Can anyone tell others how to do research, how to create, and how to discover new things? It is almost certain that this is impossible. For a long time, I have been working hard to learn mathematics, understand mathematics, seek truth, prove a theorem, and solve a problem. Now I have to work hard to make clear how I do this work. The important part of the whole work process is Mental work, that is hard to explain - but I can at least try to talk about the part of manual labor.
Mathematics is not a deductive science - it is a cliché. When you try to prove a theorem, you not only list the hypotheses, but then start the reasoning. The work you have to do should be trial and error, constant exploration, and guesswork. You have to figure out the truth. At this point, you are doing the same as the technicians in the lab, but there are some differences in their accuracy and amount of information. If philosophers have the courage, they may also look at us like a technician.
I like to do research, I want to do research, I have to do research, but I don't want to sit down and start doing research - I can delay it and delay it. Even though I am infinitely attached to my work, I am still reluctant to start doing it; every job is like a fight. Is there nothing I can (or must?) do it first? Can't I cut the pencil first? In fact, I never used a pencil, but "sharp pencil" has become synonymous with all the techniques that help to delay the concentration of energy. It can mean reading materials in the library, sorting out old notes, or even preparing for the class to be told tomorrow. The reason for doing these things is: once these things are over, I can really do it. Undisturbed.
When Carmichael complained that he was a graduate school director who could spend less than 20 hours a week on research work, I was surprised that I still feel very strange. In those years when I was a big achievement, I might spend an average of 20 hours a week thinking about mathematics, but the situation is much less than 20 hours. These few exceptions, only two or three times in my life, came when my long thought ladder approached the apex. Although I have never been a graduate school director, I seem to have only three or four hours of work per day. This is the real "work"; the rest of the time I use for writing, teaching, commenting, and exchanging opinions. For identification, lectures, editorial work, travel. Generally speaking, I always come up with various ways to "sharp pencils." Everyone who does research work has fallen into a leisure period. During my leisure period, other professional activities, as low as and including teaching lessons, became an excuse for my life. Yes, yes, I may not have proved any new theorem today, but at least I explained the sine theorem very thoroughly today. I didn't eat a day.
Why do mathematicians study? There are several answers to this question. My favorite answer is: Be curious - we need to know. This is almost equal to saying "Because I am willing to do this", I accept this answer - that is also a good answer. However, there are other answers, they are more realistic.
We teach mathematics to future engineers, physicists, biologists, psychologists, economists, and mathematicians. If we only teach them to solve the exercises in the textbook, then the education they receive will be out of date before they graduate. Even from a crude and secular business perspective, our students are prepared to answer future questions, even questions that have never been asked in our class. It’s not enough to teach them everything they already know – they must also know how to discover things that have not yet been discovered. In other words, they must be trained in independent problem solving – to do research work. A teacher, if he is never always thinking about solving a problem - answering a question he still does not know the answer - psychologically, he is the skill of the students who do not intend to teach him to solve the problem.
Doing research work, one thing that I am not good at and therefore never like is competition. I am not very good at grabbing honors in front of others. Another way I strive to be the first is to leave the mainstream of research and find a small and deep drowning water that belongs to me. I hate spending a lot of time to prove a famous conjecture but not getting results, so what I do is to sort out the concepts that are missed by others and to clarify the results that are fruitful. Such things can't be done often in your life. If the concepts and those questions are really "correct," they will be widely accepted, and you are likely to be more capable in your own development. And more discerning people are behind. This is fair, I can stand it; this is a reasonable division of labor. Of course, I hope that the subnormal invariant subspace theorem is proved by me, but at least I have made some contributions in introducing concepts and pointing out methods.
Another aspect of not engaging in competition is that I don't agree with the speed of rushing to compete. I asked myself, what is the relationship between the recent beautiful results and one or two years? There is no relationship at all. I say this to myself, but even for myself, such an answer sometimes doesn't work. For those who have a heart that is different from me, such an answer is always wrong. When Lomonosov (the simultaneous invariant subspace of the exchange compact operator) and Scott Brown's (about the subnormal operator) spread, I was excited like I was The second operator is like a theorist, eager to know the details quickly. However, such an exception is rare. So I can still live in the center of my life most of my life.
There is also writing. I sat down at my desk, lifted a black ballpoint pen and started writing on a standard paper of 81/2x11 square. I wrote a "1" in the upper right corner and started: "These notes The purpose is to study the influence of the perturbation of rank 1 on the lattice of .... After writing this natural paragraph, I marked a black body "A" on the edge of the manuscript paper, and then began to write the B paragraph, page number and Paragraph words form the reference system, and can often be written in a hundred pages: 87C means 87 on page 87. I put these manuscripts into a three-ring note and put labels on the ridges: approximation theory, lattice, integral operator, and so on. If a research project is successful, the notebook becomes a paper, but whether it is successful or not, this notebook is difficult to throw away. I often put dozens of books on the bookshelf next to my desk. I still hope that those unfinished notes will continue to receive new supplements. I hope that those notes that have become published will be found to imply some neglect. The precious sprout of the new ideas, and this new idea is precisely what is needed to solve a large and unresolved problem.
