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Fields of mathematics that expand the mind

Anonymous Tue 15 Feb 02:32:13 2022 No.14230542 View Reply Original Report

Quoted By: >>14230796 >>14230932 >>14231217 >>14233497 >>14233809

What fields of mathematics do you find force you to think for a long time even when trying to solve the simplest problems? For me, it's combinatorics. Related courses are one of the few times I've really ever had trouble studying mathematics

Anonymous

Anonymous Tue 15 Feb 2022 04:29:53 No.14230796 Report

Quoted By: >>14230811

>>14230542
for me its got to be those ring homomorphisms

Anonymous

Anonymous Tue 15 Feb 2022 04:34:57 No.14230811 Report

Quoted By:

>>14230796
interesting. i am only taking ring theory as a course now. let's hope it will be fun

Anonymous

Anonymous Tue 15 Feb 2022 05:39:02 No.14230928 Report

Quoted By:

Just wait until you see the Burnside ring.

Anonymous

Anonymous Tue 15 Feb 2022 05:40:57 No.14230932 Report

Quoted By: >>14231175

>>14230542
Functional Analysis.
Partial Differential Equations.
Linear Algebra.

Anonymous

Anonymous Tue 15 Feb 2022 07:52:47 No.14231175 Report

Quoted By:

I don't like math where you have to spend a lot of time thinking. I'm a lazy person. I prefer math which can be solved quickly with geometric intuition like for example >>14230932. But anything with algebra or combinatorics on the other hand I dislike very much.

Anonymous

Anonymous Tue 15 Feb 2022 08:15:12 No.14231217 Report

Quoted By: >>14233164 >>14233209

>>14230542
Probabilities. Easily the most confusing subject. Similarly, expectations, likelihoods, estimators, all of it. There's a lot of subtlety going on and the low level stuff tends to brute force you through concepts without letting you really breathe and comprehend what's being laid out imo.

There's a lot of unintuitive work in it that you just have to carefully work through and then check is correct.

Anonymous

Anonymous Tue 15 Feb 2022 08:26:52 No.14231234 Report

Quoted By: >>14231872

Number theory

Anonymous

Anonymous Tue 15 Feb 2022 15:45:23 No.14231872 Report

Quoted By:

>>14231234
based

Anonymous

Anonymous Tue 15 Feb 2022 15:55:07 No.14231896 Report

Quoted By: >>14233164 >>14233179

Homotopy groups of spheres. Once solved, humans will conquer the currently known universe.

Anonymous

Anonymous Tue 15 Feb 2022 22:29:10 No.14233164 Report

Quoted By:

>>14231217
yeah. monty hall problem is some shit. there's plenty of questions you can ask in probability that are tricky because your intuitions rub against it. currently im taking a course on probabilistic combinatorics and every result is non-constructive!

>>14231896
why do you say that anon?

Anonymous

Anonymous Tue 15 Feb 2022 22:35:21 No.14233179 Report

Quoted By:

>>14231896
Dude, you are literally a meme that Daniel Rubin makes fun of in his channel. Le homotopy groups of spheres guy.

Anonymous

Anonymous Tue 15 Feb 2022 22:42:16 No.14233199 Report

Quoted By:

Galois theory.
I will never forget the proof of the impossibility of angle trisection, even if I get Alzheimer when I grow old.

Anonymous

Anonymous Tue 15 Feb 2022 22:46:47 No.14233209 Report

Quoted By: >>14233236 >>14233484

>>14231217
probability is really easy when you know the definition of it

Anonymous

Anonymous Tue 15 Feb 2022 22:56:38 No.14233236 Report

Quoted By: >>14233244

>>14233209
What the def

Anonymous

Anonymous Tue 15 Feb 2022 22:59:24 No.14233244 Report

Quoted By: >>14233484

>>14233236
number of possibilities that meet the event's constraints divided by the number of equally llkely possibilities

Anonymous

Anonymous Wed 16 Feb 2022 00:29:51 No.14233484 Report

Quoted By: >>14233787

>>14233209
Yes, when you brute force through the math it is very clear, but your initial intuition is wrong most of the time.

