# Tag Info

3

A few suggestions for things that might improve the situation: (1) Buy the books that are well-illustrated. This will indicate to authors and publishers that illustrations are important. Another very well illustrated one is Trefethen's book on approximation. See also this question. (2) Learn to use the graphical tools in packages like Matlab and ...

2

Set theory is absolutely necessary to learn more advanced mathematics. It is needed for just about every branch of mathematics, if not every branch. In my opinion, it would be a good idea to start learning some basic set theory notions at least. It will definitely show up in classes like real analysis, complex analysis, and probability just to name a few. ...

2

A vector field $F|_S\colon S\to\mathbb{R}^n$ is an assignment of $n$-dimensional real vectors to points in a subset of $\mathbb{R}^n$ so it's really just a vector-valued function $F\colon\mathbb{R}^n\to\mathbb{R}^n$ restricted to a subdomain $S\subset\mathbb{R}^n$. Note that the domain and codomain of $F$ have the same dimension, but $S$ can possibly be a ...

1

It could also be a more "well-known" result, which is normally found in big textbooks but is sufficiently independent that you can treat it thoroughly in a dozen of pages. This happens often for example in computational number theory, where many algorithms use elementary notions (so not much background material needed) but are based on sufficiently ...

3

Completely understand a single "thing" (the solution to a specific problem in a paper for example) and explain it in detail in your -own- words would be my suggestion. This may include filling out the details of the paper that have been left to the reader.

0

Two fantastic books that every mathematician should know are Counterexamples in Analysis by Olmsted Counterexamples in Topology by Steen and Seebach, already mentioned in the comments.

0

Alert .. highly iconoclastic answer follows :) . I believe you have to just have the "gift" for it. You either do or don't. It's like music. Unfortunately I think that's the truth. (Consider too Douglas Hofstadter's writings on how he had a gift for math -- BUT -- only to a certain level. Very astute!) In terms of just getting through, it's ...

2

If you have already learnt group theory, I may suggest you to go through the book 'theory of algebraic numbers' by Pollard & Diamond. It's a really good treatise to start off. You don't even need to know the definition of ring to read this book. Everything is given there in a very well setup. After having finished that book, you may pay a look at 'A ...

1

Don't go it alone. Don't be afraid to talk to your professor/TAs when you're having trouble understanding something. (Actually, this is a good rule for any class.) Don't be afraid to collaborate with your classmates, either -- doing so is silently discouraged in most high school classes, where homework is a big part of your grade, but it should be ...

1

I can't comment or even upvote, so I'll put this as an answer. Several people already told you this, but it can't be stressed enough: Solve a lot of problems! After that, solve even more. Solve problems from the textbooks, try to solve problems which you've heard about from your professors. Try to solve the famous ( P = NP ? ) problem - in this case you will ...

3

The best advice I have ever gotten: Do the exercises without looking at the suggested solutions. This will force you to think critically. In the beginning this will probably be very time consuming, but do not give up, because after a while the payoff will be huge. If you really get stuck, discuss the problem with someone else. In this way, you will have to ...

2

The best advice I got while an undergrad Math major was to take programming/software courses. After picking up a minor in Computer Science (and then a graduate degree in the same), I'm now a full-time software developer and loving it! Once you get calculus out of the way and have time for a few electives, try your hand at discrete mathematics: ...

2

Get some old test questions from previous years for the same professor. Hate to say it, but after wondering why the dumb frat guys were scoring so well in tests, I came upon this 'secret'. Turns out professors are lazy and recycle old questions. You can beat yourself over the head doing hundreds of example questions from books, but there's nothing that works ...

20

Apart from all the good answers that the other guys provided, I have one suggestion: Use pen and paper! In other words, do the exercises (or new concepts) instead of studying them. Imagine a day in the future when you are reading a question or studying a new concept. Then you start a conversation like this with yourself while looking at the textbook: "Nah! ...

30

I completed two bachelor degrees in mathematics and physics in Vienna. I don't know how it compares to an non-European bachelor degree, but I think my experience may be of help. I can't give advice on (calculus) textbooks but maybe I can give you some general advice about studying mathematics in university: Be precise: precision and adherence to the ...

2

Okay I'm not all that good with the advice. But I will tell you this. I am not a fan of Stewart's book. Trust me if you really want to grasp the roots such as Mitch Knight suggests above I strongly urge you to consider a few other texts as supplements. These are my recommendations. You won't go through them all, obviously. But if you can properly grasp a ...

