Another math problem! (Not HW)

[Julian]

yeah, the guy who doesnt change clothes@3pm

oshi, how do you know it's me/who are you

[Max™]

Gerard ’t Hooft, Theoretical Physics as a Challenge

I don't expect to make money doing this, I do it for the satisfaction of knowledge itself, and because the hole in physics bugs the SHIT out of me.

Plus it's great seeing things in a deeper manner, you feel almost as if you're more connected to everything than you did before you started.


The other day I was thinking about modeling something like a path integral formulation, but with each observable history of the Universe taken as a sum of 3 dimensional "now" states related by proximity across a 4th dimension, and each possible history that could have emerged from the specific way the laws of physics emerged from that singularity (differing only on a coarse grained level, the finer differences would be subsumed by a single history, providing the uncertainty we observe) would form a bundle of histories related to any observed history by proximity across a 5th dimension.

Then I realized that if you took all of the subtly different ways in which physics could have emerged, they as well would form bundles related across that 5th dimension to the history bundle we find ourselves within, and that the sum of all possible bundles would be a 5 D hypersphere that emerged from the same singularity we did.

When I saw it in my mind I almost cried.

[Trajan]

Anyone have a mind = blown moment the first time they learned something new in math they feel like sharing? I figure math is full of those moments.

Mine would have to be the fact that you can manipulate the divergent infinite series 1 + 2 + 3 + 4 + ... to get a "sum" so that 1 +2 + 3 + 4 + ... = -1/12 (Ramanujan Sum). That one definitely blew my mind the first time I read about it. Ramanujan was, imo, the most underrated mathematician of the 20th century, easily one of the best.

[Max™]

*points at sig*

[Trajan]

Yea I know, Euler's Identity is beautiful.

[Kaylia]

Wtf, can anyone explain me why the right answer was cantor function? I seriously don't get it.

http://www.mcglaun.com/euler.gif

The 1 piss me off. e^i*pi = -1

physics is applied math, somehow i think if anyone's that interested in a subject so heavily rooted in math (which is something they at least like) they won't have any trouble with it, especially considering how proficient you seem to be at all of it woozie.

Theoretical physics...sure. There is the whole experimental aspect though.

[Eliseos]

Anyone have a mind = blown moment the first time they learned something new in math they feel like sharing? I figure math is full of those moments.

1/(sqrt Mu0 * Epsilon0) = 299792458 m/s

Maxwell is my hero.

[SDSD]

Anyone have a mind = blown moment the first time they learned something new in math they feel like sharing? I figure math is full of those moments.

The first time I fully understood graph theory as it's applied to my research in epigenetics.

(and when I got A's in my analysis classes)

[Woozie]

Spoiler: show

yeah I wasn't trying to make a connection. in fact, I'd say the proof you wanted is sort of a lemma necessary to prove the finitely generated theorem. I just brought it up as an interesting and related fact, and it shows that the classification Max brought up actually is relevant to your question (but with the implications reversed)

I'm currently (just a first year) phd math student. I would say to anyone who's interested in math/physics enough to consider doing a phd in them to remember that you can always study them as a hobby, and that there are some advantages to doing it that way (no grades or exams, it's much more fun when you know you're doing it for leisure, you get to pick the topics and the textbooks so you can choose books with the highest readability, etc). there are also numerous advantages to doing the phd-- chief among them accreditation for your knowledge. but one thing that is definitely not an advantage is employability. (edit: ok actually your employability will be marginally better with a phd degree,
but that margin alone is definitely not worth the amount of effort between master's and full phd)

there's a cold, hard truth, that academia trains orders of magnitude more people than it needs to replenish its own ranks. and a phd in a non-professional program mostly trains you to be an academic. I was naive when I applied to phd programs, thinking that going to a top 20 program would make me a good candidate for academic jobs. and the truth is I probably could find a job in academia after graduating, but it would mostly be teaching with little or no research and probably wouldn't be tenure track. I've decided to pick a specialization with employment prospects outside of academia, like numerical analysis (basically anything applied with lots of emphasis on using computers). so not only will I have job options outside of academia, but my options within academia will be better because of supply and demand.

