
#19
Jun606, 12:54 PM

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I wanted to specialize in number theory, but then I read a very discouraging book by Guy; now I'm not so sure anymore.




#20
Jun606, 01:08 PM

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#21
Jun606, 01:54 PM

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It is very frustrating and discouraging that while any of the problems can be understood by a child, I am unable to make the most minuscule contribution to even one.
Et tu, [tex]\mathbb{N}[/tex]? 



#22
Jun606, 02:32 PM

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3) AP courses and colege admissions.
Although I have poo poohed AP calc courses as not comparable to college courses, and often taught by unqualified teachers, there are many reasons to take them anyway in high school. At some high schools they may be the best courses offered, and they may be taught by the best teachers available. Politically, they are frequently seen by college admissions officials as evidence that the student has challenged him/herself by taking the most difficult courses available to him/her. Thus trying merely to get a good mathematical preparation and avoiding bad AP courses can backfire at college admissions time, since the admissions officials are not knowledgeable enough to see these ill conceived courses for what they are. The same goes for high school students taking college courses while still in high school. I have advised some very smart kids to stay at their high schools and establish their math backgrounds with good solid high school level algebra and geometry, augmented by extra reading and projects, only to see these same kids turned down by their first choice colleges, in favor of weaker students whose high school resumes featured college calculus courses. It is very hard to detect real ability outside ones own field, or even in it, and college admissions officials are not that great at it. They tend to like easy identifiers, like newspaper editor and AP courses, prizes, and college courses taken in high school. They do not know how deep a thinker a kid is since they have not taught them, and often they are not scientists themselves, and would not know how to recognize a budding scientific mind first hand anyway. Fortunately after you do get in college this kind of myopia lessens since you are being evaluated by scholars who (hopefully) see your work, but it never fully goes away. When applying to colleges try to get letters from people who will write positive ones, and who understand the game well enough to know how to do this honestly but skillfully. I have unwittingly sandbagged some outstanding kids early in my career, by simply writing truthful, unvarnished letters that appeared weak compared to the overblown and ridiculous ones written by less academically knowledgable people. As a college math professor and researcher of course I have seen, met, heard, and even worked with some of the smartest people in the world, Fields medalists and so on, so few high school kids no matter how bright, are going to really blow me away. But admissions officials reading my letter do not give me credit for being on a different level from high school teachers and guidance counselors, and so those people may be better letter writers for this purpose. Of course by now I know how to tell the truth more persuasively. Once you get in college, the people teaching you will usually be actual mathematicians themselves, and will know exactly how well you are grasping the material, and can write letters for grad school that will help you accordingly. Before college, take all the instructional opportunites available to you, just do not expect them to live up to the hype they may enjoy. Spend as much time as possible immersed in the subject itself, and with like minded people, to keep the love alive, but be aware of your resume. 



#23
Jun606, 02:37 PM

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Thanks J77. Your input is just what Courtrigrad, and no doubt others, seems to be asking for.




#24
Jun606, 02:42 PM

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Jbusc, topology is such a basic fioundational subjuect that it does not depend on much else, whereas differential geometry is at the other end of the spectrum.. still there are inrtroductions to differential geometry that only use calculus of several variables (and topology and linear algebra). Try Shifrin's notes on his webpage.http://www.math.uga.edu/~shifrin/




#25
Jun606, 02:45 PM

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I am curious to see how hurkyl describes himself  at times he has also struck me as perhaps a mathematical logician.
maybe i just assumed all the moderators on the "physics forum" were physicists. 



#26
Jun606, 03:27 PM

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dragonfall, the way to solve problems is to make them easier, try changing the hypotheses of some of those problems, and just keep changing them until you get a problem you can solve. then try to work your way back up a little bit.
25 years ago Bob Friedman and I showed that most Riemann surfaces with involution have different matrices of skew symmetric periods. About the same time Ron Donagi conjectured that if two such gadgets had the same period matrices, then the quotient Riemann surfaces must both be 4:1 covers of the line. No one has been able to show this more precise result yet, but many have tried, and I still hope to. The problems you are talking about have stumped everyone in the world for decades or longer. Such books are not meant as a problem set for young mathematicians. Of course if you solve one fine, but if not, you are in very good company. 



