Should I Become a Mathematician?

[matqkks]

There are various definitions for mathematics and mathematicians. For example:
Paul Erdos said mathematics is a machine which converts coffee into theorems and proofs.
Lord Kelvin talking to his engineering students at Cambridge asked the question 'whom do you call a mathematician'. Like most lectures he goes on to answer, 'A mathematician is a person who finds
int(exp(-x²))dx between the limits +infinity and -infinity is equal to square_root(pi)
as obvious as you find 2x2=4'.
Another definition for mathematics is 'science of patterns' and a mathematicians is someone who is a pattern searcher.
Remember Plato had written on one of his archways 'Let no man ignorant of geometry enter here'.
I am sure many of you have your own definitions.

 

[qspeechc]

I found this site with some free e-books.

http://www.math.uiuc.edu/~r-ash/

 

[Tom86]

Thanks for that. Looks like some good stuff there...I'm going to have a read through the Pari tutorial later.

This has probably been posted before but those more algebraicly inclined may find this link useful:
http://www.jmilne.org/math/index.html

 

[elfboy]

if you want to become proficient at math prepare to spend atleast two hours a day deriving stuff and exepect to become frustrated. It also helps to have mathematica but don't rely on it as a crutch.

 

[CoCoA]

I finished most of Fulton's book Algebraic curves and did about half of the exercises, except I did quite get his presentation of resolution of singularities. Any suggestions on materials for that?

 

[arshavin]

I am curious if other people have same issues, on desire to do maths. In my case, motivation to study fluctuates alot, on some days I have intense interest and can work for hours. Then there are times where i can't be bothered to do anything, even when i know the stuff is supposed to be interesting. I'm in undergrad, so this means my coursework is very inconsistent

 

[mathwonk]

coca you might try walker's book for resolution of singularities. or i could send some notes, or put them on my website.

 

[mathwonk]

the basic idea for resolving singularities, is to look at a curve that resembles the union of the x and y axes, hence has a "singularity" at the origin, because there are two "branches" passing through one point, and separate those two branches so they no longer intersect there.

Riemann just reached into the plane and lifted the two branches out and replaced the origin by two points, getting an abstract curve that did not cross itself.

more later

 

[thrill3rnit3]

mr. mathwonk

is What Is Mathematics? An Elementary Approach to Ideas and Methods by Courant

a good book? have you taken a look at it?

 

[mathwonk]

it is perhaps the best book on math for non mathematicians. i have a copy and i think it is excellent. read it and learn from a real master.

 

[CoCoA]

Mathwonk,

Thank you for your explanations on resolving singularities. Fulton's graphics don't do it justice (at least in the new PDF, I don't know about the original), but I found some nice graphics on the Internet such as:
http://www.math.rutgers.edu/courses/535/535-f02/Movie5.html
http://www.math.purdue.edu/~dvb/algeom.html

But I have 3 questions:
Fulton first gives an affine blow-up, then a projective blow-up of multiple points. Is the affine case actually used, or is it just a segue into what is really done in projective space? And are multiple singularities really resolved all at once? I feel the blow-up of multiple points at once may be difficult to algorithmize. And lastly, is the topic of quadratic transformations used in practice? I am willing to acknowledge its plusses and minuses, but actually understanding it is giving me a headache.

 

[mathwonk]

resolving singularities is perhaps best understood by imagining how the singularities arose.

if you have a curve in space, and you project it into the plane from a point, any two points of the curve which lie on a common line through the center of the projection, will go to the same point in the plane.

thus to desingularize the plane curve you want to revrse the process, replacing the one point by the original two points, or replacing the collapsed image of the line through the center of projection, by the line.

so we get the process of "blowing up" singularities, or replacing a single point by a line. this is most naturally and easily done by returning to a higher dimension, so as I recall fulton defines blowing up abstractly, in a product space, then re embeds the object into projective space.
but if one wants to remain in the plane, then one cannot raise the dimension so must resort to blowing up some points and then blowing down also some lines, just so the final result will still be in a plane.

of course it is impossible to desingularize most plane curves and have the non singular version also be a plane curve, so in that setting, where a plane curve is wanted, we settle for reducing the complexity of the singularities, obtaining a plane curve birational to the original one, but with only singular points that look (infinitesimally) like the intersection of a family of lines with different slopes passing through the point.

