Introductory materials

[Jesus Briales]

Hi everyone,

I'm trying to understand better the theory as well as typical mathematical tools (calculus) underlying optimization in manifolds.

The related book "Optimization Algorithms on Matrix Manifolds" by Absil et al is certainly a great piece of work.

However, even though I love maths, my background is in engineering rather than mathematics, so I'm often struggling

to get a complete intuition about the most basic concepts in the book (e.g. atlases, embedded and quotient manifolds, etc.).

I wonder if there are any other introductory materials I could use in addition to the book above.

I'm specially interested in video materials, maybe from some recorded undergraduate course, tutorials at some conference or similar.

Thanks for sharing!

- Jesus

[pierr...@gmail.com]

I am not aware of video materials, but a few sets of slides are accessible from the "Related events" section of http://sites.uclouvain.be/absil/amsbook/. (More recent slides exist, but they are not meant to be intelligible without listening to the talk, hence I am not making them publicly available.) -PA

[Nicolas Boumal]

Hello Jesus,

I do not know of courses or tutorials about optimization on manifolds that exist in video, unfortunately. That would be nice to create at some point.

For the specific concepts you brought up (atlases, submanifolds and quotient manifold), I expect that there should exist good introductory courses for differential geometry on either coursera, or edX, or MIT OCW, or others, that would cover this in accessible terms. I do not know of any in particular to recommend, but if you find one you liked, it would be great if you could tell us about it here.

About atlases in particular, I would also add that they are never used in practice, only in theory. Of course, they are a necessary tool to understand even just the definition of manifolds, but using the somewhat higher level notion that "a manifold can be linearized at every point, into a tangent space", you can get a long way in gaining high level understanding.

Best,

Nicolas

[Jesus Briales]

Thank you both for the directions, I will have a look and, as Nicolas proposed, if I find any interesting resources which are particularly useful for this introduction we can share them here.

Regards,

Jesus

[tfl...@gradcenter.cuny.edu]

For anyone else who may be interested in video lectures, I found this series to be very helpful https://www.youtube.com/watch?v=7G4SqIboeig&list=PLFeEvEPtX_0S6vxxiiNPrJbLu9aK1UVC_. The first 10 lectures cover material from topological manifolds up to Riemannian metrics. It's from the perspective of a physicist and I believe it's very well paced for those coming from backgrounds other than pure math.

[Nicolas Boumal]

Thank you for posting this link to the lectures -- I just listened to the one about connections, and it is indeed excellent material.

Best,

Nicolas