Other Should I Become a Mathematician?

[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.

 

[proton]

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.

wow, that's pretty funny. I've always wondered why you liked pikachu. i thought it just had to do something with him being cute

 

[mathwonk]

yes, I've been at my school since 1977, and became a full prof about 1989. in those days we had no procedure for automatically bringing people up for consideration for promotion, and so it just happened whenever someone thought of it.

thus you could linger unnoticed for a while unless you complained or inquired as to what was going on. consequently a lot of people were left unpromoted for longer than they should have been. we have remedied that now, and everyone is given timely consideration, or at least the ability to remind us and demand it, every year.

by the way, so pikachu is just one character, like bilbo? not a whole race, like hobbits?

 

[mathwonk]

ok i found out they are not as harmless as appearance suggests, able to store electricity in their cheeks, and make lightning attacks. well that seems about right.

so they are not exactly schmoos, if you know what that is.

 

[Doctoress SD]

Hey, I am doing a Major in Physics, but since I got here I have been excited about Maths, like wow, I discovered Topology and I was hooked. I didn't know Topology, it is a huge area. I would also like to finish reading my book by Penrose.

I would have done a Maths minor, but my University doesn't offer Maths, it specialises in Chemistry and Maths was dropped due to the increase in "Micky Mouse Subjects".

So I just have to be happy with my own background reading.

SD

 

[Doctoress SD]

...

by the way, so pikachu is just one character, like bilbo? not a whole race, like hobbits?

You don't know that, yet you have Pikachu as your avatar? There are more than one Pikachu's by the way. But Ash's Pikachu is the most well known and famous. It kind of like having dogs which are animals, then you have breeds of Dogs, well you have Pokemon and Pikachu is a breed of a certain group of pokemon. Like you have rock, fire, electricity and water. Kinda like sets, Pikachu is a subset of electricity which is a subset of pokemon.
Only Ash's Pikachu is a subset of all the pikachu's.

I am a bit worried as to why I know this. They annoyed me so much, my little brother was obsessed with them.

 

[ValenceE]

guess you might have a closer relationship with your brother than you thought...



VE

 

[mathwonk]

wait, I am confused, fire, water, electricity, ash, pikachu,...

im lost, this is so much more complex than the singularity theory of the discriminant locus of principally polarized abelian varieties.and i cannot handle the idea of a university that dropped MATH because it was becoming so mickey mouse.i mean just WHAT is wrong with mickey mouse?>? if not for my comic book reading, (see mystery of man eater mountain), i would never have developed the creativity needed to do pure math research.

back me up here zapper.

 

[The|M|onster]

I will be taking my first course in Abstract Algebra this fall. The textbook that we will be using is:

Contemporary Abstract Algebra (6th ed) by Joseph Gallian

Does anyone have any experience with this text? Also, can anyone recommend a good text to use as a supplement?

 

[uman]

I have been reading Algebra by Michael Artin. I like it a lot although I haven't looked at any other algebra texts (except Dummit and Foote, which I wasn't prepared for) so I don't have much to compare it too, but I find he makes the material interesting and there are a lot of good exercises.

I'm only about halfway through the second chapter, so take my advice with a grain of salt, but from what I've seen so far it's a solid book. Try checking it out from the library.

 

[mathwonk]

good algebra books include:

birkhoff and maclane, shifrin, artin, dummitt and foote, jacobson, van der waerden, lang, hungerford.

some people like herstein, but i found it deceptively slick, but the problems are useful.

oh yes, and i wrote several which are free on my website, and also james milne has several free ones on his website, and also lee lady, and many other people.

http://us.geocities.com/alex_stef/mylist.html

 

[Doctoress SD]

I am learning Programming from scratch over summer. I will be using Tordran or something like that. And C and Python.

I have tried downloading compilers but they don't want to work. I think I am having most difficulty knowing what to do with the blank screan in which I am expected to write codes.

 

[DavidWhitbeck]

I am learning Programming from scratch over summer. I will be using Tordran or something like that. And C and Python.

I have tried downloading compilers but they don't want to work. I think I am having most difficulty knowing what to do with the blank screan in which I am expected to write codes.