#27
Jun606, 03:41 PM

P: 1,239

is there any difference in majoring in math in a liberal arts college vs. a bigger college. I am going to a LAC this August.
Thanks 



#28
Jun606, 05:02 PM

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I'm a mathematician / computer scientist by training (just BS) and employment. I've just made a hobby out of doing a tremendous amount of selfstudy! I even sometimes read textbooks as "light reading".




#29
Jun606, 08:15 PM

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Wow. You would do a superb PhD if you have the inclination, but as you are already earning a living that would be a sacrifice.
You have the innate power and creativity of a PhD level mathematician. This is unusual with only a BS. 



#30
Jun606, 08:29 PM

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Courtrigrad,
Well I went to a big university (Harvard) and found majoring in math there stimulating by the exposure to top people and high standards, but discouraging through the impersonality and lack of hands on guidance. So it took me a long time to find my way, but because I never gave up, eventually the high standards were still in my brain and became helpful far down the line. In my opinion you could possibly do even better and sooner at a LAC (which one?) with some personal guidance from people who actually get to know you. And even if the Fields medalists are not teaching there, still the level of faculty is so high now everywhere, I believe you will be very well served. The best early teaching I received was at Brandeis, a small research university, much more personal than my undergrad experience at Harvard. Later I went to Utah and got great grad school guidance and finally returned to Harvard as a postdoc, the ideal status for me at Harvard. I.e. Harvard is at such a high level that the instruction was more appropriate for me as a postdoc than an undergrad. One of the best research algebraic geometers/ topologists in the country (Robert MacPherson) went to a small liberal arts college, Swarthmore. I think it is hard to find a college in the country now where there is not more offered than one person can easily absorb. What do you take in 4 years, 32 semester courses? and the Harvard catalog contains over 3,000 courses. It might possibly help to go to a college witha grad program. E.g. Wesleyan is to me a typical liberal arts college, and has 26 undergrad math courses and 29 grad math courses, more than anyone could possibly take. The difference with going to Harvard is there you will also have the chance to take graduate algebraic topology or algebraic geometry as an undergraduate, but how many people need this at that stage? 



#31
Jun606, 10:19 PM

P: 1,239

Yeah, I was waitlisted by Wesleyan and Oberlin (hopefully will get off). I am currently planning to attend Denison University in Ohio. My current goal is to become an applied mathematician; perhaps to a "3+2 preprofessional program" (i.e. 3 years at Denison and 2 years at Columbia) or stay 4 years at Denison and apply to graduate school. My only concern is whether I will be able to study the core essentials properly (in college). Or maybe I have to do Apostol by myself, and follow the likes of Stewart in college.




#32
Jun606, 10:59 PM

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I looked at their webpage and it looks as if they have a very active department. there was a conference there on group theory and one of their graduates placed first in the nation in a math contest recently.
It looks like an especially strong place in applied math, and also has a presence in groups, functional analysis, and knot theory. The faculty picture is also fun looking. I think you will enjoy it there. This will be a place where there are not a lot of advanced grad courses, but the treatment of undergraduates should be outstanding. It looks like a very promising place indeed. good luck, and as Bill Monroe told my brother to tell me " tell him, don't hang back, come right up and introduce himself". (My brother was Bill Monroe's fiddler in college.) keep in touch. 



#33
Jun706, 12:33 AM

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jbusc, here is a gorgeous book on maifolds, from lectures by a fields medalist and great expositor. try it. and give it some time. if you can read this you will really learn something.
Topology from the Differentiable Viewpoint princeton univ press. John Milnor Paper  1997  $26.95 / £17.50  ISBN: 0691048339 76 pp. This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem. 



