Instructor

Prof. Swapneel Mahajan

Semester

Autumn ‘21

Course Difficulty

The course was definitely very theoretical and rigorous. It entailed detailed proofs.

Time Commitment Required

Classes were heavy (content wise) so it required around 5-6 hours per week (excluding class hours).

Grading Policy and Statistics

14 AA
2 AB

Sir was very generous about grading and continuously told us not to worry about our marks and just learn.

Attendance Policy

Attendance was compulsory even in the online semester (30% of the grade depended on it).

Pre-requisites

Having a basic understanding of linear algebra (matrices, vector spaces and so on).

Evaluation Scheme

Attendance: 30%
Presentation 1: 30%
Presentation 2: 40%
(Presentations: solve any 2 questions from the bank allotted and present the solution to the rest of the class - so it was done in a group, and each member of the group had to solve 2 questions)

Topics Covered in the Course

Everything was approached from the lens of category theory.

1. Manifolds (with a brief introduction to metric spaces)
   1.1. Topological spaces
   1.2. Topological manifolds
   1.3. Smooth manifolds
   1.4. Submanifolds

2. Smooth functions and tangent vectors
   2.1. Algebra of smooth functions
   2.2. Derivations of an algebra
   2.3. Tangent vectors and derivations
   2.4. Tangent bundle and vector fields
   2.5. Cotangent bundle and one-forms
   2.6. Bundle of tensors and tensor fields
   2.7. Differential forms

3. Vector bundles
   3.1. Smooth vector bundles
   3.2. Sections of a smooth vector bundle
   3.3. Operations on vector spaces
   3.4. Operations on smooth vector bundles
   3.5. Interactions of the functor of smooth sections with operations
   3.6. Dual of the category of smooth vector bundles

4. Connections and curvature
   4.1. Covariant derivatives wrt vector fields
   4.2. Connections on vector bundles
   4.3. Connections and matrix-valued one-forms
   4.4. Operations on connections
   4.5. Differential forms on a vector bundle
   4.6. Curvature
   4.7. Parallel transport

Teaching Style

Sir had already pre-recorded his lectures and would play them in class. He would pause them at certain points and induce discussions.

Tutorials/Assignments/Projects

We only had two presentations and the difficulty level depended on the questions you chose.

Feedback on Exams

None

Motivation for taking this course

Differential geometry was a concept that I applying to my project in the systems and control department - not all of is of course, but a subset of it. As a result, I believed it was essential to get a rounded view of the field. Therefore this course seemed like a good starting point.

Course Highlights

The course is a really interesting course wherein the learning aspect is more emphasised over the evaluating one. The approach (via category theory) is indeed a unique one and nicely ties up all the concepts together.

Course Importance

As mentioned earlier, differential geometry is applied extensively in control. It also sees its applications in general relativity (curvature and all the topics around it).

How strongly would I recommend this course?

I would recommend people to take the course if they are interested in mathematics and the rigor associated with it.

When to take this course?

In my seventh semester. The course can be taken in your third year or your fourth year.

Going Forward

Several syscon courses related to differential geometry and optimal control can be done (SC624, SC633, SC618) and courses offered by the physics and mathematics departments can also be checked out.

References Used

The professor used his own notes to teach.

Review By: Bhavini Jeloka