GaussBonnet

Proofs are only an example of what grade school teachers don't know. They are obviously missing all of the modern applications as well. Whether you want to call those applications "math" or something more specific ("engineering", "cryptography", etc) is a matter of taste I guess. The point still stands that kids are learning hardly anything of theory OR practice in their grade schools, and this is due to the teachers not knowing those things either.

We can talk about why the teachers don't know these things, but I think the real issue is American culture still sees math and science as something for pitiful, nerd-virgin dorks. You can try to sell math as this sexy, dangerous thing you use to win wars, but ultimately you will have to reckon with the image that most Americans have of mathematicians and scientists, which is illustrated in shows like the Big Bang Theory: they're ridiculous laughing stocks who use science as a kind of cope for not being socially successful.

The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with MULTICOLORED (whoa!) text. There is no communication between modern mathematicians and grade school teachers in America; they may as well live on separate planets. Math classes in American grade school are little more than busy work meant to weed out people who hate busy work. Infer from that what you will.

You aren't wrong. Until extremely recently, algebraic geometry was a field by and for the purest of the pure mathematicians. These days algebraic geometry has made its way into string theory.

https://en.wikipedia.org/wiki/Gromov%E2%80%93Witten_invarian...

A couple of his contributions:

1. A far reaching generalization of the Riemann-Roch theorem, which is now called the Grothendieck-Riemann-Roch theorem. As with a great deal of his work, it's about drawing conclusions about global structure from local data:

https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem

2. Creating the machinery used to solve the Riemann hypothesis for finite number fields.

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

His work wasn't focused on solving particular problems so much as it was on finding the right language with which to describe problems. The philosophy is that, with the right language, your proofs should become obvious. This is somewhat in the same spirit as Leibniz's quest for the 'Universal Characteristic'.

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

The phenomenon you are talking about is essentially that of a homomorphism, or homomorphic structures. That is, structures that appear superficially different but share an underlying common structure.

The concept of a 'functor' was invented to describe a higher order 'homomorphism of homomorphisms'. An example most people miss is the total derivative in multivariable calculus: the chain rule implies that the total derivative is a functor that maps the composition of differentiable functions on a manifold, to matrix multiplication (of matrices acting on the tangent space).

You might also be interested in various 'dual' concepts, like that between tangent spaces and cotangent spaces in differential geometry.

For algebra, I'd recommend Pinter's Book of Abstract Algebra.

The first 10 pages of the following link may be helpful; it shows probably the simplest concrete nontrivial 2-form:

https://math.berkeley.edu/~wodzicki/H185.S11/podrecznik/2for...

The first example there is: given a base point X and two vectors V,W based at X, the 2-form gives the "signed" area of the parallelogram spanned by V and W. Determinants (which measure n-dimensional parallelograms), when viewed as functions of their column vectors, have all the properties of differential forms.

Differential forms are a bit like generalized determinants and in a sense specify a way to measure something like an abstract volume in the neighborhood of a point of a manifold, in such a way that the Jacobian needed for changing coordinates is "built in".

The point of differential forms is that they give a way to express geometric theorems in a coordinate free way. Coordinates are seen as obscuring the pure geometric content of theorems. They are sometimes necessary artifacts of doing concrete calculations, but the idea is that geometry shouldn't depend on a choice of coordinates.

The important ideas can be found in pages 9-10 in this link:

https://www.math.ucla.edu/~tao/preprints/forms.pdf

Note halfway down page 9 we get a really clean equation for how to change variables (change coordinates) in an abstract way.

Also note the simple form that the general n-dimensional stokes theorem takes in terms of differential forms at the top of this page:

https://en.wikipedia.org/wiki/Stokes%27_theorem

That they allow the expression of substantial theorems in concise form is a clue that they are the "right" way to do differential geometry.

Background in pure math here. Generally researchers learn about what problems are fashionable as they talk to others in the field. As a PhD student, your advisor should give you ideas for problems to work on. Solutions to old problems tend to open up new lines of inquiry. It's a potentially infinite process, with the only limit being the capacity of the human mind.

The odd perfect number problem has been unsolved for thousands of years:

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

Why would anyone want to know the answer to this problem? The truth is that the motivation for most pure math research is purely aesthetic. You could also ask arts or english departments "How can you apply your paintings or novels?!?". Some mathematicians (mostly geometers and topologists) are inspired by problems in physics, but most aren't. In truth you never know what structure or theorem might have some future application; number theory was totally "useless" until modern cryptography made (some of) it useful.

For just a tiny taste of one small area of modern math, I dare you to click on any of the links here:

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

We are flush with structures to investigate. The idea that "most discoveries have been made" is nonsensical in a domain where the discoveries to be made are literally infinite. To give a hint as to the infinite nature of mathematical inquiry, you probably are familiar with the idea of a function mapping a number to another number. A good deal of modern math is involved with much higher order functions; we can have functions that map functions to numbers, functions to functions, functions to spaces, and so on and so on. And then we can consider functions between those functions (and so on). Category theory is an attempt to give a framework to some of these "meta" relations. It should be obvious there is no limit to these structures and no limit to the number of problems one could pose about them.