#34
Jun706, 02:57 AM

P: 1,157

During my maths degree, we used Calculus and Analytic Geometry by Gillett, and Calculus by Boyce and DiPrima, the latter I still look at from time to time. If you're reading off your own back, starting ODEs from scratch, I'd suggest the dynamical systems book by Boyce and DiPrima: Elementary differential equations and boundary value problems, as a good starting point. Possibly coupled with a more application based book like: S. Strogatz: Nonlinear Dynamics and Chaos. (or K. Alligood, T. Sauer and J. A. Yorke, Chaos: An Introduction to Dynamical Systems.) I'm sure others will like and dislike (I've heard B&DiP talked down before) the choices, but they are only entry points, with the Strogatz book bridging the gap between elementary calculus based texts and the books I recommended in my previous post. If you could be more specific about the content of the courses, that would help. Obviously content varies from institution to institution. Also, you present level of education may help  I presume you've just started university or are finishing high school? edit: Just to add, if you want a book you'll use time and time again: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Table by Milton Abramowitz and Irene A. Stegun 



#35
Jun706, 02:01 PM

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I liked the beginning ode book by amrtin braun for my class, exactly because it featured applications, and hence entertained and motivated the class. It discussed using ode's to date of paintings and detect forgeries, predict populations of pairs of interacting "predator prey" species like sharks and food fish or hares and wildcats, troop deployment in battles with illustrations from WWII (Iwo Jima), and lots more such as "galloping gertie" the famous tacoma narrows bridge that blew down years ago. It was very well written.
Boyce and DiPrima is a time tested, often used, and well liked standard book at my university too, indeed THE standard book on ode, but I was looking for a good alternative that cost a lot less. Sadly, as soon as a book becomes a standard, the price now shoots above $130. I got my copy of Braun used for $2. Braun is also more entertaining for me, but I think you cannot go wrong with BdP. I would suggest studying ode sooner than some of the other topics on your list, like reals and complex analysis. Also as wisely mentioned above, it seems prudent to go with the flow, and not be too rigid in your planning at this early stage. And the calculus book by Sternberg, if you mean Advanced Calculus by Loomis and Sternberg, it is very abstract and advanced, treating calculus essentially as functional analysis. Of course once you have finished Apostol, it will probably be fine, but I suspect the view Loomis gives in the first half of calculus is not essential for an applied mathnematician. I like it though (I took the course from Loomis in the 1960's from which this is the resulting book. The only thing I learned was that the derivative of f at p is a linear map differing from f(x)f(p) by a "little oh" function, which is of course the main idea.) There is another newer book by Sternberg and Bamberg, math for students of physics that sounds intriguing, but I have not seen a copy. In the 1960's Bamberg was the absolute most popular and entertaining physics section man in a department which was otherwise bleak and forbidding for its physics instruction. (I still remember his list of useful constants: Planck's constant, Avagadro's number, Bamberg's [phone] number...) 