so there are many somewhat inessential elements to resolution of singularities that are there in order to remain in a certain category, i.e. algebraic varieties, or projective varieties, or plane curves.

quadratic transformations are essential if you want to do everything in the plane. such a transform is the composition of three point blowups and three line blowdowns.

of course this complicates the nature of the process, at least abstractly, but quadratic transforms are very concrete and explicit, i.e. in some coordinates, just (yz, xz, xy). this map collapses the line x = 0 to the point (1,0,0) e.g.

note also that repeating this transform gives (x^2yz, y^2 xz, z^2 xy) = (x,y,z), so the transform is self inverse. This means that not only does the line x=0 all map to the point (1,0,0), but conversely, the point (1,0,0) maps somehow to the whole line x=0.

what this means is that if a curve passes through (1,0,0), say the point p of the curve is there, then the transform of that curve will have point p somewhere on the line x=0 but it could be anywhere. It depends on the position of the tangent line of the curve at p. I.e. two curves both passing through (1,0,0) hence intersecting there, but having different tangent lines there, will no longer intersect after this transform is performed, their points which did correspond to (1,0,0) will be at different points of the line x=0.

the resolution process is local, hence the affine version contains its essence, but one wants to work also on projective curves so the process is projectivized.

riemanns version is merely to yank the whole curve out of the plane, compactify its smooth points as a compact manifold, then re embed the smooth version back into the projective space. there is also an algebraic version of that process, due to zariski, called normalization of the curve, which desingularizes it in one stroke.

just take the coordinate ring of the curve and pass to its integral closure. bingo, the associated curve whose coordinate ring is that integral closure, is non singular, and birational to the original curve.

this is kind of a long story, and i don't have time to teach a whole course in desingularization of curves here and now. hang in there, it will become clearer. i myself benefited from the concrete treatment in walker via quadratic transforms, and i have also written up this story, but my notes are also quite lengthy, and not yet posted online.

it is interesting that the process of projecting curves down from a higher dimensional space to a lower one, can not only introduce singularities, but also remove them! the difference is whether the center point of the projection lies on the curve or not. I.e. projecting from a singular point of the curve can reduce the complexity of that singularity.

this is explained in joe harris' book on algebraic geometry, a first course.

in fact a quadratic transform does this. the map above given by (yz,xz,xy), can be viewed as first mapping from the plane to P^5 by the functions (yz,xz,xy, x^2,y^2,z^2), and then projecting down to P^2 by omitting the last three coordinates.

the point (1,0,0) maps first to (0,0,0,1,0,0) which lies on the center of projection (we project successively from (0,0,0,0,0,1), then (0,0,0,0,1), then (0,0,0,1)), hence this point is "blown" up by the process.

 

[mathwonk]

the process is geometrically quite simple: just form the product of the affine plane with its tangent plane. then map the non singular points of the curve C into that product by sending a point q to the pair (q, unit tangent vector to C at q). this map is not well defined at p if there are more than one tangent direction at p, but we can take the closure of the image, obtaining a curve that may have more points corresponding to p than before. i.e. distinct "branches" of the curve at p, where the curve has distinct tangents become separated by blowing up. eventually this desingularizes the curve.

 

[qspeechc]

More e-books.
http://www.math.harvard.edu/~shlomo/

 

[mathwonk]

cocoa, well? does that help at all?

 

[maze]

mr. mathwonk

is What Is Mathematics? An Elementary Approach to Ideas and Methods by Courant

a good book? have you taken a look at it?

it is perhaps the best book on math for non mathematicians. i have a copy and i think it is excellent. read it and learn from a real master.

I second this. Even if you know most of the stuff in this book, it is still very worthwhile to read. Quality stuff right there.

 

[mathwonk]

there are few people who know all of that material.

 

[maze]

there are few people who know all of that material.

Yeah definitely. But even if you are a baller mathematician and already know all the material in Courant's book, it would still be a worthwhile read. I would say a good analogy is the Fenyman lectures in physics, which are enlightening for both the freshman physics major and professional physicist alike.

