I am interested in learning more about Partial Differential Equations and its possible applications to Differential Geometry. My personal take on a traditional rigorous course in PDE is as follows: the function spaces that are defined seem made up, the actual PDE's that are being studied seem contrived and there doesn't seem to be any rhyme or reason as to why one would care about all these things.
Ofcourse, this is all wrong! The function spaces are not contrived at all, the PDE's that are studied (Poisson equation, Heat equation, Transport equation, Wave equation etc) arise extremely naturally while studying real life problems. Variants of these problems also arise while studying interesting questions in Differential Geometry.
By the way, don't go by the names of the PDE's; just look at the PDE's themselves and see if you feel it has any connection with say the flow of heat on a rod or the propagation of waves. Chances are that if the PDE's were not called by those names, you would never have thought they have any practical use.
My goal is now to approach PDE's in a completely different way (different from how it is traditionally presented in a rigorous mathematics course). I want to start with some simple PDE and explain the non-rigorous heuristics about how that PDE arises in a real life situation (for example, the Poisson equation with Dirichlet Boundary condition arises naturally while trying to study the displacement of an elastic membrane subjected to a transversal load). In the process of studying this problem (or some similar problem), I will gradually define Distributions and Sobolev spaces. I will then directly move to the discussion of Finite Element Method and explain how to obtain approximate solutions. I also hope to show you some numerical simulations based on this technology (I intend to use AI for this; I have no intention of actually writing any code to implement FEM right now).
I feel that done this way, the discussion will be less dry. Furthermore, when the technical things come up (such as the Elliptic Regularity Theorem for Laplacian etc) one will feel much more motivated.
I will then go on to extend this discussion to Differential Geometry (the Laplacian will become the Laplace Beltrami Operator). I will try to illustrate the idea of curvature through this discussion. I will also discuss numerical methods (Finite Element Method) on Riemannian Manifolds (such as the sphere) to solve PDE's (such as the Poisson equation). I also wish to show some simulations based on this (again, I plan to use AI for this).
Two long term goals I have: by studying the Heat equation on a Riemannian Manifold I want to discuss
(1) Hodge Theorem (2) Gauss-Bonnet Theorem
Ideally, I would like to prove both (1) and (2), but a complete proof would be very long and extremely technical. A less ambitious goal is to give an idea behind the proof.
References:
1) "Numerical Solution of PDE by Finite Element Method" by Claes Johnson
2) "The Laplacian on a Riemannian Manifold" by Steven Rosenberg
3) "Topics in Functional Analysis and Applications" by S. Kesavan