Should I Become a Mathematician?

[pivoxa15]

well maybe. it seems to me that key aspects of good teaching include conveying enthusiasm, sensing difficulties that the student has, and encouraging them, also providing role models. i do not see how a computer can do very well at those things.

when i was a student, and even a professional, an essential part of doing my work was having someone to tell it to. i think heard someone say about the great teacher r.l.moore, that students worked so hard for him because he was just so pleased by good work.

Moore's method looks very interesting. At my university all I do in maths and physics courses is copy what the lecturer writes on the board without understanding a thing which is depressing. And when I come around to do excercies or study for an exam, its like I am learning the material for the first time so at the end of the course I haven't learned as much as I should and the marks reflect that. It also rasies the issue that its like I am self teaching myself everything which makes me wonder whether I should enrol in these maths subjects. Would it be more effective to truly self learn it at my own pace? But I guess enrolling in a course forces me to learn the material and pushes me to another level which is good.

Why do you have the 2005 Maths guru badge I thought that was won by Matt?

 

[mathwonk]

i met penrose before the thread on index notation. i am not sure i would have been foolish enough to debate his notation with him, but it would have been nice to have him explain it to me. if i understood it better i might just possibly not be so opinionated about it.

matt and i arm wrestled for the guru badge, and he beat me so badly he gave it to me as a consolation prize.

 

[pivoxa15]

What about the 2006 maths guru badge?

 

[mathwonk]

anybody taking group theory? here is a very basic question a student just asked me today:

suppose you have a group G and a subgroup H of index n. Prove there must be a normal subgroup K contained in H, such that #(G/K) divides n!think "group actions".

 

[mathwonk]

here are some little number theory puzzles:"clock arithmetic" is arithmetic where 12 = 0, i.e. whenever you ad 12 to a number you get back to the same number.

arithmetic "modulo" p, is arithmetic where you get back to the same number whenever you add p to that number. Thus the complete set of different numbers is 0,1,2,3,4,...,p-1. e.g. modulo 5, 2+6 = 3+5 = 3. Thus two numbers (integers) a,b are equal modulo p if a-b is a multiple of p. e.g. 8 and 3 are equal modulo 5, and 81 and -1 are equal modulo 41.

interestingly, if p is a prime number of form 4n+1, such as 5 or 13, or 17, ...(there are infinitely many of them), then there is an integer x such that x^2 = -1, modulo p. e.g. 2^2 = -1, modulo 5, and 4^2 = -1, modulo 17, and 9^2 = -1 modulo 41.

this follows from another basic number theory fact - "wilson's theorem" -that the product of the numbers 1,2,3,...,(p-1) is equal to -1 modulo p, if p is prime.

can you deduce the fact that X^2 = -1 hasan integer solution modulo any prime p, from wilson's theorem?

try looking at the example modulo 5, where (1)(2)(3)(4) = (1)(2)(-2)(-1) = 4 = -1, modulo 5. or the example modulo 11, where (1)(2)(3)(4)(5)(6)(7)(8)(9)(10) = (1)(2)(3)(4)(5)(-5)(-4)(-3)(-2)(-1) = (120)(-120) = (-1)(1) = -1, modulo 11.can you saee why wilsons theorem, i.e. the fact that the product

(1)(2)(...)(p-1) = -1, modulo p, will lead to a solution of X^2 = -1, modulo p?This is not so easy, so give it a try, but you may not see it at once. I probably would not have done so as a student.

 

[mathwonk]

academic advice: take time off for xmas to be with loved ones, in fact or in spirit. If you cannot be there, call your mom and dad and tell them you love them. best wishes.

merry xmas!

 

[acm]

Wilson's theorem 1*2*3*...*(p-1) = -1 mod P
In case of p= 5
1*2*-2*-1 = 4 = -1 mod 5 , I'm calling the largest term X.
1*2*3...*X*-X*...*-1 = -1 mod P
If -X*...*-1 has 2n terms, then X^2 = -1 mod PFails if there is an even prime, This is my first brush with Number theory so be gentle.

 

[d_leet]

Fails if there is an even prime, This is my first brush with Number theory so be gentle.

No it doesn't since 2 is the only even prime, and it works for 2.

 

[Ki Man]

how can you have an even prime greater than two

 

[mathwonk]

well, first and last.

 

[Eivind]

Have anyone studied at Cambridge here? I'm planning to study mathematics there, and therefore I'd like to hear what students or former students of mathematics has to say about it. Positive and negatives, was it/is it challenging enough, environment, etc.

I'm currently at my first year in upper secondary school, in Norway, hence comments from people studying abroad will be appreciated.

 

[Ki Man]

by upper secondary school you mean high school right...

