I love Maths very much. However, most of the time I can't find new questions to answer nor know what to do with an unfamiliar problem. I really think this is going to be my disadvantage in my maths career. Does any have this problem and any advice?
Well, there are millions of Math textbooks and I am certain that you haven't worked through EVERY one of them...
that skill of problem finding is apparently much rarer than problem solving. there are standard ways to go at it however, such as changing the hypoheses in a known theorem, or changing the conclusion. i.e. if something is true in dimension one, ask what happens in dimension two. or if a differentiable function has a certain property, ask what happens for only continuous functions. try to get the habit of noticing what is not said, as well as what is said. and any time someone says "we do not nkow what happens when..." there is an open question. it took me my whole life to notice it, but there is a standard theory of finitely generated classification of modules over pid's, but not more general rings, as far as i knew. on the other hand, i knew that a dedekind domain is basically q ring that is locally a pid. thus there should be some weakker classification theorem for finitekly generated modules over a dedekind domain. guess what? there is, and it appears at the end of books like dummitt and foote. but asking this question should have been obvious. notice many oif these books do not make it clear at all why such a questiion is an obvious extension of a well kmnown theory. we are not very good at teaching how to ask questions in math, and it does not appear in many places. one exception is basic algebraic geometry by shafarevich. when he explains a theorem, he ioften asks an open question extending what ahs been explained. but because a question, once asked is easier to answer, many of his questions have now been answered. e.g. in the section on cycles, he posed the problem of fiun ite generation, but clemens answered it about 20 years ago, and in the section on uniformization, he posed a question on the structure of universal covers of algebraic varieties, and now a lot of work has been done on it by kollar and others. I was so nose to the ground when ireqad shafarevich, i was only interested in the theorems there and ignored or did not even observe the open questions. so to find open problems, you have to learn to be looking for them. As briefly put above, you have to spend some time reflecting, i.e. thinking, about what you have learned. there is a famous book about this, called :"how to solve it" by george polya. read that. one tip irecall from that book, is that soklutions to problems are like grapes, they come in bunches. so when you have an idea that solves one problem, look around for another that it solves. i used to not do this, and when i published a paper with an idea, I would notice later someone else had proved much more than i did with the same idea. I was occuopied with my problem, and having worked hard and long to find the key to its solution I was satisfied. It did not occur to me to maximize the productivity of my idea. Ideas are few and far between. To expect to generate a new one for every problem is hard. So push them as far as they will go. But not forever. Try to think of a new one occasionally too. I also have had trouble doing this,a s we all have.
Well, this maybe a long shot, but you can always look at Hilbert's problems or Clay institute millenium problems... Maybe you won't solve any of those (maybe you will ;) ) but they may present you with a direction to take in your thoughts or open the door to an interesting field... Kolmogorov was also famous for posing questions and I remember seeing one book collecting them, but I can't remember the title. Best luck, and report here if you find a good answer to your question :)
If you're no professional mathematician yet, that shouldn't bother you, since you've got a lot more to learn. Otherwise, the answers are already given - work hard and you shall be rewarded.
leon1127, where are you at in your studies? What was the last math class you have taken? This question could mean a lot of different things depending on where you are at. If you still in the basics, calc 1 - 4, diff eq, lin alg, I might be able to help, assuming I understand your question correctly. If you are into your higher level math classes, mathwonks advice would be more beneficial then anything I could hope to send you.
I am senior. I have done topology, analysis and so. I am about to start some studies in Clifford Algebra later. I have tried to put some effort on numerical solution of Euler differential equation, but I couldn't find the question that has not been answered and suitable for my knowledge at the same time. I have also put a lot of effort on calculus of variations, and manifold theory. This is the stage I am at in my education. I am still trying to digest the answer that Mathwonk has given me.
Classification of modules Just in case anyone wants a more complete citation: see section 19.5 of Eisenbud, Commutative Algebra with a View Towards Algebraic Geometry for an series of exercises on the classification of modules over a Dedekind domain. But I guess Eisenbud also sinned, since he failed to stress that this generalizes the classification over PIDs...
this "sin" of eisenbud's book, is in not showing where the idea for the result came from. presumably that was not his goal. but this is the reason i emphasize trying to solve problems yourself, or prove theorems yourself, as opposed tor eading them only in books. many people here still believe that learning should come first, and of course there is much to elarn. my point however is that one of the thigns you want to learn, namely how to find problems and their solutions, cannot be elarned by merely reading and retaining more and more proofs in books. i have tried to demonstrate here over several years, that one can generate proofs of many statements merely by getting that statement clear in the first place, then analyzing it for its resemblance to other basic ideas and tools. let me congratulate you on this question, as I do not immediately recall another question concerning the all important topic of creativity in the years i have been posting here. the only secret i myself have for doing research is "analogy". I learn as much good math as possible, really learn it down deep in my pores, and then when I listen to another problem I try to compare it to something else I know in some way, to see if there is a leson from the old case to apply to the new one. hopefully that provides the "inspiration". the other 99% is all hard work. it helps to appreciate your own inspirations too. once iwas loistening to a talk, and the speaker ahd a mysterious result he could not imnterpret geometrically. When I heard hima sk the puzzle the night before, I had no idea either, but after listening to him speak for an hour, and all the juices were flowing, I saw a way to comopare it to anoither situation which ompletely explained the mystery. everyone just sat with their mouths open,a nd afterwards he askede me what eh shiould do with the result. i said well it was inspired by hi talk, and it was his problem so take it. he published it afterward with no specific credit to me, and the reviewer for the paper singled out that one resukt as the mosts triking and origina in the paper. years later other smart peopel also rediscovered and opublished this result. so perhaps i would have benefited professionally from requesting credit. but at least i have the pleasure of remembering my insight. the practical side is that over the years it ahs not ahppoened all that often.