I continue to sit at my desk for as long as possible - this can be understood as, as long as I have energy, or as long as I have time, I sit at my desk like this, I try to organize my notes until a weak beat appears, such as A lemma is determined, or, in the worst case, a problem that has not been carefully studied but is clearly not hopeless. In that way, my subconscious mind can be put into work, and at the best of times, when I go to the office, or when I attend a class, or even sleep at night, I make unexpected progress. The unpredictable questions sometimes made me unable to sleep, but I seemed to have developed a way to fool myself. After I turned over and over for a while, the time was not long—usually only a few minutes—I Solved the problem; the proof or counterexample of the problem appeared in the flash, I am satisfied, turned over and fell asleep. That flash is almost always proved to be false; it proves that there is a huge loophole, or that counterexample does not object at all. But in any case, the time I believe in that "solution" is long enough to make me sleep well. The strange thing is that at night, in bed, in the dark, I never remembered that I suspected the "thinking". I believe 100% that it is a good thing, and in some cases it has even proved to be correct.
I don't care about sitting at the clock. When I have to stop thinking because of the time of class or when I have to stop thinking, I am always happy to take my notes away. I might be thinking about my problems when I go downstairs to the classroom, or when I start my car and close my garage door; but I am not angry because of this interruption (not like some of my friends said, They hate being interrupted.) These are all part of life, and I feel very comfortable when we think about it in a few hours, when I am working together with me.
Good questions, good research questions, where do you come from? They may come from a hidden cave, in the same cave, the writer discovers their novel plot, and the composer discovers their tune—no one knows where it is, or even breaks into it once or twice. After that, I can't remember its location. One thing is certain: a good question does not come from the ambiguous desire to do promotion. The opposite is true: the root of all major mathematical problems is a special case, a concrete example. A concept that seems to be very common in mathematics is essentially the same as a small concrete exception. Usually, it is this special case that reveals universality for the first time. An accurate and clear method of expounding "substantially the same" is like a theorem. The Riesz theorem about linear functionals is typical. The vector fixed in an inner product defines a bounded linear functional; the abstract concept of a bounded linear functional appears to be very general on the surface; in fact, each abstract concept is specific The way that is produced, the theorem is also.
This is one of many topics that I and Diudonne seem to have their own opinions. In Maryland, I did an academic report, which happened to be one of the many times that Didione visited there. The subject of that report was approaching. The question I chose was: I knew an arbitrary operator A on the Hilbert space, and found a positive (non-negative and semi-determined) operator P. ||AP||. I am very fortunate: I found out that there is a small specific exception that contains all the concepts, all the difficulties, everything is necessary to understand and overcome the steps they need. I have made my report close to that exception, by matrix 0100/ I was very proud of the operator on C^2 defined: I thought that I successfully clarified a good question and a satisfactory solution, but did not fall into the term of the analysis that was not related to this. Go in. Didogne was polite and friendly at the time, but apparently showed disdain afterwards; I couldn’t remember his original words, but to the extent he congratulated me on his burlesque performance, his impression of my report seemed to It is "entertainment mathematics", which is a ridiculous word in his vocabulary; he thinks my report is more than fun, but it is contrived and frivolous, I think (and continue to think) the problem is far more than that. The difference between our evaluations is due to the difference in our views. I think that for Didogne, what is important is the powerful general theorem. From this theorem, it is easy to introduce all the special cases you need. For me, the greatest advancement step is that it is very illustrative. A central example of the problem, from this example we can easily figure out all the things that are universal around the example.
As a mathematician, my strongest ability is to see when two things are "identical." For example, when I thought about David Berg's theorem (regular equal diagonality plus firmness), I noticed that its dilemma is very similar to that proof: every compact (Compactam) is Kang A continuous image of the Cantor collection, from then on, can be used without a great inspiration to use the classic expression without its proof, and the result is a new way of understanding the results of the Berg. I can cite a lot of such examples. Some of the most prominent examples occur in the duality theory. For example, the study of the tight Abelian group is the same as the Fourier series, just as the study of Boolean algebra The study of the disconnected Hausdorff space is the same. Other examples are not the same ones: the classical method of successive approximation is the same as the Banach fixed point theorem, probability theory and measure The same is true.
When this is linked to see the problem, the mathematics is clear; thus, the problem is removed, the appearance is revealed, and the essence is revealed. Has he advanced the development of mathematics? Are those great new ideas just to see that two things are the same? I often think like this - but I am not always sure.