Math Thread

[hopeandjoy]

I'm going to fail the AP Calc test and have to take Calc 1 in college.

Sometimes I wish I wasn't majoring in a science.

 

[Music Dragon]

I guess I never really introduced myself in my previous post. I'm currently studying engineering physics, though my master's degree is going to be in mathematics (which, I realized after three years, is in many ways more enjoyable than physics). Right now I'm working on my bachelor's thesis in combinatorial game theory, and I am very, very stuck. (Game theory is great for studying disjunctive sums of games, but the game I'm studying never seems to break down into sums...) Hopefully I'll have something to show for it by the end of this semester, but I'm pretty stressed about it.

I'm aiming for a teaching career, so I do some tutoring as well. Right now I'm teaching the first-year students multivariable calculus (and learning a lot as I go). Tomorrow I'm going to introduce them to the method of Lagrange multipliers. How exciting!

 

[Murkrow]

This screwed me up so much in one of my exams last year. It's even worse when there's a computation that looks really horrific, so I'm split between

• doing the computation
• looking for a clever way to simplify the computation
• moving on to a different question

On the bright side, it's less demoralizing than staring at a question where I have no idea how to even start.

Oh um I'm in my second year of studying (mostly applied) maths at a university and talking about maths on the internet is another one of my hobbies for which my interest is much greater than my skill.

Something cool I learned recently is that the number of riffle shuffles needed to nearly randomize a deck of 52 cards is about 7. Between 5 and 7 shuffles, the distribution of the order of the deck suddenly changes to be very close to discrete uniform (that is, each order of the deck is roughly equally likely). This sudden change is known as the 'cutoff phenomenon' in Markov Chains if anyone is interested. The result for riffle shuffles was proved in the 1980s by David Aldous and Persi Diaconis and I'm told that the proof is quite elegant. I think it's pretty cool that the solution to a real world problem that is quite easy to understand (how many times do I have to shuffle) involves a surprising result and a nice proof. It's like the convergence of everything I love about maths.
Here's a link if you want to know more: http://www.ams.org/samplings/feature-column/fcarc-shuffle

On the other hand, I do like proof questions that gives you something to work to. "Prove that X is true"
Though that does scare me away from going into academia. If I'm asked to prove something in a homework I assume the question isn't incorrect and so know that it is true. Even if I can't prove it myself I'm at least sure a proof exists. I can't imagine how hard it is to be an actual mathematician who comes up with new proofs. I can imagine trying to prove something for ages only for it to end up being false in the end.


I have a bit of an interest in Markov processes. They seem to have a lot of applications. I can't study it at university since I don't have the prerequisites (I took a bunch of pure modules instead of stats), but I do occasionally watch some lectures on them that MIT put on youtube.

GUYS
This is one of the most fun and educational Numberphile videos ever. At least in my opinion! Few things have blown my personal mind like back when I learned and properly started to grasp how stupidly simple computers are and how all this near-omnipotent complexity we're working with every day is built on top of the most inane things in layers upon layers of increasing abstraction. It's amazing. And this video demonstrates this fact beautifully... using dominoes.

I'd definitely do something like that if I had lots of dominoes and patience. There's probably lots of other zany ways to to computation as well, though I can't think of any myself.
People seem to love making pointless but impressive machines in Minecraft, doing it with dominoes should be a thing since it's much more difficult, slow, and can't be done by writing a computer program to do it for you.
Also I have some dominoes that can attach to a "track" of sorts. As in, they can fall over as usual but they can't slide anywhere, using something like that could prevent accidental knocking over bits you don't want moving. Though if they wanted Guinness involved using something like that might have been considered cheating.

I guess I never really introduced myself in my previous post. I'm currently studying engineering physics, though my master's degree is going to be in mathematics (which, I realized after three years, is in many ways more enjoyable than physics). Right now I'm working on my bachelor's thesis in combinatorial game theory, and I am very, very stuck. (Game theory is great for studying disjunctive sums of games, but the game I'm studying never seems to break down into sums...) Hopefully I'll have something to show for it by the end of this semester, but I'm pretty stressed about it.

What about maths do you like over physics? I always sort of regret not doing more physics!
Though I also get the same feelings relating to computer science so in the end maths is a bit of a middle ground that I'm glad to be studying after all.


