity and Schauder estimates (Tuesday notes), The Riemannian uniformisation theorem (Thursday notes)

Senorita, bade bade deshon mein aise chote chote HW dete hain...


 To be handed to me on

 Homework (subject to changes; please check regularly)


4th Jan

No HW due


18th Jan

HW 1


8th Feb

HW 2


15th Feb

HW 3


24th Mar

HW 4


5th April

HW 5