ask someone with a phd in math anything >phd in math 300k

give me a quick rundown on your thesis

For those of us who have a nagging feeling they like math but were too fucked up in HS to get it, can you do a brief escalating explanation of the diff levels of maths, from arithmetic through algebra, calc, geometry etc up to the highest level? That would give me/us a conceptual framework for what has always been the equivalent of a jumbled mess of Chinese. I'd like to know what they all are and how they relate and build off each other bc I was farting in class when I was supposed to be learning.

Thanks man.

>>8727895
lol lionpath

>>8727895

I like that the picture implies that your Ph.D. has been boiled down, in this one pic, into an all-important web 1.0 graphic "ticked this box, ticked that box", and now it's filed away forever in the appropriate institution as a demonstrable fact: "yes, person x received x degree from this institution as a consequence of doing abcde".

The picture is personally relevant in another way. /sci/ will laugh but I fucked up part of undergrad, and then learned enough grit to go back and finish. I can remember getting the final grades: I felt a great relief, and delayed gratification at having properly completed undergrad. And the screen which proved it to me, was a screen very much like the OP's screen.

Anyway, standard questions OP, what did you study and talk about, how did they initiate your ass, was there champagne etc.

>>8727932
algebraic number theory <- elementary number theory -> analytic number theory

linear algebra over fields (vector spaces) -> linear algebra over rings (modules)
commutative algebra <- rings -> fields -> galois theory <- groups
homological algebra

calculus on manifolds <- multivariable calculus <- integral and differential calculus -> real analysis -> complex analysis

lots of different geometries: euclidean geometry, projective geometry, hyperbolic geometry

geometry + calculus = differential geometry

geometry + number theory = arithmetic geometry

geometry + algebra = algebraic geometry

discrete math: combinatorics/graph theory

statistics/probability

the list goes on...

>>8727949
probably shouldn't leave out

differential topology <- topology -> algebraic topology

>>8727932

I'm not the OP but this is kind of a "cute" post by /sci/ standards. I'm going to take it at face value and attempt a response though.

Basic math includes things like arithmetic, basic ideas in geometry, and various tricks about these things. They can be combined to do patterens and stuff!

then there's a step up. some slightly harder/ more abstract ideas. think about triangles and circles. Trigonometry. Rules for symbol manipulation and why they work, algebra, basic algebra. Think about how number systems actually work, and what makes certain numbers different from one another, in several different ways. Eventually there comes logic and the idea of, starting from some basic premises, you are forced into some other statements. This is (American) high-school level stuff, and calculus can be taught around this point. The thing about learning calculus is that you have to be able to keep track of 5-6 things at a time, which is something that most kids just can't do. This is what makes calculus a "basic adulthood" activity that many adults struggle with.

Then comes the rest of undergrad math, more generalization, more abstraction, and beyond this, more of the same, but even more elaborate. I will not endeavor to comment further on the latter.

Basically, one idea is that math is "getting" (meaning: the stuff that we can discover and know) so complex that humans can't be relied upon to do it any more. This is a serious idea that has some weight to it - certain proofs have been carried out by computer over the past few decades, and are either too tedious or strange enough as to not be understood immediately.

>>8727962
>This is a serious idea that has some weight to it - certain proofs have been carried out by computer over the past few decades, and are either too tedious or strange enough as to not be understood immediately.
this is somewhat misleading

long and massive calculations are carried out by computers but the proofs based on those calculations are still completely put together by humans, no computer is generating new proofs of things as far as i know, and only relatively elementary mathematical proofs have been successfully checked within a theorem prover (i.e. Coq)

>>8727944
i can't believe i read all that drivel.

>>8727938
nah many universities use the same thing. it's by PeopleSoft