It's the only time in math where I've had to actively fight against intuition. That's what makes it challenging. Most other math is pretty ease to breeze through on intuition and then refine it later. Probabilities are not like that at all.

>>14233244
The subtlety is figuring out all the possibilities of the numerator and denominator. Like the basic concept is simple, you're literally just looking at areas or counts restrained by other areas and counts, but getting those areas and counts can be unintuitive.

Anonymous

Anonymous Wed 16 Feb 2022 00:34:04 No.14233497 Report

Quoted By: >>14233505

I think it's when you see familiar definitions or objects used or extend in unexpected ways.
For example, the krull dimension. fourier analysis on groups, differentiation on banach spaces, generalized stokes, tangent spaces, analytic continuation, galois theory, central limit theorem, brownian motion, etc.

Anyway >>14230542 I think combinatorics requires a lot of seeing things in the right way many times. You stare down a problem forever and then you have a flash of insight. Invigorating but scary at the same time.

Anonymous

Anonymous Wed 16 Feb 2022 00:36:15 No.14233505 Report

Quoted By:

>>14233497
And after measure theory, fuck we put a measure on everything and integrate. Also, Zariski topology and the spec(R) construction.

Anonymous

Anonymous Wed 16 Feb 2022 01:48:24 No.14233732 Report

Quoted By: >>14233783

I think most of the basic theories are basic precisely because they expand your mind. Probability only seems counterintuitive because it deals with such seemingly concrete situations, like Monty Hall or Boy/Girl paradox, but the abstraction shows you ways to construct confusing results.

Calculus broadens your worldview when you first encounter it because it's everywhere in a way nothing you see before it really is. Obviously precalc is in everything, but you don't see it until you learn calculus and start using it.

That said, the fields that I think really changed the way I think at a base level? Combinatorics definitely required a new kind of thinking, but I saw it in high school and it set in pretty quick, the advanced problems are still hard to solve but the setting makes sense.

Real abstract algebra took a lot of getting used to for me, biggest takeaway I can pick out is how contextual objects are and that you can't rely on your intuition. An object can be anything depending on what context you put it in. This probably won't resonate because I'm autistic and have autistic little ways I explain things to myself, but realizing that objects are just objects and not vectors or group elements or numbers or polynomials until you put them in a set with that algebraic structure was a shift for me. Maybe I'm just retarded, but the first time I saw formal series in algebra and got told that we can't talk about convergence I was just confused about why we were talking about them at all.

Then measure theory solves some real-world paradoxes like the fucking turtle thing and how the probability of a single even can be 0 but that doesn't mean it's impossible, shit like that. Understanding what the continuum really is and how your human notion of distance actually relates to it was big. Other than that, real analysis didn't expand my mind, just taught me how to check the details of statements I already understood. Useful, but not mindblowing.

Anonymous

Anonymous Wed 16 Feb 2022 02:05:41 No.14233783 Report

Quoted By: >>14233787 >>14233798

>>14233732
probability is counterintutitive because it doesn't exist

the entire concept relies on the defitinion of an experiment, which has to be infinitely repeatable and at the same time have a pre-determined set of outcomes. Not even a coin toss follows that rule in real life.

Anonymous

Anonymous Wed 16 Feb 2022 02:06:59 No.14233787 Report

Quoted By:

>>14233783
and from that definition all you're really doing is choosing areas, picking parts of a big pie so to say, like this dude said
>>14233484

Anonymous

Anonymous Wed 16 Feb 2022 02:08:35 No.14233791 Report

Quoted By:

It's definitely combinatorics. Counting correctly is just plain hard. It filters a hilarious amount of mathematicians.

Anonymous

Anonymous Wed 16 Feb 2022 02:10:56 No.14233798 Report

Quoted By:

>>14233783
Nothing in math "exists", man, we're all fucking around with linguistics games here. Probability is just as real as measure theory, how you choose to interpret the information it provides you with will determine how intuitive you find it.

Anonymous

Anonymous Wed 16 Feb 2022 02:13:30 No.14233809 Report

Quoted By:

>>14230542
Convergence proofs, especially in measure theory, and also complex analysis and vector calculus in general.
I find combinatorics and algebra to be much easier.

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