What is the geometric approach to mathematical research?

[kay bei]

TL;DR Summary

Maxwell, Dirac and Mikhail Gromov have been known to think Geometrically when doing their theoretical and mathematical research. What does this mean? and How is it different from the Algebraic approach to thinking.

I read this article History of James Clerk Maxwell and it talks about Maxwell and Dirac also at some point. It is said that Maxwell thought geometrically, and also Dirac said he thought of de Sitter Space geometrically. They say their approach to mathematics is geometric. I see this mentioned everywhere. Mikhail Gromov has said he thinks and solves problems geometrically. How is someone supposed to think of Manifolds, and abstract spaces geometrically? I don’t understand this process they use whereby they use geometric thinking to help solve a problem. Are there any basic and more advanced examples which show how someone uses this type of thinking to solve a problem? Please help me understand this. In an interview a mathematician (a fields medalist, I can find the article for you), explained that some mathematicians think algebraically, and others think geometrically and it was said that Gromov thought geometrically. Gromov has said in his interview with AMS that John Nash's thinking style influenced him and that Nash also thinks geometrically, he mentioned also Stephen Smale. I just don’t know the difference in these thinking styles (alebraic vs geometric). How does one develops this without any good examples comparing both approaches side by side on basic and difficult problems. I’ve never had a teacher show the difference. Hopefully someone can clarify this. I am asking because I would prefer to purchase a textbook which takes one approach which suits my thinking style. I may find it easier to think geometrically vs algebraically so it would be beneficial to know what is this difference. There is even a textbook on abstract algebra : a geometric approach. This is so confusing? Is it true that some mathematicians in the field of abstract algebra think in a geometric way to solve their research problems? Do you notice this also among mathematicians? Hopefully someone can help clarify this geometric vs algebraic approach mentioned everywhere.

 

[fresh_42]

This is hard to answer. I think it refers to thinking in entire objects and not so much in technics, coordinates, or algebraic structures.

What are the complex numbers for you? Are they

• $r\cdot e^{i\varphi }$ or $a+ib$? analytical
• a point in the complex plane, or on the Riemann sphere? geometrical
• a polynomial in $\mathbb{R}[x]/(x^2+1)$? algebraical

What are linear subspaces for you? Are they

• linear combinations of a subset of basis vectors? analytical
• a plane in space or a line in the plane, a hypersurface? geometrical
• the kernel of a linear transformation? algebraical

See, one can look at mathematical subjects in many different ways, although they all represent the same thing. A geometrical point of view is an imagination of how parts fit into the whole image. It is neither driven by technical descriptions, nor categorial points of view. Instead, mathematical objects become an entity in their own right.

 

[jedishrfu]

When I think of things I often use geometric means to compose my thoughts. Sometimes you can come up with a drawing that embodies your idea and that guides you to a solution or deeper understanding.

Consider the Flatlands book where 2D visualization is extended to 3D and beyond.

https://en.wikipedia.org/wiki/Flatland

Of course, there are some concepts that can't be thought of geometrically. Great mathematicians will come up with something that helps them work the problem and it will likely be geometric with some kind of mapping to what they are trying to solve.

In vector analysis and other applied fields, it's common to switch back and forth between a geometric view and an algebraic view.

Basically, this is a difficult skill to teach. You will have to extend your abilities to do so.