a lot of people give similar warnings to this and then go on to say "you need to be ABSOLUTELY SURE about going in to academia before you do it, because it's SRSBSNS." but the truth is, if you get funding, it's not a bad lifestyle, and it does provide temporary shelter from getting a real job-- which won't happen anyway if you're graduating now. I'd say the only people who really need to take the decision seriously are people who have some other options that are really good, and which they might no longer have a few years from now if/when they drop out of a grad program.

and if you think it would make you bitter to work in a job that isn't related to your phd training you also might want to reconsider. because you can't expect that you will become a professor. I believe there's a bigger trimming between phds -> professorship than there is between bachelors -> grad school.

edit: also, even if you do become a professor, there's a very big likelihood that most of your time will be spent teaching stuff far below your level of training. very few professors work in top research institutions, and if you aren't doing that then most of your time will be spent teaching calculus or maybe even algebra or arithmetic. so even if you become a professor you are still highly unlikely to use your actual training as an academic or even teach classes related your subfield of specialization

Very informative post. I've been talking with professors and others about career related stuff a lot lately, and I've asked Neo about it in the LHC thread. Getting into theoretical math and physics seems virtually impossible. There was presentation on careers in physics, math, and other sciences at the APS meeting. The vast majority of people with a degree in physics or math will go into the industry. Every time I told someone there that I wanted to do theoretical physics, they were all like "Ohhhhhhhhh no, you don't want to do that. Trust me, you don't want to do that". After talking to a few people about it and going to a few presentations on it, I figured theoretical physics may be better off as a hobby.

So I decide that I'd rather be a theoretical mathematician than an applied physicist. So I started going to the math department and learning a lot about math careers. When I told them I wanted to go into theoretical math, they were all like "Ohhhhhhhhh no, you don't want to do that. Trust me, you don't want to do that". So yeah, I really need to figure out exactly what I want to do. I'm just trying to learn as much about the job markets and the actual jobs themselves for now before I figure out where to go. But I still think I'm more likely to pick physics than math, despite the fact that I'll be closer to a PhD in math than in physics.

Edit: And yeah, about the finitely generated groups: I must have misinterpreted your post. My bad.

[Woozie]

I cannot live without my TI-89 Titanium. I cannot do simple math to save my life, I generally only use it for plugging in numbers to make sure I didn't make some silly mistake (which often happens). The automatic unit conversions and the constants it has are badass, it even has the Rydberg Infinity Constant.

I'd die for my TI-Voyage 200. If my apartment was on fire, was laced with C4, and there was an earthquake, volcano, and tornado hitting at the same time, I'd still risk my life to run into my apartment and save my calculator.

[Woozie]

Holy crap, thanks Max. That site is awesome. Why haven't you ever posted that before? You know how many people here are teaching themselves physics, you should have posted that a long time ago lol. Any time I ask a professor what books I should study to get into theoretical physics, they say something along the lines of "how about applied physics?". I mean, I guess it's not really they're fault. If they don't study theoretical physics, they probably haven't read a variety of books on these subjects and thus can't recommend any to me. Though if I bring them the book, they'll often look through it for me and tell me what they think of it. But by then it's too late because I've already bought the book, so if it sucks I'm stuck with it.

Edit: Now that I think about it, there should be some sort of website or something dedicated to reviews of academic textbooks. A lot high level books don't have thorough reviews on sites like amazon and stuff.