I did a course on game theory last semester. It was interesting but it wasn't quite my thing. I've got to wonder though how you come up with payoffs? We learned lots of methods to work out who things like what player has more sway or something like that but a lot of questions we were given provided payoffs for us to use. I'd imagine there are some situations where working out what the payoffs are would need you to measure something not easily quantifiable?
(I'm probably talking nonsense and sounding stupid to you, aren't I?)





I have to chose what subjects I want to learn next year now. I have a vague idea here and there but it's going to take a while for me to decide entirely. The annoying thing about looking through descriptions for courses is that if I don't know the words it uses it makes me think it's too hard for me. But I don't know them because I haven't learned them yet! When I do know the words I assume they're too easy. I don't know what I could do!
It might be easier if I had any idea at all what I want to do when I grow up.
The kind of maths I enjoy constantly changes too. One day I prefer algebra, then it's topology, then it's number theory. So far the only thing I've decided I'm definitely going to do next year is cryptography.



EDIT: Also, another maths thing I want to say. I know the proofs and I understand them, but no matter how hard I try I just can't get my head around the fact that:
1) The number of real numbers is greater than the number of rational numbers
2) The rational numbers are dense in the real numbers
At the same time. Both individually make sense to me. Putting them together make my head hurt.

 

[Mustardear]

On the other hand, I do like proof questions that gives you something to work to. "Prove that X is true"
Though that does scare me away from going into academia. If I'm asked to prove something in a homework I assume the question isn't incorrect and so know that it is true. Even if I can't prove it myself I'm at least sure a proof exists. I can't imagine how hard it is to be an actual mathematician who comes up with new proofs. I can imagine trying to prove something for ages only for it to end up being false in the end.

Reminds me of a quote from Julia Robinson, which perhaps you've heard, when asked to describe what she does during the day:
"Monday: Try to prove theorem
Tuesday: Try to prove theorem
Wednesday:Try to prove theorem
Thursday: Try to prove theorem
Friday: Theorem false"

 

[Music Dragon]

On the other hand, I do like proof questions that gives you something to work to. "Prove that X is true"
Though that does scare me away from going into academia. If I'm asked to prove something in a homework I assume the question isn't incorrect and so know that it is true. Even if I can't prove it myself I'm at least sure a proof exists. I can't imagine how hard it is to be an actual mathematician who comes up with new proofs. I can imagine trying to prove something for ages only for it to end up being false in the end.

Don't let that put you off! Most people seem to believe that being a mathematician means being really really smart until poof, theorems. But the truth is that it doesn't take a sudden flash of inspired genius to produce results. You can actually tackle the unknown in a somewhat systematic fashion. After analyzing a problem for a while, you might come up with a super-interesting conjecture that you're unable to prove, or you might be able to prove something that seems rather trivial and insignificant, or - as you said - you might spend ages trying to prove something that turns out to be false in the end. But all of that is still progress! This is true of the advancement of any field of study, I think. You don't have to just wake up one day and invent Galois theory. These things happen in small steps, and with lots of collaboration. Anything you discover, anything at all, is a step in the right direction.

What about maths do you like over physics? I always sort of regret not doing more physics!
Though I also get the same feelings relating to computer science so in the end maths is a bit of a middle ground that I'm glad to be studying after all.

I don't rightly know! I guess I just started noticing after a while that I was skimming over all the physics courses because I was so engrossed in mathematics. They're both fascinating areas of study, really. Physics is a description of the natural world that we live in, and that's certainly worthy, but I think mathematics goes beyond that. We use mathematics as a tool to solve and understand physical problems, but you can also extend it to things that don't have to exist in reality. It is an art of abstraction and generalization.

I did a course on game theory last semester. It was interesting but it wasn't quite my thing. I've got to wonder though how you come up with payoffs? We learned lots of methods to work out who things like what player has more sway or something like that but a lot of questions we were given provided payoffs for us to use. I'd imagine there are some situations where working out what the payoffs are would need you to measure something not easily quantifiable?
(I'm probably talking nonsense and sounding stupid to you, aren't I?)

No, not at all. But I think you're talking about classical game theory, as opposed to combinatorial game theory! The classical theory is too close to economics for me, really. I'm always put off by the thought of applying mathematics to practical things...