#36
Jun706, 03:31 PM

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Becoming a mathematician, part 4) College training.
I suspect it does not matter greatly which college you go to, as they all have their strengths and weaknesses. Places like Harvard or Stanford or Berkeley offer famous lecturers on a high level, incredibly advanced courses, and brilliant highly competitive students. For many of us, this can be more intimidating than inspiring. And often the famous professors are simply unavailable for conversation outside of class. In the early 60’s at Harvard, I found the lectures were wonderful, if I got the best professors, and then they walked out and I never saw them again until next time. Office hours were minimal and if I tried to see some of them, they were frequently busy or uninterested. Even intelligent questions in class seemed as likely to be met with sarcasm as a helpful answer. I suspect things have changed now with people like Joe Harris and Curt McMullen there, who are great teachers as well as researchers, and who enjoy students. Of course there were outstanding teachers like Tate and Bott there in the old days too, but not everyone was like them. As a result, I had to go away and get back my enthusiasm for math at a more supportive place. It is helpful to go somewhere where you will enjoy your time, enjoy the courses and the other students, and get help from professors who think students matter. Today this is more common everywhere, even at famous universities, than it was long ago, but ask around among the student body. And be prepared to work very hard. Some if not most of my own undergraduate frustrations could have been lessened, possibly solved, by better study habits. As to what courses to take, this is tricky and complicated by the almost worthless AP preparation most kids get today in high school. In general an AP class is a class taught by someone with nowhere near the training or understanding of a college professor, although they may be a fine teacher. But to expect a calculus course taught by an average high schol math teacher to substitute for a honors introduction to calculus taught by Curt McMullen or Wilfried Schmid or Paul Sally, is ridiculous. Nonetheless, so many students have bought this ridiculous idea that Harvard and Stanford do not even offer an honors introduction to calculus anymore for future math majors. There simply are none out there who have not had AP calculus in high school. Thus the student entering from high school is faced with beginning in one of many choices of several variable calculus courses. The most advanced one, the one taught a la Loomis and Sternberg, realistically requires preparation in a very strong one variable course a la Apostol, but which Harvard does not itself offer. So the only students prepared to take it are those elite ones coming in from Andover or Exeter or the Bronx high school of science, but not the rest of us coming in with our inadequate AP courses from normal high schools. Thus the jump from high school to college has been made harder by the existence of AP courses. So in my opinion, even with AP calculus preparation, it is often helpful for a prospective mathematician to try to begin college in an introductory, but very challenging, one variable calculus course, modeled on the books of Spivak or Apostol, if you can find them. These do exist a few places, such as University of Georgia, and University of Chicago, which still offer beginning Spivak style calculus honors courses. To quote the placement notice from Chicago: “The strong recommendation from the department is that students who have AP credit for one or two quarters of calculus enroll in honors calculus (math 16100) when they enter as first year students. This builds on the strong computational background provided in AP courses and best prepares entering students for further study in mathematics.” (I am not positive, but I assume that 16100 is the spivak course. But do your own homeworkl to be sure.) The point is that AP preparation provides no theoretical understanding, so plunging students into advanced and theoretical calculus courses of several variables, as they do at Harvard and Stanford, by beginning in Apostol vol 2, or Loomis and Sternberg, without background from Apostol volume 1 or Spivak, is academic suicide even for most very bright and motivated students. If you go to a school where there is no Spivak or Apostol vol. 1 type course, where the calculus preparation is from Stewart, or some such book, you are perhaps getting another AP course, only in college. Then you have to choose more carefully. Many such college courses will indeed be no more challenging than a high school AP course, and should not be repeated. Just ask the professor. They know the difference, and will help you choose the right level course. Either get in an honors section, or an advanced course suitable to your background. And join the math club. Try to find out who the best professors are, and do not be scared off if weak students say a certain professor is tough. You may not think so if you are a strong student. Once you get there, try to sit in on courses before taking them, to see which professors suit you. Student evaluations are notoriously hard to interpret correctly. The professor with the worst reputation among students, Maurice Auslander, was in my grad school days at Brandeis my absolute favorite professor. He cared the most, offered the most, and taught us the most. He also worked us the hardest. Once you get a semester or two under your belt, it will get easier to find the right class, as hopefully the colleges own courses prepare you for their continuations, although this is not guaranteed! There is no way to force one professor to included everything the enxt one expects, nor to exclude material he/she loves that is outside the curriculum. Do your own investigating. Ask the professor what is needed for his/her course and try to get it on your own if necessary. After leaving the honors program temporarily as an undergraduate, I got back in by studying on my own over the summer from an advanced calculus book (David Widder), to make up my theoretical deficiencies and survive the next course. Everyone should study calculus, linear algebra, abstract algebra, ode, and some basic topology. If you have no background in proofs from high school, you will need to remedy that as soon as possible. It is best to do this before entering, even if they offer a “proofs and logic” course. Such courses are often offered to junior math majors, whereas they are needed to understand even beginning courses well. For this reason it is extremely helpful to read good math books on your own that contain proofs. Today especially it is important to know some physics even if if you only plan to do math. Much of the inutition and application of math comes from physics. Even if you only want to do number theory, sometimes viewed as the purest and most esoteric branch of math, many of the deepest ideas in number theory come from geometry and analysis and even statistics, so nothing should be skipped. Work hard, read good books, seek good teachers, and try to have fun. College is potentially the most exciting and fun time of your life, and the one where, believe it or not, you have the most freedom and free time. 


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