47

It's an abbreviation of quod erat demonstrandum, which is the Latin translation of a Greek phrase meaning "which had to be proven". To the ancient Greeks, a proof wasn't complete unless the last sentence in your proof was basically the statement of the theorem. Putting QED after that sentence was their way of saying, "and that's what I was trying to prove, ...


39

Here are some general pointers for gaining intuition in topology: Learn lots of examples early, and use them to guide your understanding. Take the definition of a topology, for instance. The original motivation for this definition comes from familiar topological spaces, such as the real numbers or, more generally, $\mathbb R^n$ or, more generally still, ...


35

The bad news is that mathematical notation can be regarded as an incredibly dense kind of compression: you take a very subtle and carefully evolved idea and choose some symbol to represent it. The "quick meaning" of a symbol will typically lose all the subtlety. There's a worse problem: typesetting is a pain, so it's pretty common for mathematicians to use ...


33

For $\$2,300$? No, that's ridiculous. I'd honestly just recommend wikipedia, this site, etc.; but if you want a single text, I'd recommend the Princeton Companion to Mathematics. It's written (and written well) by accomplished and knowledgeable mathematicians, and it's very readable.


32

From A Comprehensive Dictionary of Mathematics by Roger Thompson: "quod erat demonstrandum" (Latin) -- This stems from medieval translators' habitual tendency of translating the Greek for "this was to be demonstrated" to the Latin phrase above. This appeared originally at the end of many of Euclid's propositions, signifying that he had proved what he set out ...


30

For general topology, it is hard to beat Ryszard Engelking's "General Topology". It starts at the very basics, but goes through quite advanced topics. It may be perhaps a bit dated, but it is still the standard reference in general topology.


28

These are all symbols that are usually learned in high school or introductory math in college. There is a wiki page that gives names and purpose for most symbols: https://en.wikipedia.org/wiki/List_of_mathematical_symbols Wikipedia may not be a reliable source for some topics, but with math it is usually very good for a run down in an unknown or new topic.


22

I suspect you have vastly underestimated the breadth and depth of mathematics if you think you can "learn every single math topic pretty well" from an encyclopedia or any other source(s), but I commend you for your enthusiasm and ambition. What you can do is use an encyclopedia (online or otherwise) or tome such as The Princeton Companion to Mathematics (or ...


18

Stephen Willard, General Topology This book is less complete than Engelking, but still contains enough material to make a good reference book. It is also quite cheap, as a Dover book.


14

No. Not only is the Cram101 series not useful for learning, but it's a scam. Here is a review of the "outline" for Billingsley's Probability and Measure: I thought it was some comments on the Billingsley's book, Probability and Measure, but it is just a small notebook with a kind of a (very bad) dictionary of probability terms in the margin of each page. ...


14

Vectors: The area of a parallelogram embedded in a three-dimensional Euclidean space can be calculated using vectors. Let vectors $AB$ and $AC$ point respectively from $A$ to $B$ and from $A$ to $C$. The area of parallelogram ABDC is then $$\left|AB \times AC\right|$$ so that the area of a triangle is half of this, giving $$A_{\text{triangle}} = \frac{1}{2} |...


14

This is a deep question raising a lot of issues about mathematical education. I could discuss it at length, but I'll try and keep this to some practical advice: Learn as you go, one symbol at a time. There are countless (but not uncountable) mathematical symbols, and each with different meanings in different contexts. For example, $\pi$ is usually taken to ...


12

The following 3 volume set (translated from Russian, edited by A. V. Arhangelskii) deserves to be mentioned among references for general topology, too. It is part of Encyclopaedia of Mathematical Sciences series. General topology. I. Basic concepts and constructions. Dimension theory. Encyclopaedia of Mathematical Sciences, 17. Springer-Verlag, Berlin, 1990....


11

Please forgive what should be just a comment, but is too long to post as such. The OP's plaint brings to mind a passage from Thomas Mann's novel Royal Highness: What he saw made his head swim. A fantastic hocus-pocus, a witches' sabbath of abbreviated symbols, written in a childish round hand which was the obvious result of Miss Spoelmann's ...


