Should I Become a Mathematician?

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  • #1,651
tgt
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your question is hard. but i say, as in the nba, go with the love you feel for the topic. a phd is indeed a long hard road, so it is essential to be committed to your topic and to have a supportive advisor. i chose based on the attraction i felt for the material presented in lectures, and my ability to understand and connect with the advisor who taught the course. i still had to pass through more than one such experience before i found the maturity and commitment to carry through the job of completing a thesis.

ideally you should feel, this material is speaking to me, and this lecturer is speaking to me.
Is it true that Phds in other fields can be much less work? i.e I over herad a guy talking on the tram about his Phd which he only started one month ago and had already done 30,000 words. However he did have a lot of background knowledge prior to starting it. It was on the current Middle Eastern situation.

The word length is 100,000 words for a complete thesis? But how would you count the equations and symbols? Surely they would factor into the word count?
 
  • #1,652
mathwonk
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there is no word limit for a math thesis. some are only 30-40 pages, some are 300. riemann's entire lifes works comprise only about 400 pages.

the definition is something like "non trivial original work", and i have ehard of theses where "original" could mean a new proof of an old result, not necessarily a new result never proved before.

but it is very hard to do. one trick some people have used well is to find an old result from an earlier time, and clean it up, make the proof more solid, or add something to it.

others at the opposite extreme, take a very new result, and extend it or apply the ideas to a related but different situation.
 
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  • #1,653
tgt
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there is no word limit for a math thesis. some are only 30-40 pages, some are 300. riemann's entire lifes works comprise only about 400 pages.

the definition is something like "non trivial original work", and i have ehard of theses where "original" could mean a new proof of an old result, not necessarily a new result never proved before.

but it is very hard to do. one trick some people have used well is to find an old result from an earlier time, and clean it up, make the proof more solid, or add something to it.

others at the opposite extreme, take a very new result, and extend it or apply the ideas to a related but different situation.
So it would be a lot easier for geniuses. Didn't Grothendieck did the equivalent of 6 thesis by the time he was meant to earn his Phd. Nash's game theory was only 20 pages?
 
  • #1,655
mathwonk
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may i remind us all, that people like kontsevich are not in need of our advice here. most of us should not take him as our absolute model. if we do, we will likely not finish our degree in our lifetimes. it is fine to be inspired by such people, but it is more realistic and healthy not to judge ourselves against them.
 
  • #1,656
Oh, I agree. I was just continuing tgt's line of thought of amazing theses and doctoral students -- possibly informing him of one such person who he did not know about. What a bear it would be to consider Kontsevich as the model. But, I do celebrate his genius; "we" have so many of those!
 
  • #1,657
Well, once you all become mathematicians, could you please create Quaternion Analysis, (and hey, go for Cayley/Graves/Octonion Analysis if you are feeling really brave) because Complex Analysis is just not cutting it. Us folks really need you mathematicians to help us out on this one.
 
  • #1,658
mathwonk
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surely that is an old topic, already done? there is surely a lot of noncommutative analysis out there. if not, thanks for the suggestion/motivation. how about some details as to what is missing from complex analysis, and what the phenomena are that cry out for quaternionic analysis?
 
  • #1,659
surely that is an old topic, already done? there is surely a lot of noncommutative analysis out there. if not, thanks for the suggestion/motivation. how about some details as to what is missing from complex analysis, and what the phenomena are that cry out for quaternionic analysis?
Certainly there has been much work done on the Clifford algebras, the algebras in general, hypercomplex numbers, etc. but I have never really seen a single publication dedicated to quaternionic analysis as I have real and complex analysis. I wasn't aware of the term 'noncommutative analysis', which pretty much sums up what I was looking for, and reveals my ignorance. I suppose noncommutative analysis would pretty much cover everything I was interested in and more, so I'll look into it. I am a physics student and not a mathematician, so do please forgive my lack of awareness. Thanks!
 
  • #1,660
mathwonk
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i may have made up the term. but analysis on linear spaces applies to linear operators which are non commutative, and the term non commutative geometry refers I believe to mathematics which is essentially non commutative analysis. so if non commutative analysis returns few hits, try non commutative geometry.
 
  • #1,661
i may have made up the term. but analysis on linear spaces applies to linear operators which are non commutative, and the term non commutative geometry refers I believe to mathematics which is essentially non commutative analysis. so if non commutative analysis returns few hits, try non commutative geometry.
Noncommutative analysis and nonncommutative geometry both turned up quite a lot, though geometry much more so. Thanks.
 