Have you tried gcc (GNU C Compiler)? Regardless of your platform it should work. As for the blank sheet thing-- try hello world first. That is see if you can write a program that just prints out the line "Hello World!", that's the classic first program. Actually if you get a self study focused programming book it should have plenty of exercises for you to do.

 

[Quincy]

wait, I am confused, fire, water, electricity, ash, pikachu,...

im lost, this is so much more complex than the singularity theory of the discriminant locus of principally polarized abelian varieties.

I've always wanted to explain pokemon to a mathematician! (sarcasm intended)

In a fictional world, there are creatures called "pokemon," a lot like animals in the real world. Each of these "pokemon" have a certain "type." Three of these types are fire, water, and electricity. People in this fictional world collect pokemon and some have their pokemon battle other people's pokemon (cruel, yes i know, but the pokemon don't mind). They battle by having the pokemon use certain attacks against the opposing pokemon; all these attacks have names and there are hundreds of them. I won't go into the other battling mechanics, as they are too complicated, and involve math by the way.
Pikachu is one of these pokemon; it is of the electric type. In the animated show called Pokemon (english-dubbed from the japanese and original version of the animated show), the main character's name is Ash. He has a pikachu, making it the most famous pikachu, though there are indeed many other pikachus in the fictional pokemon world. Pokemon is in the top 5 of the longest running animated shows in the U.S.A. (9 years) and has the most episodes of any other animated show (509 episodes). Presumptively directed towards children, many pokemon episodes have been censored due to sexual, violent, and mature content.

 

[mathwonk]

this does remind me in a perverse way, of schmoos, conceived by al capp, no doubt well before your time. schmoos were bowling pin shaped animals which were extremely delicious when prepared in any of a wide variety of ways, and which enjoyed presenting themselves to hungry humans as pork chops, bacon, or any other variety of meat chops, all ready sliced for eating. check them out on google and see if their history survives.

yes in fact they have their own wikipedia article.

 

[mathwonk]

if you are asking yourself what schmoos and pikachu have to do with becoming a mathematician, remember my dictum that math is all about rampant creativity, at least before the hard technical part begins.

 

[The|M|onster]

I have been reading Algebra by Michael Artin. I like it a lot although I haven't looked at any other algebra texts (except Dummit and Foote, which I wasn't prepared for) so I don't have much to compare it too, but I find he makes the material interesting and there are a lot of good exercises.

I'm only about halfway through the second chapter, so take my advice with a grain of salt, but from what I've seen so far it's a solid book. Try checking it out from the library.

good algebra books include:

birkhoff and maclane, shifrin, artin, dummitt and foote, jacobson, van der waerden, lang, hungerford.

some people like herstein, but i found it deceptively slick, but the problems are useful.

oh yes, and i wrote several which are free on my website, and also james milne has several free ones on his website, and also lee lady, and many other people.

http://us.geocities.com/alex_stef/mylist.html

Thanks, guys. I checked out a copy of Artin from the library and started going through those links. There's some good stuff on that geocities page, Mathwonk.

 

[d1ff30m0rf1zm]

Mathwonk,

What book do you recommend for a basic course in Lebesgue integration? Currently I am using Lebesgue Integration on Euclidean Space by Frank Jones. Also, can you tell me what a course using this book [or your preferred book] would look like, i.e., topics by week? The syllabus outlined here: h ttp:// w ww.maths.no tt.ac.uk /personal/jff/G1CMIN/ seems to be roughly equivalent to the first 3 or 4 chapters of LIoES.

Thanks.

 

[mathwonk]

i'm sorry but that is one of the many topics i know next to nothing about, another being lie groups.

but do not despair, as i have many friends who are experts in analysis, and i will forward their recommendations.

one recent favorite text for that course is by wheeden and zygmund, zygmund being the famous classical analyst in that pairing.

another favorite for a long time is a text by royden, which i myself did not greatly like, but the first couple of chapters seem excellent, since he tries to take a hands on concrete approach, with simple, clear maxims for beginners. i would get it from the library and copy the first couple chapters, as to me the rest is abstract crapola. but who am i to judge?

of course all experts, but few students, like rudin. if you must choose rudin, and again i recommend going to the library for this, i suggest big rudin not baby rudin, since big rudin is a good book, with stuff you do not get everywhere, but little rudin has stuff you do get elsewhere, only it is harder to read it in baby rudin.

all books by george simmons are readable. i also like calculus of several vbls by wendell fleming, which includes lebesgue integration, a wonderful book.

i and experts seem to agree, that the book by riesz and nagy is excellent, but very old fashioned.

if you only want one recommendation, and you want MINE, knowing i am not an expert, i recommend fleming.

