*Problem solving, from homework problems to unsolved problems, is certainly an important aspect of mathematics, though definitely not the only one.Later in your research career, you will find that problems are mainly solved by knowledge (of your own field and of other fields), experience, patience and hard work; but for the type of problems one sees in school, college or in mathematics competitions one needs a slightly different set of problem solving skills.*

(More generally, you may find other links on my career advice page to be of use to you.) Dear sir , I learn mathematics as a hobby because I did not have a chance to study college-level mathematics at any good educational institutions.

Do one best learn mathematics which one does not know anything about , say symplectic geometry , best by starting from page 1 of a book on symplectic geometry , and work out all (or most) the proofs and problems as he read.

And I greatly appreciated the article on your page where you say that someone does NOT have to be a genius to be a mathematician.

Does that mean you don’t consider yourself a genius or you don’t really see a distinction between yourself and others who apply themselves and are ambitious?

译文url: My name is Leif and I wanted to ask you on your opinion on something-forgive me if this seems strange or random.

As someone who has apparently fit the profile of “child prodigy” and “genius” (e.g.I am not saying I don’t believe some people are naturally more gifted at certain things or develop stronger skills or have stronger talents than others.I just think it’s possible to achieve levels of excellence and brilliance even in your weak areas through consistent practice. After all, they made their share of mistakes and setbacks but did not give up and they were not superhuman or genetically enhanced as far as I know..the fact that you graduated high school and university and earned your doctorate so young), I’m wondering if you think you possess something that only a few others have in terms of intellectual ability or not.I have always believed that if someone applies himself and puts in enough time, effort, concentration, and perseverance they can accomplish whatever they set their mind to.As you are still several years away from having to attack research-level mathematics problems, your current skill in solving such problems is not particularly relevant (much as the calculus-solving skill of, say, a seventh-grader, has much bearing on how good that seventh-grader will be at calculus when he or she encounters it at the college level).The more important consideration is the extent to which your problem-solving skills are improving over time.And I don’t think that’s arrogant or unrealistic.I wanted to get your honest opinion. Hey Leif, This book might be useful in pursuing the answer for your question: Disclaimer: I just read the summary and reviews of that book. I’m, at the moment, too busy with studying Maths stuff. Tao, I am a high school student, I loved math got good grades in my middle school years.But I find math hard and i often make many mistakes now.That’s why I disagree with this post by astronomer Julianne Dalcanton which i found linked from your page where she doesn’t believe most people can reach the level of Feynman-Einstein-Hawking smart. And isn’t part of their fame due to circumstance and perhaps even chance-not their intellectual ability but I mean their status and the fact that their discoveries happened to be earth-shattering or were given more attention by the public at-large beyond the scientific community?Surely some of the scientists working today will make equally groundbreaking or insightful discoveries or develop innovative theories and thus can fairly be labeled “genius” or as having the same level of smarts? I think I have developed a stronger aptitude for language than for math and due to suffering from depression in high school and middle school I didn’t push myself nearly as much as I could and lost much of my motivation.

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