Should I Become a Mathematician?

[mathwonk]

the best and the cheapest!

i myself also benefited from greenleaf's intro to complex variables.

most books had too much theory and not enough examples for me until greenleaf.

the point is to get familiar with power series. of course cartan spends the whole first part on them, but it is more proof oriented.

greenleaf also shows you how to calculate with specific ones until you feel more at home with them.

greenleaf is definitely intro however and cartan is authoritative and deep.

 

[ehrenfest]

Perhaps I should have posted this here:

https://www.physicsforums.com/showthread.php?t=233595

 

[MathematicalPhysicist]

Not sure about the best, but it does seem to be one of the cheapest.

I myself have bought markushevich's volumes, but haven't yet used it properly, I hope after I finish my undergraduate studies i will have time before graduate studies.

anyway, in my course, the lectrurer advised on the book by sardson, or something like this, but i didn't use it.

 

[tgt]

I asked a question previously which wasn't answered.
"Just like to know how to decide on a Phd area, let alone a Phd topic. Phd is a hard degree with 3 or 4 years so the decision is substantial. However some people may even choose a topic they know close to nothing of. What do you think? How to choose wisely?"

Another question based on doing a Phd is how to choose a supervisor? Is it the case that all the student need to think about is the area where he/she likes to work and need not think about whether he/she will get on with the supervisor? So it will be a bit of a gamble?

 

[mathwonk]

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.

 

[tgt]

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?

 

[mathwonk]

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 heard 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.

 

[tgt]

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?

 

[eastside00_99]

I have skimmed through this guys thesis: http://www.ams.org/featurecolumn/archive/kontsevich.html

It won him the fields metal and it was, I believe, only 20 pages.

 

[mathwonk]

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.

 

[eastside00_99]

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!

 

[Eric Belcastro]

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.

 

[mathwonk]

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?

 

[Eric Belcastro]

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!

 

[mathwonk]

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.

 

[Eric Belcastro]

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.

 

[mathwonk]

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.

 

[quasar987]

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.

 

[Eric Belcastro]

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.

 

[mathwonk]

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.

 

[Eric Belcastro]

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.

 

[muppet]

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?

 

[mathwonk]

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.

 

[Quincy]

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?

 

[ehrenfest]

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?

 

[ehrenfest]

Taking a JH degree

What is a JH degree?

 

[Quincy]

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...

 

[uman]

Joint honors, maybe?

 

[mathwonk]

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 I am sticking to it.

 

[matqkks]

I think JH stands for joint honours.

 

[tgt]

I just like to know how academics get promoted to full professor in the US.

 

[muppet]

Sorry all- JH does indeed stand for joint honours.

I just like to know how academics get promoted to full professor in the US.

I'm slightly hazy on what the distinction is in the UK between professors and anyone else, as I don't think we have the system of tenure here?

 

[mathwonk]

In the US, in math, one starts out after the PhD as either assistant professor, or more commonly now postdoctoral fellow for about 1-3 years.

Then a "tenure track" job hopefully follows as assistant professor. One pursues ones research, practices ones teaching, and after 4-6 years of publishing and establishing a beginning reputation in ones area, one may be promoted to associate professor.

the requirements are roughly the clear sign of emerging excellence in research, and likelihood of, or realization of, national stature as an expert. this is judged based on publications, grants, and letters of reference from known experts.

Then after say 4-6 more years, (but it can be more, or rarely fewer), if one has given evidence of sufficient stature in ones field, preferably on the international level, as evidenced by reviews of publications, letters from expert referees not closely associated with the candidate as friends, one may be promoted to full professor.

the quality of ones teaching should also be excellent, or at least adequate, or that alone can be cause for failure to promote.

The research often tends to receive greater weight, probably since research can bring in grant money. But teaching also matters to students and their parents as well as colleagues, and people also take teaching seriously.

Of course it is less clearly agreed how to evaluate teaching than research. Some people look only at student evaluations, but these can be influenced by factors such as making the course too easy, or giving higher than average, or lower than average, grades. In reading evaluations, one should look for statements that the teacher was "challenging", as well as helpful, but these are not that common. some students comment even on the clothing of professors, or think that a professor is unprepared who does not use notes, when the opposite is often the case.

thus classroom visits and examination of teaching materials by colleagues are also used, as is publication of textbooks, acceptances of such books, and reviews.

tenure is usually granted about the same time as the associate professorship, and should indicate convincing evidence that the candidate is someone who will be a desirable member of the department for life, and in particular who will achieve full professor.

professors who achieve unusual stature in research or as teachers may receive further special chairs or professorships. at a place like harvard, most professors may be chaired ones, while at a state school there may be only one or two if any in a given department.

 

[Darkiekurdo]

Mathwonk, out of curiosity: are you a full professor?

 

[DeadWolfe]

http://www.math.uga.edu/dept_members/faculty.html

Appears so.