 

[mathwonk]

forgive me, i do not answer your second question as i have not taught it since 1968, when i used lang, analysis II, a very abstract book I do not recommend for a first course. i.e. i even taught the course but did not learn a lot myself.

of course 40 years later, after hearing an introductory talk by an expert i realized what lang was trying to tell me, and did appreciate it, so of course you always learn something, but it is hard to wait 40 years to find out what it was!

 

[d1ff30m0rf1zm]

i'm sorry but that is one of the many topics i know next to squat about, another being lie groups.

but do not despair, as i have many friends who are experts in analysis, and i will forward their recommendations.

one recent favorite text for that course is by wheeden and zygmund, zygmund being the famous classical analyst in that pairing.

another favorite for a long time is a text by royden, which i myself did not greatly like, but the first couple of chapters seem excellent, since he tries to take a hands on concrete approach, with simple, clear maxims for beginners. i would get it from the library and copy the first couple chapters, as to me the rest is abstract crapola. but who am i to judge?

of course all experts, but few students, like rudin. if you must choose rudin, and again i recommend going to the library for this, i suggest big rudin not baby rudin, since big rudin is a good book, with stuff you do not get everywhere, but little rudin has stuff you do get elsewhere, only it is harder to read it in baby rudin.

all books by george simmons are readable. i also like calculus of several vbls by wendell fleming, which includes lebesgue integration, a wonderful book.

i and experts seem to agree, that the book by riesz and nagy is excellent, but very old fashioned.

if you only want one recommendation, and you want MINE, knowing i am not an expert, i recommend fleming.

I remember reading Ch. 3 and some of Ch. 4 of Royden some time ago, but I didn't spend enough time on it to remember it well. I know the basics of measures from W. W. L. Chen's lecture notes. I should be able to summon Daddy Rudin, Royden, and Fleming using my dark magic without much difficulty [but it might be overkill to get all three]. I've been reading The Elements of Integration by Robert Bartle and I have really enjoyed it very much.

forgive me, i do not answer your second question as i have not taught it since 1968, when i used lang, analysis II, a very abstract book I do not recommend for a first course. i.e. i even taught the course but did not learn a lot myself.

of course 40 years later, after hearing an introductory talk by an expert i realized what lang was trying to tell me, and did appreciate it, so of course you always learn something, but it is hard to wait 40 years to find out what it was!

I see. That's fine.

I have another question. At the moment I am considering attending the HCSSiM program. However, a friend who attended tells me that it might end up being a waste of my time since a lot of the material there will be review for me [but I haven't studied graph theory]. She said its more of an introduction to proofs. Furthermore, its about $2300 [though there is financial aid] and six weeks long; that money can be used for my classes.

The dilemma is that my main goal in the next few months is to prepare for my classes at the university in the Fall. To that end I am spending, and must spend, quite a lot of time reviewing and preparing myself. HCSSiM only leaves 5 hours after classes, which isn't enough to prepare for 3 [maybe 4] rigorous classes. Another friend who has experience with such things suggests that going to a place like HCSSiM will give some kind of verification for my self-studying, and it'll set me up for getting in touch with professors. However, I have been lucky enough to have to been able to do that on my own. My uncle has been able to get me in touch with the head of the mathematics department at a very respectable university. I talked with him recently -- he was impressed, and he said he would talk to his colleagues and friends at Princeton [on the topic of mentors -- i.e., someone to verify my self-studying and serve as a mentor for [possibly] research in the future].

Do you think it would be beneficial to attend?

Thanks for your advice.

 

[mathwonk]

contact with smart people who are currently engaged in research is often very helpful. in the beginning of ones career, it is often advised.

 

[d1ff30m0rf1zm]

Indeed. So classes start in two months. Should I attend HCSSiM?

 

[mathwonk]

i cannot say, but if you attend, make sure you listen well.