This page contains information about the course MT3184 Differential geometry and Lie groups that  I will be teaching in the semester August-November 2022 in IISER Pune. 

Prerequisites for the course:

1. Some understanding of topology (separation axioms, countability axioms, compactness, connectedness, quotient spaces).  It would be good if you already know examples of spaces that satisfy certain property (but not the other). 

2. Some understanding of multi variable calculus (partial derivative, total derivative of functions between subsets of R^n, smooth functions between subsets of R^n, Taylor's theorem)

3. Some understanding of group theory (definition of group, some basic properties), some understanding of linear algebra (vector spaces, linear maps between vector spaces, some basic properties).

There are 37 teaching days (if everything goes as per academic calendar). Out of which we will use 30 teaching days for the below mentioned lecture sessions and 4 teaching days for problem sessions. The remaining 3 teaching days will be reserved for discussion of some special topics (based on interest of students). 

Ambitious/tentative list of topics  to be covered (one topic per session): 

1. (Introduction to the course and) Review of multivariable calculus. 

2. Review of multilinear algebra 

3. Definition of smooth manifolds (and some examples).

4. Definition of smooth map between smooth manifolds (and some examples).

5. Smooth "bump" functions and partition of unity.

6. Tangent space at a point on smooth manifold (some properties, some computations).

7. smooth map between smooth manifolds inducing linear map between tangent spaces. 

8. Two classes of smooth maps : immersions and submersions.

9. Two classes of submanifolds: immersed submanifolds, embedded submanifolds.

10. Whitney embedding theorem (at least for compact manifolds).

11. Tangent bundle construction (some idea about general notion of vector bundle).

12. vector fields on smooth manifolds.

13. Lie bracket of vector fields.

14. integral curves (and flows) associated to vector fields.

15. Cotangent bundle construction.

16. differential forms on smooth manifolds.

17. Lie derivative, interior derivative, and exterior derivative operators.

18. Cartan's magic formula (relating Lie derivative, interior derivative, exterior derivative).

19. Orientation on a smooth manifold. 

20. integration on oriented smooth manifolds.

21. de Rham cohomology and de Rham theorem (assuming the student already saw some cohomology).

22. Stoke's theorem in the set up of differential forms. 

23. Definition of Lie groups (and some examples).

24. Lie algebra of a Lie group (some computations)

25. Exponential map in the setup of Lie groups and Lie algebras.

26. The Baker-Campbell-Hausdorff formula (really ambitious topic). 

27. Lie group action on a smooth manifold, and the quotient manifold theorem.

28. (universal) covering space of a smooth manifold.

29. Foliations and distributions.

30. Frobenius theorem on Integrable distributions.

Note : It is yet to be decided if the topics mentioned in red color are to be discussed in class or to be left to students for further reading.  

Update: The topics mentioned in red color was not discussed due to time constraints.

Special topics (to be covered based on interest/feedback of students) 

1. ----

2. ----

3. ----

Problem sessions (viva session):

We will have 4 problem/viva sessions, one each on the following days:

1. 30th August

2. 21st September

3. 31st October

4. 23rd November