 

[qspeechc]

Courant is a great teacher right? Courant and John say in the preface to "Introduction to Calculus and Analysis":

"Mathematics presented as a closed, linearly ordered, system of truths without reference to origin and purpose has its charm and satisfies a philosophical need. But the attitude of introverted science is unsuitable for students who seek intellectual intellectual independence rather than indoctrination; disregard for applications and intuition leads to isolation and atrophy of mathematics. It is extremely importat that students and instructors should be proected from smug purism."

 

[PhysicalAnomaly]

Hey,

I'm currently doing a chem eng, maths, physics double degree and complete with honours in maths, it'd take me 6 years. I'm wondering if age matters in the field of mathematics. Most of the accomplished mathematicians seem to have PhD's well under their belt by the age of 25.

I like my engineering studies both for marketability and because I get to learn very applied areas of maths and science to a good depth. However, I'm worried that the extra 2 years it'd be a disadvantge in terms of a successful academic career.

I'm wondering if I should drop the engineering degree and just do science. The degree including the honours year would only be 4 years and I'd get to study more maths courses. Then I could get a PhD sooner and all that.

Also, I'm a bit confused by the US system. I'm from Australia and over here, a PhD can be undertaken right after a bachelors degree with an honours year. But I've heard that in the US, I'd need to do a masters before a PhD. True? How long would the bachelor's, master's and PhD take? And what are the requirements for postgrad? If I have done a research year in engineering but not in maths (only a major or double major in maths), what are the chances that I'd be allowed in?

 

[mathwonk]

i myself spent 5 years getting out of college, got a masters, taught for several years, re entered grad school, leaving at last with a phd in my 30's. i found that grad schools were happy to have talent where they could find it.

I was given a little stricter set of rules the second time, three years to finish or else. It was difficult to do, but i managed with a good advisor.

In general the masters is not prerequisite to a phd, but an alternative, i.e. usually you enroll in one or the other program. Sometimes phd students take a masters along the way, as non - thesis masters requirements are often a subset of phd requirements. then they have something to show if they do not finish the phd.

 

[PhysicalAnomaly]

It's a big relief to know that if I applied to the US to do a PhD, I won't need a masters first. That shortens it from 5 to 3 years.

And also good to know that they'll accept students that might not have made the best choices in terms of courses or majors selected during undergraduate study.

I wonder why you goofed off? Was the material too easy or too slow? No inspiring tutors? The courses at my uni treat the students as if they were little kids and absorb information at a vegetative rate. Fortunately, I have some tutors who are inspiring and also give some form of push in the right direction for students who aren't content with the pace.

I'm also still wondering how age affects academic mathematics careers. Looking at history, surely it can be no coincidence that the most prolific and acomplished mathematicians had early starts. Have you faced any hurdles that you probably wouldn't have if you started early? Problems with positions at unis and the like. And in terms of research. I've heard that the 20-30 year old period is when they produce the most important research of their lives and the rest is relegated to teaching. How true would you say this is?

Oh and one last curiosity. Who're you descended from mathematically? XD

 

[mathwonk]

i was very disoriented at harvard college, coming from a weak high school background in the south. i had hardly studied at all and had still gotten straight A's, so harvard was a big shock.

(in school, and probably even in most sciences, it is more useful to have a medium IQ and good work habits than a higher IQ and poor habits.) so i had to work for a while before finishing.

so i was older when i started my research career, and age gets you eventually because you just get tired sooner. and also it is hard to stay motivated if you do too much desultory work to earn a living. try to keep contact with stimulating people, but when you get old you may still get sick and lose energy to work very hard.

In january i am going to MSRI in berkeley for a week or two, to listen to the young people talk about what they are doing and have done. hopefully it will stimulate the areas in my brain that enjoy the same activity.

 

[mathwonk]

my known mathematical genealogy goes back to Cayley, Galileo, Newton and Tartaglia.

My real mathematical education has come from my advisors, and after that from the numerous contemporaries who have patiently explained to me their thoughts and insights.

I did find some help in learning rigorous calculus from Hardy's book, which was the alternate text in my freshman course in college.

 

[mathwonk]

I do not wish to give the wrong impression. it is very hard and unusual to come back from neglect. it is better to avoid it.