 

[Eivind]

Yup, I think so.

 

[matt grime]

Have anyone studied at Cambridge here? I'm planning to study mathematics there, and therefore I'd like to hear what students or former students of mathematics has to say about it. Positive and negatives, was it/is it challenging enough, environment, etc.

I'm currently at my first year in upper secondary school, in Norway, hence comments from people studying abroad will be appreciated.

One option that I noticed a lot of German students taking was an German undergrad degree and Part III. Perhaps something similar might be most appropriate for you too?

If you want to specfiy where you are in the system, then best is probably to give your age, school year, and the number of years of compulsory education in your country, and what age you'd start university at normally (this does vary from country to country - it is frequently 17 in Scotland for instance).

 

[Eivind]

Well, I'm currently in my 11th year of school, and has two more years to go before starting at a university, so I'm expecting to start when I'm 18-19. In Norway there are 10 years of compulsory education, and then three years at an upper secondary school, if you want to.

On the German students - they studied first in Germany and then went abroad, is that what you mean? I am not so familiar with the English terms about degrees, so bear over me.

 

[mathwonk]

A very highly recommended activity for budding mathematicians is to attend conferences in your area, even if you think they will be over your head. here is a calendar of upcoming ones from the AMS, one of which is at UGA, March 29-April 3, on abelian varieties.

http://www.ams.org/mathcal/

http://www.math.uga.edu/~valery/conf07/conf07.html

For people working in braid groups, take note of the one in Cortona, Italy held at the Pallazone, in May I think. This is the most fantastic location imaginable for a conference, as you can see from the photos.

If you might be in Paris in June, here is another fine one, on the occasion of the 60th birthday of Arnaud beauville:

http://www.ams.org/mathcal/info/2007_jun11-15_paris.html

 

[mathwonk]

and the one at the fields institute devoted to the work of spencer bloch looks great too.

 

[J77]

And for UK applied mathematicians - particularly students - the annual BAMC is being held in Bristol this year: http://web.enm.bris.ac.uk/bamc2007/

 

[mathwonk]

One of the names listed on the BAMS announcement is Phillip Naylor. I do not meet many Naylors, but apparently Little John was one, of Robin Hood's band!

 

[mathwonk]

The organizers of the UGA conference on abelian varieties in March specifically asked us to publicize it today, and invite graduate students in algebraic geometry to apply. The deadline is Feb 15, so please refer to the web address in post #402 above if you are an interested grad student or recent PhD.

Of course if you are in the area, you may find it feasible to fund your own visit. I guarantee it will be an impressive array of speakers. Phillip Griffiths for example is the Director of the Institute for Advanced study in Princeton, just to mention one of them

 

[JasonJo]

hey mathwonk i wanted some general advice on mathematics.

i am a junior mathematics student right now, and i am taking the next step up, a grad course in complex analysis and doing research in dynamical systems and hopefully attending an REU over the summer.

however, admittedly i am not as quick or as fast as some of my younger friends in college. i just switched to being a full time math major in Spring 06 and my first formal proof was less than a year ago. i came into college as an economics major.

what can i do to catch up and bridge the gap further? in terms of general techniques.

and how do you decide what to do your thesis on in grad school? i took courses in combintorics, graph theory, analysis, differential geometry, dynamical systems and abstract algebra, i just don't know how to decide.

 

[mathwonk]

as to how to progress even faster it is hard to say. I don't suppose any of my suggestions will really ring a bell. I myself decided to sleep less to have more time to study. I also ran 4 miles a day, more fitness, more energy.

how to decide what your thesis is on? Well I am one of the many whose advisor suggested a problem.

But the ideal case is that in your course of study, you learn to question what you are seeing, what would be true if? why do they not mention the non commutative case? what happens in higher dimensions? could i make progress on this problem if i simplify the situation appropriately?...

and then you find a question that entrances you, one that you care enough about to pursue through the long night of the thesis.

You have to love what you are doing to make it through the long hard work of researching it, that's what I found. I didn't love or have the talent for analysis, so I went back to my true love of algebraic geometry.

It originally attracted me in grad school because it was geometry hence close to the very intuitive subject of topology i had always liked, but involved algebra too, hence more challenging. I.e. I wanted something that played to my strengths but that did not lie completely within my comfort zone.

Of course I know now that topology is hard too, but i am just relating my youthful impressions as they occurred to me. (I realize now that it is by adapting the tools of algebraic topology that enabled algebraic geometry to be revolutionized.)

So think back on those courses you took and try to recall which ones you liked the best.

you might want to try topology and algebraic geometry too. what school are you at? what is available?

It was Alan Mayer at Brandeis, whose courses hooked me on algebraic geometry.

