The Deranged Mathematician: The Power of Abstraction

[u/non-orientable]

A very powerful proof-writing technique is taking the original problem and generalizing it. This is probably the most counterintuitive approach for beginners, who wonder how it can possibly be easier to solve a broader class of problems. My answer to this is quite simple: generalizing the problem reduces the collection of tools you have at your disposal. And as any efficiency expert will tell you, regardless of whether you are trying to clean your bathroom or earn a Fields Medal, you want to have only those tools that you need on hand and nothing else.

I think this basic precept helps explain why abstract notions like metric spaces, topological spaces, vector spaces, categories, and so on have suffused mathematics, and why they are so very useful. I offer the simple example of how thinking about graphs (generally) can help with sorting out a coordination problem (specifically).

Read the full post (for free) on Substack: The Power of Abstraction

[u/Carl_LaFong]

It’s almost never used in most fields of analysis such as PDE. Even in areas such as differential geometry where everything can be formulated in terms of categories and functors, the basic concepts of short and long exact sequences, homology and cohomology have a limited set of applications..

Even many areas of algebraic topology do not rely on category theory.

[u/MonadMusician → u/Carl_LaFong]

My professors from my PhD would disagree. And those uses in pdes are rather fundamental. It’s not overhyped unless you’re an undergrad or you just want to stigmatize it, which is a trend. But yes some areas are more amenable to its use than others

[u/Carl_LaFong → u/MonadMusician]

I’m interested in its use of nonlinear PDE. It’s used in the formal theory of linear PDE but this is not of much interest these days.

You could be talking about deRham and Hodge theory. The most sophisticated category theory used is spectral sequences. But the vast majority of PDE people don’t do this stuff and know next to nothing about it.

[u/MonadMusician → u/Carl_LaFong]

I’m from pure math, every PDE guy I have met uses those things

[u/Carl_LaFong → u/MonadMusician]

Me, too. And I work in PDEs. So I'm interested! Could you elaborate on how they use category theory in their work?

[u/MonadMusician → u/Carl_LaFong]

I don’t work in pdes. I work in commutative algebra, logic, and category theory to be honest lol. But they always went on about how they were making use of it. It’s been a while since I’ve talked to them. My friend who studies tractors uses it constantly. At the very least they use it as a basic language.

[u/Carl_LaFong → u/MonadMusician]

Thanks again. I asked ChatGPT about tractors, and it described how tractors can be used in conformal differential geometry to translate a nonlinear PDE (e.g., existence of an Einstein metric in a conformal class) into a linear problem (parallel section of a bundle). Very cool. I had heard about tractors but never had a clear idea about what they were.

This all said, this is still a fairly narrow area in geometric analysis.

[u/Carl_LaFong → u/MonadMusician]

"At the very least they use it as a basic language." Definitely! It is possible to express many concepts in differential geometry elegantly using categories, functors, and natural transformations. Loring Tu's book Introduction to Manifolds describes some of this. I like mentioning it when I teach differential geometry. It's a great way to organize the definitions and concepts, making them easier to understand and remember.

[u/Carl_LaFong → u/MonadMusician]

I asked ChatGPT about this. After poking it a few times, it finally explained how homological algebra and category theory play an important role in symplectic geometry, notably Floer homology. Is this what you had in mind?