Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for exams and because I was lead to it through a very circumvoluted path of research (from arithmetics through algebraic geometry to D-modules).

Anyway, I was told that PDE were classified into 3 families: hyperbolic, elliptic, parabolic respectively. I saw exemples for each: Wave, Laplace and Heat equations. I also saw lots of different methods to solve these examples: using symmetry to reduce to an ODE, Fourier transform, Distributions, Fourier series (in dimension 1 to deal with boundary conditons). Each time it seemed the answer was "reduce the problem to a polynomial equation or an ODE by any way you can". So my geometric brain kicked in and tried to give a unified geometric interpration to all this.

I think I understand the distinction between those equations. We are considering 2nd order PDE with real coefficient. And the idea is to reduce the classification to that of their principal symbols seen as quadratic forms on $\mathbb{R}^n$.

So I asked this kind a questions: let's consider a LINEAR PDE $Pf = 0$ where $P$ is a linear differential operator of degree $d$ on $\mathbb{R}^n$.

Question 1: Let's say a "naive elliptic PDE" is any PDE given by a differential operator $P$ whose principal symbol $\sigma(P)$ satisfies $\sigma(P)(x,\xi) \neq 0$ for $\xi\neq 0$. Is this definition any good?

If answer to question 1 is yes,

Question 2: What is the analogue of a parabolic or hyperbolic operator?

The obvious perfectly nice answer would be: "PDE's are classified by the hypersurfaces defined by their principal symbols". Unfortunatly the answer I got was something like "Don't overthink it, the classification is more heuristic than anything.". Does that mean, "there is a classification along theses lines but it is a bit more subtle" or that "things are much much more complicated in higher dimensions/degrees"?

Anyway...

Question 3: Is it at least true that the classification of PDE with constant coefficient is related to the classification of real algebraic projective hypersurfaces $\{\sigma(P) = 0 \} \subset \mathbb{P}^{n-1}_{\mathbb{R}}$?

Let's assume the previous questions aren't completly wrong for trivial reasons. Let's consider a PDE with non-constant coefficients. We should then classify those according to "families of algebraic projective hypersurfaces" in $\mathbb{P}(T^*\mathbb{R}^n)$.

Question 4: Which kind of families can we expect? Is that related to Gabber's theorem of involutivity of the characteristic variety?

I am now assuming someone answered all these questions without thinking the words "this is so completly wrong". I have a final question (at least before the next one):

Question 5: Why is this all so hard to learn/teach?