 

[mathwonk]

Matt? Want to toss in any savvy counsel here?

 

[pivoxa15]

as to how to progress even faster it is hard to say. I don't suppose any of my suggestions will really ring a bell. I myself decided to sleep less to have more time to study. So I became a vegetarian, less meat, less digestion time, less sleep. I also ran 4 miles a day, more fitness, more energy.

How long have you been a vegetarian for? Did you eat a lot of vegetables and carbs a day because running 4miles a day recquires a lot of energy. Did you find eating no meat had negative health effects? Dosen't meat help with brain function? Do you think digestion negatively impacts you during your study?

 

[mathwonk]

i found no downside to eating vegetables (and lentils for protein). It kept me thinner, but I had plenty of energy to run and work, and my brain seemed to function ok. That was the period when i did my PhD work.

Some of the smartest and most energetic people i ever met have been lifelong vegetarians. And i think vegetarians are generally healthier than meat eaters.

 

[JasonRox]

I no longer eat red meats, and that means only chicken.

I run a decent number of times. I have a group of friends where all plan on running some races in the summer, so hopefully we do well at running this term.

Being active and eating properly is the best choice anyone can make.

 

[JasonJo]

i applied to 5 REU programs, and I have the basics, really good mathematics gpa, really good undergrad gpa and good reccomendations. i don't think I'm going to get in.

so what is a mathematics student to do over the summer? there are no mathematics labs like physics or engineering students have that oppurtunity.

i was wondering if i should get an internship somewhere, but i don't mean like a Citigroup internship, but more like a RAND internship.

so if i don't get into an REU program, what should my backup plan be? mind you I am a pure mathematics students.

thanks

 

[Ki Man]

Job market as a mathematician expected to decline. Why?

 

[J77]

Most UK mathematicians I know love a real ale or two or three or several.

There's your key to fast progression...

Drink more beer!

(tho' not that carbonated stuff they have over in the US )

 

[DaBeags]

REUs

I was able to get into an REU without much more than what you've had. It's a great experience, and recommend it to anyone who wants to get a taste of math research.

 

[ianb]

Is Polya's How To Solve It book a suitable introduction to proof writing? I sort of just ordered it and I'm not sure I've made the right choice here.

 

[mathwonk]

It is an introduction to problem solving. It is worth having.

 

[pivoxa15]

i am no longer a vegetarian because preparing and cooking vegetables takes more time than broiling a steak.

i found no downside to eating vegetables (and lentils for protein). It kept me thinner, but I had plenty of energy to run and work, and my brain seemed to function ok. That was the period when i did my PhD work.

Some of the smartest and most energetic people i ever met have been lifelong vegetarians. And i think vegetarians are generally healthier than meat eaters.

I have tried being a vegetarian but what I found most discouraging is having to pick out the little bits of meat in foods like pizza and throw it away. I find that wasteful and unnatural. Cooking is also a bother.

I think I read that both Ramajuan, Newton and Hardy were vegetarians.

 

[ianb]

It is an introduction to problem solving. It is worth having.

Thanks for commenting. I have room for another book (Amazon gift certificate), and I'm thinking of "https://www.amazon.com/dp/038790459X/?tag=pfamazon01-20". I also have Rudin's book, which turned out to be an overly ambitious choice. As you can see, I'm a beginner, and I'm approaching abstract math for the first time. I'm hoping Polya's book would get me up to speed with all the proof writing/reading requisite. Am I on the right path here, or do you have other suggestions?

 

[mathwonk]

except for polya, in my opinion the hundreds of books on how to do mathematics are mostly worthless. maybe not entirely, since i recall learning a tiny amount from browsing some good ones back in high school (by Max Black?).

but mostly one just needs to practice reading and writing proofs. it helped me at the very beginning to have a little course in logic, from principles of mathematics, by allendoerfer and oakley. another good intro to logic and proofs is harold jacobs high school geometry book.

but these myriad books used in college courses with titles like introduction to higher math, or how to write proofs, or whatever, include some of the worst books we have to sort through as math professors.

the one i used last time, by vanstone and gilbert? wasn't as bad as most of them as i recall. i kind of liked it, but i don't recall it actually teaching you how to do proofs. it just had a nice selection of easy topics, was written decently, and was paperback hence hopefully not too exhorbitant.

I think i learned to do analysis proofs mostly from following a course on measure theory and functional analysis. The prof made it very clear how to do proof by contradiction, and use quantifiers, and negate statements, and choose epsilons and so on. But i do not think I learned any useful math in that course. I would much rather have had a more intuitive course in which i got a better feel for lebesgue measure and integration, than a formal course where i learned to manipulate epsilons.