We will start by building the background for geometry.
We will split the content into meaningful modules covered (approximately) every three weeks.
The plan is to break it so that the content can be covered by students in two weeks, with a meeting every week to discuss problems they may have. The third week in each module is a bonus to catch up with, and where they can work on questions to better understand the content.
You can always read/watch ahead, faster than the planned pace.
We will have an organizational meeting and discuss how we will proceed.
Summary: Covered how we will proceed.
The first two weeks, we will go through some basic differential topology, as covered by Prof. Frederic Schuller, in the lecture series posted. These are lectures 4-7 of the series https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic.
You can find the lecture notes here (https://mathswithphysics.blogspot.com/2016/07/lectures-on-geometric-anatomy-of.html) (there is a downloadable PDF).
Also look at videos 1-8 of https://www.youtube.com/playlist?list=PLBh2i93oe2qvRGAtgkTszX7szZDVd6jh1, skipping any locked videos.
A summary of week 1: After discussing a question on quotient spaces, we took a brief look at understanding invariants as functors, and discussing themes of looking at algebraic topology that way. Also some discussion on classification problems.
I will post exercises in week 1, and we will discuss them in week 3.
While doing exercises, also look at videos 9-15 of https://www.youtube.com/playlist?list=PLBh2i93oe2qvRGAtgkTszX7szZDVd6jh1, skipping any that are locked. These are pretty short, so you should be able to cover them easily.
The next two weeks, we will go through tensor field theory as covered by Prof. Frederic Schuller, in the lecture series posted. These are lectures 8-11 of the series https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic.
You can find the lecture notes here (https://mathswithphysics.blogspot.com/2016/07/lectures-on-geometric-anatomy-of.html) (there is a downloadable PDF).
I also recommend going through videos videos 16-22 of https://www.youtube.com/playlist?list=PLBh2i93oe2qvRGAtgkTszX7szZDVd6jh1
I will post exercises in week 4, and we will discuss them in week 6.
The next three weeks, we will go through the theory of differential forms and de Rham theory. There are two main resources for this week. We will figure out how to pace ourselves as we proceed. But for the first week here, just try going through both parallely.
We will quickly cover the lecture series by Michael Penn, which studied differential forms on flat Euclidean space. He generalizes vector calculus on Euclidean 3 space to general dimensions.
https://www.youtube.com/playlist?list=PL22w63XsKjqzQZtDZO_9s2HEMRJnaOTX7
See also the lectures in https://www.youtube.com/playlist?list=PLBh2i93oe2qvRGAtgkTszX7szZDVd6jh1, specifically videos 23-32. We will focus properly on the content of video 26 later. Here we will see how to transfer the calculus developed by Penn to manifolds.
Lecture 12 of Schuller is also good. https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic.
You can find the lecture notes here (https://mathswithphysics.blogspot.com/2016/07/lectures-on-geometric-anatomy-of.html) (there is a downloadable PDF).
We might use these as well. Let's see. https://pi.math.cornell.edu/~sjamaar/manifolds/manifold.pdf
I will post exercises in week 7, and we will discuss them in week 10.
We might go deeper into Hodge theory as well. Maybe discuss supersymmetry for the de Rham complex.
The next four weeks, we will take a detour and go through the basics of tensor calculus and Riemannian geometry.
Since Schullers approach to connections comes via principle bundles, we will resort to other sources for this.
I'm leaning towards this. Solid. https://youtu.be/jQkJ9E8jZhY?list=PLArBKNfJxuunOXcRxEq1lbJ2AmwaGCo7E
We can study symplectic geometry, the geometry of classical mechanics. I'll start another page for resources for this.
This might be a good way to develop better intuition on tensors and geometry, but will come with some analysis.
This could be an initial attempt to understand Spectral geometry. These notes by Yaiza Canzani are pretty good.
https://canzani.web.unc.edu/wp-content/uploads/sites/12623/2016/08/Laplacian.pdf
This might be interesting. There's a lot of interaction with curvature, and is also related to some research I'm engaged in. We will have to go through some representation theory, and study Lie theory.
What is most important is that you try and find sources which appeal to you, and do background reading on the topics as you have time.
Here are some suggested youtube series and notes.
The following is a beautifully crafted course. https://www.youtube.com/playlist?list=PLBh2i93oe2qvRGAtgkTszX7szZDVd6jh1
A good series on tensor calculus is given here. https://www.youtube.com/playlist?list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx
Khan academy has a short series. Haven't watched it myself. https://www.youtube.com/playlist?list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27_
Some physics lectures, on general relativity, where they develop the necessary math in parallel. https://www.youtube.com/playlist?list=PLUl4u3cNGP629n_3fX7HmKKgin_rqGzbx
Another similar one. https://www.youtube.com/playlist?list=PLzSstOSgYj0PIWvq6efp43BDH1sUBhmfc
A different branch of physics inspired math. Symplectic geometry, the geometry of classical mechanics. https://www.youtube.com/playlist?list=PLDfPUNusx1EoVnrQcCRishydtNBYU6A0c
Spiros Karrigianis has a great course in Riemannian geometry https://youtu.be/jQkJ9E8jZhY?list=PLArBKNfJxuunOXcRxEq1lbJ2AmwaGCo7E
Dominic Joyce has a brilliant course where he starts by quickly going over some Riemannian geometry
https://www.youtube.com/playlist?list=PL_jwwOG0kPgOIyKNoUwE35MJFEt4QjirI
Sjamaar has some notes. https://pi.math.cornell.edu/~sjamaar/manifolds/manifold.pdf