11

ATTENTION: No cucumbers were harmed in the making of the graphics! Here's a proof that I thought was kind of fun: You can try this experiment to prove $(e^x)'=e^x$. Materials: A cucumber A knife A cutting board An almost infinite amount of time Place the cucumber on the cutting board and assume its radius to be $1$ unit in length. Now, using the knife, ...


9

So is there a singular book or at least a place that I can go to reference these symbols to get their meanings? Without being abstract, there is, and it won't/shouldn't take more than a few months of applied effort. Calculus: "Calculus" by James Stewart. Read and quickly (don't waste a lot of time getting bogged down by textbook wordiness) work through ...


8

I found Category Theory by Steve Awodey to be a really good introduction to the subject. It doesn't lead directly into algebraic geometry (or, indeed, in any particular direction), but gives a readable introduction to the subject with plenty of examples and problems. When you've worked through this you could move on to a more advanced text or one which is ...


8

The Handbook of Set-Theoretic Topology is a great reference on many advanced areas of general topology.


8

"How does one do research in any field?" You are not expected to know how to answer that question at the start of a PhD program, and an advisor's primary job is to guide you through that process. The commenters are right that this is an extremely concerning situation that you should take seriously. Seek another advisor immediately.


7

Encyclopedia of Mathematics The Encyclopedia of Mathematics wiki is an open access resource designed specifically for the mathematics community. The original articles are from the online Encyclopaedia of Mathematics, published by Kluwer Academic Publishers in 2002. With more than 8,000 entries, illuminating nearly 50,000 notions in mathematics, the ...


7

Some possibilities of finding the translations which I am aware of. Wikipedia. Surprisingly, quite often, when I look at an article at English Wikipedia, I find translation of the term by checking language mutations of the article. Of course, this works better only for Wikipedias that are large enough. Wiktionary is a free online dictionary, which is built ...


7

Groups of order less than 30 are at http://opensourcemath.org/gap/small_groups.html Also, http://world.std.com/~jmccarro/math/SmallGroups/SmallGroups.html goes up to order 32.


6

You may also want to read a nice article of Conway, Dietrich and O'Brien http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf And also the paper of Besche, Eick and O'Brien http://www.math.auckland.ac.nz/~obrien/research/2000.pdf which contains a table of the number of groups of order $n < 2001$.


6

For Russian-English dictionary, try Lohwater: Russian-English Dictionary of the Mathematical Sciences. Moreover, in some cases, checking interwiki links at Wikipedia can be also helpful.


6

A really nice book about general topology is "Topology" by "James Dugundji". For those who can read portuguese I'd recommend "Elementos de Topologia Geral" by "Elon Lages Lima" - a great book.


6

The Metamath Proof Explorer "... has over 12,000 completely worked out proofs ..." [1] accessible via an indexed theorem list. [2] Each theorem has a corresponding unique label. The main Metamath page describes the project, the Metamath language, and programs and databases available for use. [3] The Metamath proofs are mechanical, and may or may not be ...


6

$s=pr$ where $p=\frac{a+b+c}{2}$ and $r$ is the radius of the inscribed circle. $s=\sqrt{r\cdot r_a\cdot r_b\cdot r_c}$ where $r_a,r_b,r_c$ are the exradii of excircles.


6

A two part paper by Marcus Baker (1849-1903) in vols. 1 and 2 of the Annals of Mathematics, readily available online, gives $110$ such formulae (warning: the Wikipedia article on triangles states that some of them are erroneous). A collection of formulae for the area of a plane triangle] [Part 1], Annals of Mathematics (1) 1 #6 (January 1885), 134-138. ...


5

Please look at "Topology and groupoids", http://www.bangor.ac.uk/r.brown/topgpds.html which is published privately to keep the price down, and an e-version for £5 is available through the above site. The first part is a geometric account of general topology, with motivation for definitions and theorems, starting with the neighbourhood axioms, as more ...


5

You can also take a look at this wonderful set of video lectures by Eugenia Cheng


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