I think it's the case that time dilation prevents you from ever seeing something reach the singularity. After all, as the mass M increases, the field strength at the event horizon is ~1/M, so we can make it as small as we like.
This makes sense to me. Linear algebra and differential equations are closely related, and the intuition you gain from that connection will help with the perhaps more confusing multivariate calculus.
It's somewhat silly to be asking this question at your age. I doubt you know any calculus (I didn't at your age), so how can you say you know any physics? Sure it might all sound cool, but why don't you give it a try before worrying yourself about these things?
Also, your performance now might...
If you go into quantum gravity, things like infinite Galois theory, cohomology, and algebraic geometry are very important tools from algebraic number theory that one wouldn't necessarily learn in an abstract algebra course.
There's also that cool connection between calculating Etale motives and...
That's a naive assumption.
Often a monopoly is the most efficient configuration of firms. This can occur when either the entry cost is high, the profits low, or especially in economies of scale, where returns increase much faster than costs of expanding.
Regulation is hard to do correctly. The...
What's the difference between being part of physical reality and being part of a calculation that describes physical reality?
Seems like a silly discussion to me.
I covered much of the same material (but much more algebra, galois theory, representation, etc, and less analysis) in two freshman level classes this year. On top of that I took some set theory and some chaos theory. In that sense it's less than a year's work, but certainly an absurd amount for...
Lattice isomorphism theorem. Assuming those are normal subgroups, (G/K)/(K/H) is isomorphic to G/H.
Of course, it's not hard to just count the number of cosets. G = g_1K + ... + g_nK where n = [G:K]. K = k_1H + ... + k_mH, where m = [H:K]. Then G = g_1(k_1 + ... + k_m)H + ... +...
I know a guy who does statistical mechanics of the brain. Stat mech is a very mathematical branch of physics, you might find it interesting. Beyond that I don't know much.
Universal algebra is very much an aspect of model theory, so I'll just define the other two.
Model theory studies the abstract notion of a mathematical theory. With model theory you can study things like the consistency of certain statements, how to construct models of more complicated...
It's similar to a convolution, which has some nice properties, but other than that they usually need to be dealt with on a case by case basis. On the other hand if you wish to perform a contour integral you can get the residues of f(z)g(z) with the Laurent series for f(z) and g(z) in certain cases.
I suppose I also used numbers so it's a number theory proof. And I had to found it in ZF so it's a set theory proof. It might as well also be called a combinatorics proof since that's how Sylow's theorem is proven.
When asked for an analytic proof of the theorem, I would've supplied the complex...
It's a really trivial application of the IVT. Monic, odd polynomials are large and positive for large x and large and negative for small x, hence they cross the axis somewhere. I hardly consider this an analytic proof.