4

I recently completed second year undergrad down at Melbourne Uni, and although we had prescribed texts, I used Stewarts calc. It's important to realise that although people stress that repetition (eg doing question after question) results in success, I disagree. Understanding is the key, persevere to understand the thinking behind the mathematics and you'll ...

1

A wonderful book is Proofs from THE BOOK by Ziegler and Aigner (the capitals are in the title - I am not actually shouting). This book contains lots of wonderfully elegant proofs from number theory, geometry, analysis and so on. The proofs are almost always at a low-enough level for a good undergraduate student to understand (the idea is that they are simple ...

1

You may try to understand the key points in the AKS primality test. The book Primality Testing in Polynomial Time by Dietzfelbinger is very nice. There is also the more recent, and probably simpler, Primality Testing for Beginners by Rempe-Gillen and Waldecker. See also the survey It is easy to determine whether a given integer is prime by Granville. A ...

1

It's a bit of a challenge but perhaps you could summarize the activity following Yitang Zhang's recent paper on the gaps between primes. A friend of mine is doing so as his Master's dissertation; it's doable and it'd get you up-to-date. You'd have to black-box a few results as an undergraduate though.

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One possibility is to study the so called "Euclidean proofs" for Dirichlet's theorem on arithmetic progressions, which are sometimes possible and are much easier than Dirichlet's methods form analytic number theory. However, it is an interesting question to say what an "Euclidean proof" should be, and for which arithmetic progressions $an+b$ it works. For ...

0

My 2 cents (Tips): When you read a solution, try to figure out why the person decided to do that. Getting to a solution is never as simple as it is stated. The writer didn't just read the problem and "aha! this is how you do it!" it took some inspection and trial and error. Recognize patterns. No problem is unique -- you can always apply past knowledge to ...

0

I think the most important thing is to find right problem set to work on. Too easy or to hard will not help much. Your strategy 1) is not recommended, you will not improve your problem solving skill by just reading the solution without much tries and fails. So my suggestion is to use strategy 2), but start from easier problems, the ones that you can solve in ...

1

One of the best and biggest site for math contests is Art of Problem Solving. On its forum there are thousands of problem and they are nicely divided in category, so you can practise problems of specific area. Another quite good site is Imo Math. All previous IMO math problems are there. Also if you spend some more time you can find other problem. Also ...

3

Here are some things to keep in mind: Not everybody solves every problem, nor solves every problem quickly. It is likely that you'll learn the most from trying to solve problems. You can probably learn helpful strategies about solving problems from "coaching" texts, but you are likely to learn (and retain) a lot more from practical experience. Don't "try ...

1

I realize I'm late to this party, but I'd note that the other answers try to describe a protocol best for the recipient. What's best for you? I fly on planes a lot (usually within the United States). I study both Computer Science and Abstract Mathematics (primarily separately). Often, when I sit down, the person sitting next to me will ask me what I do. ...

2

I think it was Einstein who said, "If you can't explain it to a six year old you don't understand it yourself." I find that it is always best to stick to the very basics, omitting superfluous details that matter only to someone working in the field.

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A colleague of mine once said: "I do stuff with numbers that you put in a short and wide box to make the world a better place." (Frame Theory: generalization of Linear Algebra with overcomplete bases) See this blog entry for more: My research explained, using only the 1000 most common English words (Duston G. Mixon)

1

I work with two algebraic objects that are closely related called algebraic groups and Lie algebras. These objects can act on spaces (like three-dimensional space) by transforming them in a nice way, and I study these actions. Can you make a nice analogy that illustrates what a "space" and what "transforming in a nice way" looks like in everyday ...

7

Layman's terms first; you can try to guess the mathematical problem. There are two tennis players. They decide to play each other with a new set of rules: They play point by point. The referee counts the points. The first time that one of them is three points ahead he's the winner. Now you'll notice that this game could go on forever, if they play ...

0

Look at some Richard Feynman video. He is great at explaining and visualizing concept, he makes imagination works for the layman, not just pure logic. However, when asked some difficult question he can't respond in layman terms, he says "I can't explain it, because it is not familiar to anything you have seen and don't have the necessary vocabulary to ...

4

Simply describe outcomes. Tell why it matters in the big picture. Folks that want details will certainly ask.

2

Despite not being a mathematician myself, I would like to contribute a bit as in my field I share the same problem often (depending on the project I am working on). What I have learned is that what works best is to bypass the entire description of what you're actually doing and just jump straight ahead to applications even if you will never get anywhere ...

7

"There are only two kinds of math books: Those you cannot read beyond the first sentence, and those you cannot read beyond the first page." -- C.N. Yang One of my favourite sayings from a Nobel laureate mathematician. In my view, the ability to communicate complex mathematical ideas in a simple way is a very special gift that some people have and others ...

