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1

I recommend Mosteller's "50 hard problems in probability," because having taken stats, you presumably already know a little probability theory. Mosteller's book helps you understand how little you know :) and teaches you some interesting new ideas as motivated by some relatively practical problems. (By the way, "Hard" is in the eye of the beholder: the first ...


1

Some few remarks to your questions: Am I right in first clearing my concepts in basic complex analysis (for analytic) and algebra( for algebraic)? Yes, you're right. A solid basis of the fundamental concepts of algebra and complex analysis at least at a moderate level will enrich your studies in number theory. Let's have a look for example at ...


1

Fractal's Everywhere is a great book to investigate fractals with. It's also not very expensive (the dover edition.) Strayer's Elementary Number Theory is a good intro to number theory. As well as George Andrew's Number Theory (slightly more combinatorial in flavor).


0

Hard! Consider the following simplification. You are in 3-space. You have a 2-space membrane that you can push 3-D objects through. You see 2-D cross sections of the 3-D object as you push the object through the 2-D membrane. OK, now add 1 to the above narrative: You are in 4-space. You have a 3-space membrane that you can push 4-D objects through. You ...


4

Well, I can't say how much more time a mathematician put in his work more than other scientist but I can tell you something. The father of one of my colleagues is a mathematician. He is now 91 years old and still working on math every day except Sunday for 5 hours. Maybe is just a passion involved. When you are determined to do something and it doesn't ...


3

Could the source be the blog of Terence Tao? Quote: There will of course be times when one is too frustrated, fatigued, or otherwise not motivated to work on one’s current project. This is perfectly normal, and trying to force oneself to keep at that project can become counterproductive after a while. I find that it helps to have a number of smaller ...


2

I heard here that "it takes roughly ten thousand hours of practice to achieve mastery in a field."


22

I often wonder the same thing, and for concerns and interests around these lines found myself reading "The Cambridge Handbook of Expertise and Expert Performance." (I would suggest reading this more if you are interested - it is a well-supported book and says lots of things like practicing well for a few decades along with a supportive home environment is ...


14

I recommend reading Allyn Jackson's short biography of Alexander Grothendieck. Here's a highlight to give you an idea of how much Grothendieck worked: Even among mathematicians, who tend to be single-minded and highly devoted to their work, Grothendieck was an extreme case. "Grothendieck was working on the foundations of algebraic geometry seven days a ...


14

I do not know how many hours did "great" mathematicians spend on doing research but I don't think it is of any importance. The most important think IMHO is to not worry to much about the result of your work, not compare yourself with others (from the past as well as from the present). Find what is really interesting for You in mathematics and have fun !


0

I think that the only answer I could give is that as long as you like what you're doing, there's no wrong way to study. If you try to change your method and by doing that you don't have any pleasure, stop ! Of course there are boring and tough periods, but if you enjoy it more than it sucks... You'll do it !


0

I think the best advice I can give to a prospective student is maximise your future entropy, that is take a wide selection of courses initially from all sections of math- pure,applied,statistics, computer science/programming, cryptography, that will leave your options open for future specialisation when your find your niche what you really want to do, also ...


0

I'll hazard an answer. For both plans and videos, notes etc... much can be found from MIT's OpenCourseWear. I would say your general method of studying is wise except you probably already realize it's a bit heavy on lectures. I also wouldn't say it's necessarily wise to just work all the problems. Whatever you do, it needs to stretch your mind. Try asking or ...


0

I wholeheartedly agree with Georges Elencwajg: It is very appropriate for your first introduction to cohomology to be de Rham cohomology. You will learn a lot more multivariable calculus, many tools from differential geometry, and you will meet a lot of the basics of algebraic topology in an easily motivated form. From what I gather about your background, ...


5

Read about both simultaneously in the same book: Differential forms in Algebraic Topology ! (Bott was one of the best twentieth century geometers, and it shows in this extraordinary book)


2

The two are fairly closely linked, and it wouldn't be a bad idea to study them concurrently. That said, I think if you have to pick one, go with algebraic topology first. The first reason is historical. Before the modern theory of differential topology was developed, algebraic (and combinatorial) methods were employed. It is usually much easier to calculate ...


