Should I Become a Mathematician?

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  • #1,851


Hey everyone. I've got a bit of a question.

I think it would be accurate to call myself a jack of all trades. My quantitative skills are verbal skills are quite similar when compared on an intelligence test; however, in comparison to most other students at my college, my verbal skills far exceed most others, simply because it seems like they have had a serious lack of education in that area. So far, at my liberal arts school, where study in all fields is necessary, I have been able to receive A's across the board.

I am currently debating whether or not I would like to pursue a mathematics or physics major. My passion lies in these two fields, and I also love to write. Unfortunately, I question whether or not I am talented enough to pursue a science or math major and still perform well. I thought Calc I and II were jokes last year. My intro physics class this year is quite intuitive for me. I am also enrolled in Calc III and a discrete mathematics course this year. The later is a joke while the former is definitely challenging for me, as is it for the rest of the class. This is quite discouraging for me; I'm used to quickly grasping concepts. If my limit for quick understanding lies at such a basic level of math, I question whether or not I am fit to continue.


Granted, my school has this fun thing called grad deflation, the opposite of what most schools have. As a result, homework problems and tests are absurdly difficult. While this is good for me in the long run, it sure makes things tough now. hmm... might also be important to note that multivariable calculus used to be taught in two semesters and is now squeezed into one, resulting in quite a challenging class. Perhaps my ability's appear dampened to me simply because of the rigor of the course.

Next semester I am definitely taking linear algebra; however, in order to continue to take future math classes, I would need to take a course called principles of analysis, which is typically infamous for being the toughest course required of a math major. The kids who breeze through Calc III find it very difficulty. I question how I will fair.

While someone can always say I will just need to work a bit harder, I don't think this is too possible as this point. I have been blessed and cursed with a learning disability. Things take me a long time; however, I can complete many tasks others do not have the aptitude to complete. I already devote 30 hours or more to Calc III and week and see my professor multiple times as well. Because the college of the holy cross is a small school, we lack many of the resources of larger schools, meaning that tutors are scarce.

What do you guys think my options are? I love math. Should I sacrifice my perfectionist mentality and concede that I might not receive an A, or should I simply peruse something I enjoy slightly less - but still love - and perform well?

Do what makes you happy- you only get older and life gets shorter-
I am a new mom with no time at all on my hands yet I manage.
If your parents money is the issue then apply for a student loan-
i am in debt bc of mine yet my world is still in equilibrium and everything is ok!
Linear algebra was fun when you think about it and not just memorize.
Calc 4 is the same way- and then you enter what you are talking about- advanced calc analysis in one or several variables- topology- abstract abgebra (my fav!)
These classes are MEANT to be challenging. Sometimes I would spend ten hours (while entertaining the little one lol) trying to figure out the puzzle of the proof- how to prove a sequence converges monotonically to----- lol whatever else-
and I too- have limited resources- our campus tutors are not qualified and I do not have a sitter to attend any extra study sessions- but-
I love it- so I pursue it-
Please do the same- do not be discouraged!
'Perfectionist mentality' - do you know what great minds in the past were farrrrr from perfect- they were DIFFERENT and PASSIONATE!
You will find words only get you so far- do what CHALLENGES your mind not what comes easy to it- good luck!
 
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  • #1,852


i would say this definition is only a small step in a long chain of work going back to the greeks who showed the area of a circle was a number that could be neither less than nor greater than pi R^2 essentially by showing that is was a limit of quantities that differed from pi R^2 by less than any given amount (any epsilon).

so many many people for hundreds and thousands of years gave arguments essentially equivalent to what we have as the epsilon delta definition of limit. i.e. limits were well understood by the masters for a long time before they were stated in the form we have now, and their use of them is roughly equivalent to ours.

i would say the discovery of the method limits by the greeks stand far above the much later precise statement of that method. the statement came from analyzing the method, not the other way round.

Yes I concur :)
I hate the outlined epsilon - N notation- genius, yes- still I am amazed at things like the rhind papyrus (obviously not applying to limits)- so old, so simple (now--maybe :) )- yet so important- see even then they thought math was all the 'mysteries' and 'secrets' of life... ;)
 
  • #1,853
thrill3rnit3
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mathwonk (or anyone)

Have you read this book called Geometry by Kiselev (Russian)? There's actually two books. My math teacher recommended them to me. Have you read that book, and if so, what do you think of it?

here's the link to the english translation version

http://www.sumizdat.org/

nobody has read Kiselev's Geometry in here??
 
  • #1,854
mathwonk
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you can be first!
 
  • #1,855
thrill3rnit3
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yeah...I guess

i ordered both books from Amazon. The book also had pretty good reviews. I guess I'll give them a shot.
 
  • #1,856
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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.
 
  • #1,858
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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
 
  • #1,859
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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.
 
  • #1,860
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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?
 
  • #1,861
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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 cant 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
 
  • #1,862
mathwonk
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coca you might try walker's book for resolution of singularities. or i could send some notes, or put them on my website.
 
  • #1,863
mathwonk
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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
 
  • #1,864
thrill3rnit3
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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?
 
  • #1,865
mathwonk
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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.
 
  • #1,866
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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 [Broken]
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.
 
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  • #1,867
mathwonk
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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.

niote 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.
 
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  • #1,868
mathwonk
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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.
 
  • #1,870
mathwonk
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cocoa, well??? does that help at all?
 
  • #1,871
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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.
 
  • #1,872
mathwonk
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there are few people who know all of that material.
 
  • #1,873
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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.
 
  • #1,874
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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."
 
  • #1,875


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?
 

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