What is the highest-ranking math concept?

One can’t compare that way. That’s like saying, “what’s the highest ranking literature style?”

There are diverse pathways in math, and while mathematicians may argue over which is more important or more difficult or the most taxing to understand, there isn’t even an objective way to compare them. One might argue the structures for understanding data, one other might argue arcane topological topics, another theorems in Game Theory.

Fermat’s last theorem.

^ Agreed – to date!

Sir Andrew John Wiles proved it in 1995 using methods .that sound like something from a Star Trek episode. “Elliptic Curves and Galois Representations”.

Others think this is the top so far. Wile’s Proof of Fermat’s Last Theorem‘_proof_of_Fermat’s_Last_Theorem .
“The proof itself is over 150 pages long and consumed seven years of Wiles’ research time.[1] John Coates described the proof as one of the highest achievements of number theory, and John Conway called it the proof of the century.[2] For solving Fermat’s Last Theorem, he was knighted, and received other honours.”

One thing is certain. In the future more even more difficult problems will be solved by people standing on the shoulders of the previous greats. That is how our species progresses.

In future there might even be a “Mariah or Phinotpi Theorem.”

One way of looking at math is to say that it is all about abstraction. It seems to me that the field of abstract algebra is the highest level of abstraction. I don’t know the proof of Fermat’s theorem, but from what I have read, I am pretty sure that abstract algebra was used.

A very loose definition of abstract algebra is that it studies the operation of objects on one another. One application is the operations of addition and multiplication on numbers. Another application would be the twisting operation on a Rubik’s cube. These may not seem to have anything to do with one another, but abstract algebra is able to use similar language for studying them both.

Just about any field of mathematics involves objects acting on one another and so is subject to the application of techniques from abstract algebra. For example, Fermat’s theorem is part of the field of math referred to as number theory, which covers the study of whole numbers. Under number theory there is a specialized field called algebraic number theory, which applies abstract algebra to number theory.

Since it is Pi Day, one might say the highest concept in mathematics is Transcendental Numbers.

Can’t get much higher than some good Transcendental Meditation.

@zenvelo So is the “highest concept” in philosophy Transcendentalism?

I would have to know what you mean by “highest concept” since the term doesn’t make much sense to me, but I might argue that fractals are one of the most important concepts since they are naturally occurring physical phenomena.