1

I can suggest two nice books covering basics of the above subjects. Hopefully they give you some perspective. Topology and Geometry, by Breadon is book primarily on algebraic topology but it treats the subject using differential manifolds, so you probably enjoy reading its 2-3 chapters. Here is the link. Foundations of Differentiable Manifolds and Lie ...


1

I am also an entering junior in HS, and I was looking for a similar project a couple months back.I decided on making and AI to play the game theory gunner game. The AI learns over time by recording the results of previous games, and changes it's strategy accordingly. This concept is simple, and also very similar to how life adapts(as commented upon by ...


-1

My Answer follow by question: "whether, when and how the fight against the scourge called cyber crime and corruption, its contribution can provide game theory methods and, generally, techniques of operations research and Invention of operational researchers? ".


1

Richard Dawkins speaks about Evolutionarily Stable Strategies in his book The Selfish Gene. The example of the hawk x dove strategies is really nice, but there are others. Hawk strategy (in any species) is agressive, it always fight; dove strategy (in the same species) is coward, it always flee. He shows how none of those strategies survive alone, and an ...


1

You have practical applications in economics, insurance (how to price a policy to screen for bad drivers) and finance (how to structure deals with the right incentives), natural sciences... (among others)


2

I find one of the very interesting applications is in nature itself, how living things naturally find optimal strategies in their interactions. For instance some occurences of the game "rock-paper-scissors" can be found in nature. The wikipedia page on Evolutionary Game Theory already contains nice material. The "most classical" application of game theory ...


3

UNDER NO CIRCUMSTANCES SHOULD YOU SUBMIT A PAPER TO ONE JOURNAL WHILE IT IS UNDER CONSIDERATION AT ANOTHER!


1

There are many ways to build better understanding. One way that is important is to work through many examples of concepts, and as you do so, to connect them to the theory. Just learning the theorems, and the various syllogisms etc. that connect them, is not sufficient (for most students) to yield a real understanding. Similar, just working problems in ...


0

Not an answer but more of an extended comment: I tell my students to forget the textbook at first, just try to solve the exercises/problems and then if you get stuck at a problem then go read the textbook. It is a deliberate exaggeration but the point is that reading a book or article will give one a false sense of understanding. Only when we get stuck and ...


3

I think what Poincare calls "certain order" can also be called the mathematical idea (behind the subject you are studying). To understand a mathematical idea, the following items are important: A mathematical idea is a dynamic creature and it continually evolves according to its applications. Take for example "continuity". It started as a notion for ...


3

In my experience learning mathematics is a lot like learning a language. You need that basic vocabulary, but in order to really have a conversation you need a deep understanding of what all the words really mean and how they fit together and interact, and all the subtleties therein. Once we are proficient at a language we no longer worry about what each ...


3

This isn't a complete answer by any means. A couple of weeks ago there was a conference based on the work of William Thurston. There were several references made to an idea he used (I believe) of having levels of understanding. When you first meet something, you can read the theorems, and get a first level of understanding. But as you come back to it, in ...


1

Experience of others Many top notch mathematicians say that excellent teachers played a significant role in their success; the book Mathematicians: An outer view of the Inner world made that clear to me. Here is a small sample of some of the things that are recorded there: "Interaction with others has always been an important source of ideas for me." - ...


-1

I agree with you. I read a story about a very smart an who took a few years off in the middle of his career to think and he said himself, that life would have not afforded him that period of clarity. More important then success itself, is one's own ability to control destiny enough, to take the time to think as hard as we work.


1

To answer your question succinctly, a first course on linear algebra should cover the basic computational tools: row reduction, determinants, and eigenvalues. A more advanced course should force the students to come to terms with more abstract language (vector spaces over an arbitrary field), and it should contain a sophisticated treatment of the spectral ...


0

The fundamental problem here is that math and science education in school is at too low a level. In schools they teach at a level appropriate for retarded children. This is because when it comes to math and science, the attitude taken by society is that you only need to know the minimum necessary to get a job and if you want to know more, you can learn it at ...


4

I can warmly recommend getting a Nintendo handheld device and a copy of one of the "Professor Layton" titles. A famous series of puzzle games and logic problems. The games are wrapped in an entertaining story too. Try a demo version first, if you don't believe me. :-)


7

If you're forgetting certain topics in math, then I suggest the best method is to do math problems daily or at least 5 days in a week. Usually when you learn math and stop doing it for over 2 weeks or more, you start to forget, lost your confidence, etc. (Personal experience). If you're looking for problems to do, I suggest: Buy a book from AOPS Visit ...