31

As a laymen myself, let me shed some light on this, and perhaps prevent this potentially awkward conversation from happening. The problem is you're trying to explain your research in a few sentences and induce wonder and so on. There is no solution. You'd have a better response if, instead of explaining what you're doing or how you're doing it, you told ...

1

I work in a branch of mathematical logic. Although I'm a mathematician, logic is multidisciplinary, and is also studied by philosophers, computer scientists, and linguists. The essential idea in logic is the relationship between syntax and semantics. Syntax refers to symbols and language. Most of mathematics is a linguistic exercise: we manipulate ...

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On my field: I study curvature. I study higher-dimensional shapes. I study a mix of calculus and geometry. But not the geometry of flat things like triangles and squares, but smooth curvy wavy things. On (one aspect of) my subfield: I study shapes that are like soap films. Like, if you dip a wire in a soapy water, then the bubble will ...

5

My problem is in some sense the opposite of the one described by the OP. Although my background is in pure mathematics, my current work is in statistics. The problem I have is that when I tell people what I do, they think they know about the subject when they really don't. In a way, that's a much harder issue to deal with--to dispel preconceived notions ...

4

Thanks for asking this question. As a layman (but fan of math) -- and one with recent experience asking mathematicians what they do -- I think I have a pretty relevant perspective to offer on this. Let me first suggest what not to do (not that I assume you do this, but just in case, and for anyone else). Almost every mathematician or scientist I've asked ...

16

First, I think it's important to consider the person's motivation for asking this question. If they're just asking to be polite, then I think your answer, or basically any answer, would be fine. They're not going to take away much from whatever few words you say, so those words aren't really important. If they truly are interested in your work, then it's ...

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I'm a little disappointed by the comments. Granted, it's hard to explain mathematics, but having the attitude that you're not even going to try is not doing mathematics PR any favors. We can't in good faith expect the public to fund our research if we're not even going to try to tell them what we're doing with their money. First, I hope you won't take this ...

16

Last time I tried to explain it went like this: "So, mathematicians sometimes like to have a theory on hand from which you can derive as much of mathematics as possible, something to check your intuitions against when the stuff you're studying gets really flighty. There are a few alternate formulations of such a theory, and I study one of the weirder ones." ...

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Sam sat with his eyes closed for several minutes, then said softly: "I have many names, and none of them matter." He opened his eyes slightly then, but he did not move his head. He looked upon nothing in particular. "Names are not important," he said. "To speak is to name names, but to speak is not important. A thing happens once that has never happened ...

1

Get Friedberg, Insel and Spence's "Linear Algebra" 4th edition, sit down, and do every non-computational problem in the first three chapters. It's my favorite math book. It introduces linear algebra via an axiomatic approach that you'll probably see often as you go on in math. It's rigorous, it provides intuition and a meaningful framework to linear ...

2

The tension between following abstract rules as against intuition has been present in mathematics for centuries if not much longer. From the time of Newton and Leibniz onwards mathematics became more algebraic due to the calculus. For example, the eighteenth century mathematician Lagrange played a critical role in moving away away from diagrams towards ...

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No, don't abandon your love of analogies and your search for connections to the "real world". But a caveat: be guided by it, not shackled to it. A few more remarks. (1) Linear algebra can be presented sevaral different ways: computationally, conceptually, geometrically, physically, etc. It sounds like you've encountered a mismatch between your course and ...

1

Linear algebra (and also functional analysis to some extend) are fields where it's still possible to have geometric interpretations. Linear transformations (in particular, matrices) can represent reflections, rotations and scalings which transform vectors. You lose some exact graphical interpretation when you move from 2- and 3-dimensional vector spaces to ...

2

Linear algebra is by no means about computations over concepts. There's actually a rather precise dichotomy that approximates that between computations and concepts in linear algebra, namely that between matrix algebra and the theory of abstract linear transformations. It's from the latter perspective linear algebra most naturally displays its ability to ...

2

You've written this post as a leading question, you'll get much too bloomy answers. My advice is to search for the motivations behind the introduction of this and that concept - knowing that these exist will let you concentrate on the plug and chuck you need to get your answers. I want to add that not all mathematical object are physical things - e.g. the ...

2

If you want single variable stuff, Abbott's Understanding Analysis is a good start. It is sort of like a baby Baby Rudin. If you want both single and multivariable analysis, my personal favorite is Fitzpatrick's Advanced Calculus. The texts by Rudin are also of course "the standard" in analysis. If you want to look at complex analysis, I like a book by ...

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