More successful are the people who just worked steadily all their career. As one man told me, it is hard to recover mathematics if you let it get cold, try to keep the ideas warm.

 

[mathwonk]

let me make a suggestion: do not worry if you are a genius, or future fields medal material. just work some every day, and keep moving forward. if you have any ability, any passion for your subject, you will gradually progress toward your goal, and will achieve some success, indeed more than most people.

at some few times, you may feel you are on the cusp of a breakthrough, and then you may need to work more intensively, and not rest until you have achieved your local goal. but in general just keep trying, and enjoying your work, and you will achieve more than you may have thought possible, in time.

 

[arshavin]

The courses at my uni treat the students as if they were little kids and absorb information at a vegetative rate.

PhysicalAnomaly which uni to you go to?

 

[tgt]

Mathswonk, they say a man's perspective of the world changes with time. i.e Ali saying that if a man viewed the world at 50 the same way as he did at 20 then he has wasted 30 years of his life.

How has your view of mathematics changed over the years?

 

[PhysicalAnomaly]

Mathwonk, your story is both interesting and inspiring. It's good to know that even after a fall, it's possible to rise to great heights. But yes, I will try to avoid such a thing. I don't think I have your ability to turn disasters into successes.

That's an amazing ancestry indeed! Cayley! I hope to have a mathematics ancestry one day. But as much as I'd like to place the far end of the ancestry up on a pedestal, you're right - the teacher of here and now are probably more important.

A field's medal? *dies* I don't even dare hope for one. Dream, yes - hope, no. But I'm not going to study mathematics for a prize. I would like however to come up with stuff that's new and significant and to be the master of my field (puns possibly intended).

I'm still wondering though, if studying two extra years on an engineering major to go with it would be an advantage or a disadvantage. I would think that the pro would be that I'd have experience in different field and also with applied mathematics in the most applied fashion possible. But the con would be that I'd be a lot older than everyone else doing a PhD and a postdoc etc. What do you think?

Arshavin, I go to an Australian university and they're reknown for their poor maths programs - or so I've been told by several academics and postgrad students who have been overseas. I've experienced two courses first-hand, and I'd have to say that compared to what I was doing in A levels, it was a walk in the park. The one on vector calculus spent half the semester covering the material of a prerequisite and then gave us statements of Gauss' Theorem and the Kelvin-Stokes Theorem with no explanation at all as to how they were formulated or even what each side meant. We were just taught to plug in values. Towards the end, the lecturer was solving one 5-10 minute problem in one 1 hour lecture. I learned more both in terms of depth and volume from books.

That's a bad thing though. The complete lack of a challenge meant that I concentrated my efforts towards other fields of mathematics or other sciences altogether to the detriment of those courses. I did well on the exams and my knowledge of those courses are sound... but I know I could have done better in the finals of the first one if I had put more effort into it. It just retards your interest in the particular field of mathematics. There's no excitement, no enthusiasm, no challenge...

I've found a good tutor and I'm hoping that he'll show us what is expected of us if we're to be on par with US and UK students.

 

[||spoon||]

Hey, just wondering which Australian university you are studying at? I am currently at The University of Melbourne (1st year), probably majoring in mathematical physics, and so far I have found my maths courses to be quite good...

Also i believe, please don't quote me on this :), that an honours year is equivalent to a masters overseas, or at least a certain type of masters. The new melbourne model has a high emphasis masters, as i think this model attempts to replicate foreign curriculums. More information on all that at uni website. As above, not 100% on this.

-Spoon

 

[mathwonk]

from the web there seem to be quite good and very active people at both melbourne and sydney, with active seminars, and lots of publications in areas of very current interest. ( i looked in more detail at the sydney group.)

in the tough climate for jobs that has been the case for over 30 years, it is hard to believe there could be a major university anywhere without very strong staff. where there are good teachers there is potentially a good learning opportunity. you just have to ask for it.