-1

If you are really not quite sure that you want to pursue mathematics, then a gap year might help you make that choice. My experience in first year was students started with a mixed range of knowledge, partly due to having come through different education systems. Somewhat counter intuitively, many of those students who had the strongest background that ...


-1

If you are going to a top five university for math, take a look at the courses they offer to freshmen. I can only speak for myself, but having just finished my freshman year, I took Calc(apostol), linear algebra(apostol), multivar(apostol), and abstract algebra(dummit and foote). It seems to me that the courses you are looking to take might be possible your ...


3

If you read nothing else of this answer: The best way to learn to think like a mathematician is working with mathematicians, and above all having your mistakes corrected by them. End of. I feel like if there's any time I could get ahead and have a chance to learn how to think like a mathematician, it is now. It doesn't matter whether you're ahead of ...


1

Going to the college will give you access to smart colleagues and smart teachers, and I think that social environment and those interactions will be hard to duplicate studying on your own. Also, probably access to a better library.


3

The University of Cambridge (the leading UK mathematics university) would strongly discourage mathematicians taking a gap year. They say: 5 Gap Year Only a small minority of mathematics students take a gap year. Some of those who do take a gap year apply for a deferred place before they leave school. Although in many subjects the extra ...


1

The right reason to take a gap year is that it would probably let you spend more time on math than college would. The wrong reason to take a gap year is because you think you are not prepared and need the year to prepare. Passion will overcome lack of background, and in fact you'll find that your background is not that bad compared to others. If you go ...


11

Some things to keep in mind: No matter what, don't lose your self-driven attitude. This is is probably your greatest strength, and is rare to come by. Always engage in some form of self-study. Will your offer of acceptance to your top-5 institution still be there next year if you don't go this fall? If not, are you willing to risk losing that ...


12

I started my math degree without having ever taken past Algebra 2 in high school. I studied and took it seriously, the kids who thought they already knew everything didn't. Your success will be determined by your energy, commitment, and readiness to learn. Head starts don't matter. I think you'll be just fine.


17

You're self-motivated; the kids you mention as a general rule pushed by parents trying to compensate for what they feel is a lack in their lives. Some will excel; some will flame out as a lot that drives them is 'being a wonder kid,' and being admired for it. At a good college, things equalize fast; and this can turn into an emotional barrier for them. I ...


1

The college would not have accepted you if they did not think you were ready; if they did not think you would pass. Their courses are designed to mould people in your specific position! Also, you mention your peers who -essentially- claim that they are geniuses on their Facebook pages. Well, people say a lot of things on the internet, and not all of them ...


2

The point not raised I think in the above varied comments is: what is the nature of mathematics, and how should one go about doing it? I believe that for any human activity one needs to discuss methodology, but there is not so much of a literature on this. Here is an article on this, for you to agree or disagree with. You can also look at the Prefaces to my ...


8

I once wrote a detailed answer of some topics in geometric and combinatorial group theory which would be suitable for a talk or a master's thesis. This is the branch of group theory which deals with (loosely) actions of groups, presentations, a bit of algebraic geometry, etc. The post can be found here. Special mention goes to Dehn's problems (decision ...


5

If you're into programming, try implementing some algorithms in computational group theory.


4

I have no idea what constitutes the appropriate level, but here some interesting topics: The Monster group (its construction, history, representations and moonshine, ...) Higman's PORC conjectures (results on the number of groups of order $p^n$ for $n\le7$, the counting techniques involved, reasons for current suspicion of likely falsehood, ...) Coclass ...


-1

You're way ahead of most U. S.-born college students. One time I asked a chemical engineering student what $\sqrt{-25}$ is. Wanna know what his answer was? 5. Let's just hope he doesn't decide to switch to electrical engineering. So if I was you, I'd be more worried about having too many programs to choose from. If you've got some family already in America, ...


5

First, don't write off the Ph.D. so casually. I know plenty of successful Ph.D.s whose undergraduate GPA was far lower than yours, whose transcripts included "C"s and "D"s. If you're interested enough in math to think you want a Ph.D., then there's probably a way to do it. Second, doing the master's thing can be a good idea to help you sort all this out. ...



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