  • #1,662
mathwonk
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the common idea is that a complex vector space is nothing but a real vector space plus a linear operator called J, such that J^2 = -Id. J of course is multiplication by i.

So one can imagine a quaternionic space as a real vector space plus a group of operators ±I,±J,±K,±L, such that I = identity, and J^2 = K^2 = L^2 = -Id.

etc....??? I.e. one asks for functions such that their linear approximations perhaps commute with action by this group of operators??? and then tries to understand them???


So analysis that carries a family of linear operators along is the topic.
 
  • #1,663
quasar987
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Hi mathwonk and others,

What is the Princeton companion to mathematics like? How relevant is it to an undergrad?, grad? researcher?

Is it worth buying, etc.
 
  • #1,664
the common idea is that a complex vector space is nothing but a real vector space plus a linear operator called J, such that J^2 = -Id. J of course is multiplication by i.

So one can imagine a quaternionic space as a real vector space plus a group of operators ±I,±J,±K,±L, such that I = identity, and J^2 = K^2 = L^2 = -Id.

etc....??? I.e. one asks for functions such that their linear approximations perhaps commute with action by this group of operators??? and then tries to understand them???


So analysis that carries a family of linear operators along is the topic.
Ahh. That is about as lucid a tie in from quaternions to vector space as I could ask for. I am going to actually write that down in my notebook and keep it in mind, as I am studying vector spaces now (Hermitian operators, pauli spin matrices, etc.) and keep wondering what the specific correlation would be.
 
  • #1,665
mathwonk
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of course for quaternions you know that also JK = L, KL = J, LJ = K, and KJ = -L, etc.....

and thank you for the kind remarks.
 
  • #1,666
of course for quaternions you know that also JK = L, KL = J, LJ = K, and KJ = -L, etc.....

and thank you for the kind remarks.
Yes. Thinking in terms of operators / vector spaces is really what is new to me. Now that I am starting to connect the dots, the vector space approach is starting to make more sense to me, which is good, because quantum mechanics seems to make explicit use of it.
 
  • #1,667
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I've been wondering about the scope of knowledge one can expect to obtain in such diverse subjects as maths and physics. Taking a JH degree in both with the intention of doing a PhD in mathematical/theoretical physics, there's great volumes of material from both subjects I won't formally study as an undergrad, particularly in pure maths. Do the researchers here find that in the course of their jobs they have opportunities to traverse "the road less travelled" and pick up stuff they may have missed as undergraduates? In part, I'm thinking about topics in pure maths. But I'm also thinking a lot right now about MSc courses and it strikes me that even in the most demanding courses on the market it's impossible to accquire a detailed body of knowledge that covers all of the areas I could see as potentially relevant to the sort of thing I hope to research. Given that a PhD is generally on a very specific topic, how much do you broaden your horizons once you start having to earn money? What opportunity is there to learn existent knowledge as well as contribute to it?
 
  • #1,668
mathwonk
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very little. learn as much as possible beforehand. teaching the same subject over and over makes it very hard to learn new subjects.

however early in my career i made it a rule to always have a learning seminar every year, going through some useful paper with interested friends and colleagues. i have not done it every year, but it was still very useful when i did so. just find someone who is willing to listen to you expound what you want to learn and go at it.
 
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  • #1,669
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I am interested in starting this discussion in imitation of Zappers fine forum on becoming a physicist, although i have no such clean cut advice to offer on becoming a mathematician. All I can say is I am one.
You're a mathmatician? With all due respect, why is your avatar a pikachu?
 
  • #1,670
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You're a mathmatician? With all due respect, why is your avatar a pikachu?
Is there a specific reason why you would expect a mathematician not to have a pikachu as his avatar?
 
  • #1,672
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Is there a specific reason why you would expect a mathematician not to have a pikachu as his avatar?
It's just very unexpected and surprising...
 
  • #1,673
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Joint honors, maybe?
 
  • #1,674
mathwonk
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i thought the pikachu was the patron saint of mathematicians. Is it not so?


But to be honest, from the limited choices of avatars here I first tried "the punisher", as a cool comic book character, and then I felt it might scare off students with questions, so I then chose a less threatening looking icon I had never seen before. It seems to be a pikachu, whatever that is.

so the idea was that a guy with rude answers should pretend to be nice at least in his icon.

thats my story and im sticking to it.
 
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  • #1,675
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I think JH stands for joint honours.
 

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