[joft]

The 1 piss me off. e^i*pi = -1

the argument for writign it the way it usually appears is that it includes 5 of the most important numbers in math, e, i, pi, 0, 1, and also the operations of addition, exponentiation, and the idea of equality (the key example of an equivalence relation). if you write ti the other way you only get 4 numbers (no zero) and you also lose addition

re: Woozie about career stuff, there's always the argument of learnign it for your own enrichment and then taking a career in industry. you can prepare yourself for industry while getting the degree by learning programming languages. that's basically what i'm doing, and i'm staying open to the possibility that my eventual research may be successful enough that i might get a decent appointment in academia.

that said, grad school can be pretty demanding. even if you're a top student in undergrad it will probably take some adjusting. all of your classmates will be just as smart and motivated as you are, and many moreso. you'll have classmates from india and china who are far more prepared than yuo because their educational systems are better pre-graduate level. there will be a lot more homework than you've ever done before.

for me, the crux of the matter is that i'm being paid to learn. i could probably get paid a lot more doing other stuff, but i really like learnign anyway. i'll eventually be eligible for higher paying jobs once i either finish the phd or drop out with a masters, and even if my career ends up being mostly unrelated to my training it will still have been a good experience i believe.

[ozz]

oshi, how do you know it's me/who are you

I creeped on the RL pic thread...i'll say hi to you next week

[Julian]

Oh my god, are you seriously going to make me wait an entire week? I'm so incredibly curious now.

[Kaylia]

the argument for writign it the way it usually appears is that it includes 5 of the most important numbers in math, e, i, pi, 0, 1, and also the operations of addition, exponentiation, and the idea of equality (the key example of an equivalence relation). if you write ti the other way you only get 4 numbers (no zero) and you also lose addition

It makes some sort of sense, but it still feel articial to write it this way. You're summing a 2 dimensional number to a one dimensional number. The way I see it, it's like saying (-1,0) + 1 = 0, instead of (-1,0) + (1, 0) = (0,0). In the end, it's the same thing, but someone not familiar with complex number will miss the meaning of such equation. Our notation allow us to do this (because the 0i are implied), but a lot is lost imo. It's not impressive when you look at it from R² perspective, and it feels someone is attempting to trick us into thinking it's an amazing equation, when in fact, it's just the additive inverse in disguise.

If it was, exp(iPi) + exp(i*0) = 0, or exp(iPi), I wouldn't have any problem with it, but as it is, this equation feel wrong if you don't understand the implication under it. The "= 0" equations have a certain beauty when you can see the symmetry between both term, but in this case, it feels more like a definition (series) than anything.

[joft]

i guess it depends on your metaphysics but the math is legit. you can regard the complex numbers as a field in their own right (without thinking of them as a 2 dimensional real vector space) and "1" as being the multiplicative unit of that field, 0 the additive unit, etc. so you don't have to think of it as summing a 2-diml number to a 1-diml number, that's just one possible interpretation/construction of the complex numbers.

it makes the most sense topologically to think of the complex plane, but that is arguably a more complicated object (a topological field) than a plain old field without any topological considerations

for your entertainment: Local field - Wikipedia, the free encyclopedia

[ozz]

Oh my god, are you seriously going to make me wait an entire week? I'm so incredibly curious now.

umm...I'm really tall and I was the last person to come in yesterday?

[Max™]

Posted that t'Hooft site a few times in various threads, it's why I've been studying math almost exclusively basically for the last uh... I dunno, a while, since early this year I think?

The Euler identity is an example of how weird my math background is, totally accustomed to dealing with tensor spaces and Einstein Notation type stuff, yet I first saw the Euler formula like, a month ago?

Learned what I needed, the rest just kinda didn't occur to me, lol.

[Kaylia]

I'm not sure which edit of my post you saw, because I edited the wording 30 times, but I understand that.

If you know you're strictly dealing with complex number, and that every terms (including the 0 and 1) are complex number...then sure (it's still an ugly notation, but it's correct). However, I'm ready to bet most people think of this 0 and 1 as real number, not complex number. That's why i'm saying this equation is misleading.

[Julian]

umm...I'm really tall and I was the last person to come in yesterday?

You didn't happen to sit like, to the front-left of me, did you? <_<