the basic principle of problem solving is the opposite of that for problem finding, instead of making the problem more general, make the problem more specific, i.e. if your problem is too hard in dimension 2, try it in dimension one. if the infinite case is too hard, say a problem in a vector space over Q, try it over a finite field. if the non abelain case is too hard, try the abelian case. if it is too hard in a general topological space, try a metric space, or even R^n. or try the compact hausdorff case. and a lesson for me and other profs to keep in mind, save the creativity for your research, students are notoriously uncreative on tests. that clever question you loved so much to think up, is probably impossible on a timed test. oh yes another reference for creative problem solving is the wonderful book by jacques hadamard, which i have mentioned here before, something like on discovery in the mathematical sciences, or the psychology of invention in the mathematical field. he emphasizes there the importance of preparing the ground or "preparing the unconscious", for its assistance in discovery, by hard work in advance. in my case mentioned above, the speakers problem had been posed the night before and i had thought about it until the next day to some extent. then i listened to the talk. also i had spent years thinking abut the analogous situation, and then at the end of the talk it popped into my mind to connect two things which actually had always been connected since abel and riemann. just no one thought of doing it for this question, and there was one little extra wrinkle i knew and included that we often ignore - the connection between algebraic geometry and non - algebraic geometry. i.e. to answer this question my thoughts literally went "outside the box" of objects we were assuming was the context for the problem. of course i could not have done this, as many people here point out, without the knowledge of what was out there. but the interesting thing was, there were many people in the audience who knew that too, and much more than I, yet no one else thought to do it. of course in other settings those people showed their mettle by displaying much bigger insights than my little remark. so "every dog has his day", big or small, and i enjoyed that one. most of the others had not heard the question the day before as i had, and were not as consumed or fascinated with it.
here is a copy of hadamard's delightful and unique book: The Psychology of Invention in the Mathematical Field Jacques Hadamard [30 Day Returns Policy] Bookseller: thriftbooks.com (Seattle, WA, U.S.A.) Book Price: US$ 1.78 notice again the principle that really good books are often very cheap.
here is a copy of polyas book: How to Solve it G. Polya Bookseller: Usedbooks123 (Sumas, WA, U.S.A.) Price: US$ 3.48 [Convert Currency] Quantity: 1 Shipping within U.S.A.: US$ 3.88
To Mathwonk When you encounter a problem requires extra knowledge that you dont have, how do you act? Do you study them in depth or just know what you need to know to continue the original problem? Or more fundamental question, how do you know what knowledge you lack? Is it just experience? I am considering those books that you suggest. Speaking of book, I have one last question. From your many years experience in US maths industry, how do you gain access to a book that is very expensive?
What I do in this case, since there's a local university nearby, I check the university's library to see if they have the book I'm looking for. If you don;t have a university or college nearby you could always check at a city library.
i am lucky that i have access to a large university library, which also links to other university libraries with interborrowing privileges. but books are not a big problem for me as mostly i am informed by papers which are distributed free. the internet has almost all papers i need. i.e. rather than needing papers i dont have, i have more papers than time to read them. at the research level, there are not many books which ahve up to the minute information. e.g. the book which is most up to date for my specialty, was written with my help, rather than helping me learn the topic. i.e. i am more expert than the existing books, and have been asked to write books on my specialty. so i have no need of books except in areas where i am a novice.
i asked that question because my school has a very small maths book collection, especially the topics that I am interested, namely variational calculus. Moreover, I am nowhere near expert in this topic. I have been inspired by the work of Mariano Giaquinta. it gives me not just formulation, but also insight on the topic. Unfortunately, series of Ergebnisse der Mathematik und ihrer Grenzgebiete is very expensive. I can't gain access to it easily. Thanks everyone for answering my question.
well those books are expensive. i still suggest you go to your library and see if they have a reciprocal arrangement with a larger library that would allow you to borrow the book from a library that has it. i remark that even as a professor, the pay scale was so inadequate for so long, that I was unable to afford to buy books for about 25 years. even about 5 years ago, nearing 60, I took on teaching a voluntary seminar course at my school because the reward was no pay, but a 300 dollar book allowance. this allowed me to finally to buy two 150 dollar books i had wanted for my whole professional life. In the past 4-5 years, with both sons out of college, i have finally begun to be able to buy books costing no more than 30 dollars or so, when i want. one good thing is that for most students, there are really ooptiions of buying much cheaper books that cover the same material. i do not know your status in studying variational calculus, but the volume 2 of richard courant has a achapteron it beginning from the euler equations that apparently gave the subject its start. fundamentally, it is just anoither name for differential calculus, but on infinite dimensional spaces, like spaces of paths. euler treated this problem, and essentially computed the derivative of the lkength function for paths. thus this derivative will be zero, i.e. eulers equtions will be solved i suppose, when the path is a s short as possible. hence one can use eulers equations i suppose, to show thw shortest path in the plane joining two points is a straight line! this sort of thing demonstartes again the valoue of elarning mathematics conceptually in the first place. i.e. if one elarns the theory that a derivative is a best linear approximtion, rather than a number, then variational calculus is just another variation on that theme, instead of a new subject. so libraries are a way of life, even for professors and profesionals. interesting world we live in, when a basketball player gets a new free pair of 150 dollars shoes for every game, and a profesor and student must borrow the books they need to try to advance the scientific stature of the nation and the world.