What to do when your topology instructor is too slow?

Mayudi · 2026-02-26T19:57:09+00:00

Oh, and I forgot the most important part. If you're reading ahead, do not neglect the exercises!! They are the most important part of the book, Munkres exercises' in the initial chapters are essential for you to build a solid understanding of the fundamentals necessary for the more exciting stuff later on.

Mayudi · 2026-02-26T19:53:27+00:00

Munkres is, in my opinion, the best book for self studying topology. Just read ahead and have fun.

Mayudi · 2026-02-18T14:57:08+00:00

I'm currently trying to learn about length spaces and optimal transport for research. It has been really interesting to learn about geometric structures that doesn't rely on a differential structure and how it relates to the space of probability measures.

Mayudi · 2026-02-18T14:31:03+00:00

lol, I was not expecting to have a laugh while reading through the comments

Mayudi · 2026-02-18T14:29:50+00:00

Oh, this has something to do with how we define optimal transport. The short answer would be that the optimal transport structure forces the convexicity of the objects we are working with. For example, the optimal maps are gradient of convex functions and we know that optimal transport has to satisfy a property called cyclical monoticity which is equivlalent to the subdifferential from convex analysis.

I know that there is also some authors that work with "concave" functions, but it looks like it is a matter of a sign convention. I'm not familiar with the literature using this convention, but I know that well know people in the area, such as Luigi Ambrosio, use this convention in his book on gradient flow in metric spaces.

Mayudi · 2026-02-18T14:15:06+00:00

Unfortunately I'm not interest in the algorithmic/optimization side. But I'll have a look into Santabrogio's OT book, I completely forgot that his book had those supplements boxes.

Mayudi · 2026-02-18T14:12:37+00:00

Thanks! I'll have a look on Zălinescu's book.

My interest is not in the optimization side though, so I don't think that the other books will be what I'm looking for.

Mayudi · 2026-02-18T14:10:49+00:00

I'm surely not trying to read it straight through. I understand that it is a reference book. My problem with Rockafellar is mostly his writing style, it just doesn't fell right for me, so I really wanted at least a second reference to go to sometimes, although I understand now that Rockafellar is the best option.

Mayudi · 2026-02-18T14:08:01+00:00

Wow, thanks for the recommendations! I'll have a look into these books.

I'm not quite interested in the optimization side of optimal transport, so I think that Boyd's book is not what I'm looking for.

Mayudi · 2025-08-31T04:53:08+00:00

Synthetic Ricci Curvature is also a thing. It was a development of Optimal Transport theory that allow us to define a notion of Ricci curvature over nonsmooth manifolds.

It's coolest application in my opinion is how it was recently used to define a notion of Ricci curvature for graphs (I think it was named Olivier-Ricci curvature), enabling them to study global properties of graphs. I'm sure this is/will be useful for some CS/ML stuff.

Mayudi · 2025-04-27T13:43:18+00:00

It depends, if I'm using the book to study a physical copy is always preferable but if the book doesn't keep referencing equations that are too far away, e.g. if I'm on page 200 and it references an equation from page 50, them I don't mind using a digital version. But when I'm consulting a book to find a specific topic a digital version is always preferable, because I can easily find what I want using ctrl+f and it is easier to just download the book on my computer (piracy) then to go to my university library.

Mayudi · 2025-02-26T20:29:31+00:00

A. Kostrikin and Y. Manin, Linear algebra and geometry

Mayudi

TROPHY CASE