 

[mathwonk]

my view of mathematics has changed from one of a student, who thinks that math is what he reads in books, to one of a mathematician, who thinks that math is part of nature waiting to be uncovered.

as a young student, who depended on a good memory to get through early schooling, i liked books filled with abstract definitions, and then theorems that followed from those definitions, like kelley's "general topology". now most of that stuff seems trivial to me and i never look in there for anything. but i feel differently about dieudonne's book, foundations of modern analysis. there seems to be a lot to learn in there and i have looked back many times over the years.

you can recognize this book oriented attitude in questions that appear here such as "what is the definition of a "gezundheit?" my textbook says they are always flatulent, but i have heard some people say they can also be piliated?"

then someone answers with a quote from wikipedia, or somewhere. i occasionally try to respond by saying these words mean whatever you want them to, you just have to be precise about it, but i often feel misunderstood by people who think questions are answered by a reference to authority rather than to logic.

anyway, at first i did not know that definitions are only made to enable us to navigate through real difficulties and real situations that arise in examples and problems.

I did not realize you learn more by calculating hard examples than by memorizing abstract theories. but if the theories are good ones eventually they also may lead to the ability to make new calculations. so the two play off each other, the problems inspire the theories, and the theories, if worthwhile, illuminate the examples.

I was also more afraid as a young student, of the as yet unsolved problem, afraid i could never solve it. later it seemed that if one just put in the work, faithfully, that a solution usually emerged in time, if not of the full original goal, at least of a good part of it.

this fear is related to the attitude that math is in the books, since in that case, where the book ends, ones confidence ends. once you begin to see the math in nature, you just keep exploring.

to help acquire this change of attitude it is crucial to stop just reading books passively, and begin to read actively, trying to work out the proof on ones own before reading it, and to think of good examples, and work through them.

It is also important to work every day. Unfortunately time spent here is lost to working, so when one is working he has less time to answer questions or hang out here. this is not meant as lack of interest or concern for friends in the community. So I will try to take my own advice and get something written up today. It is challenging to do while preparing courses, lecturing,...etc..., but essential.

good luck to you.

 

[Cincinnatus]

math is part of nature waiting to be uncovered.

That statement is fraught with philosophical controversy.

 

[GSpeight]

What do other mathematicians do if they get "in a rut" and feel like they're losing motivation?

I'm a 4th year MMath student in the UK and this term I'm taking courses in Stochastic Analysis, Differential Geometry, Brownian Motion and PDEs. I'm also supervising 1st year students (4 hours a week plus planning and marking work) which I'm getting paid for and quite enjoy (except marking) but it takes time. Also I need to start work on my 4th year project.

I find that after I get home (after 5 most weekdays) and finish marking first year work I'm quite tired and lack the motivation to work on example sheets, read lecture notes or do research for my project. Obviously leaving these things until the end of term is far from ideal!

I'll probably be taking one fewer course and have a little less marking next term but I'd appreciate any advice for keeping up my motivation and making the best use of my time.

 

[PhysicalAnomaly]

It's not that there aren't any good teachers. I feel that the problem is that the policy here is that everyone should be entitled to a degree and so people don't put much effort into it. This leads to a feedback effect where the syllabus and lecturers pamper the students which leads to them demanding to be pampered which leads to more pampering. The lecturers assign very little homework, don't explain anything that isn't strictly necessary for the student to use the tools and decrease the content and difficulty of the exams every few years. In the end, you find that the vast majority study very little, complain lots about how difficult simple concepts are and look at you strangely when you study more than what is strictly on the syllabus. The atmosphere encourages mediocrity, the workload (or lack of it) encourages students to slack and there ends up being little support for students to study ahead. I grew up in an asian country and find that a wildly competitive cohort and atmosphere makes a huge difference. Having a teacher who really knows the subject and pushes the students also helps.

But that is in general. Upon realising that depending only upon what is being taught would get me nowhere quickly, I've managed to find a few rare tutors who are enthusiastic and inspiring and are happy to answer questions and provide guidance, even on material not covered by the syllabus. Hopefully, with the help of good books and those good tutors, I'll be able to keep up with what everyone else is studying in the US and the UK.

(Btw, I'm finding Munkres' Topology to be one of the most interesting books I've ever read. It's so challenging and I can feel my mind exploding with each new concept.)

(Also, I'd like to recommend Tao's Analysis. I've looked at Rudin and feel that Tao's book is far less intimidating. Why is his book not as popular as Rudin's?)