118

I have no doubt that you there are many useful online resources for these, but many such results are available here at MSE, together with their proofs. I'll give a list of some basic results about images and preimages links to the posts, which have proofs here at MSE. I am making this CW, so feel free to add more identities and pointers to further useful ...


110

These are the sets of qualifying/preliminary examinations of US universities that I collected some time ago for the same purposes as you. (Dave L. Renfro points out in a commentary below that he compiled a similar list a decade ago, the following includes new departments, updated old broken links and removes unavailable sources). These exams are of much help ...


96

I have no doubt that many resources about Cauchy functional equations and its relatives are available. But many properties of them have been shown in MSE posts, I will try to provide here list of links to the questions I am aware of. I have made this post CW, feel free to add more links and improve this answer in any way. Several links have been collected ...


86

In my personal experience, I've found Wikipedia tremendously useful and reliable both in my studies and in my research. Rarely are there ever mistakes. Anytime you get information, especially from the internet, you should always check with at least one other source, of course. I usually use wikipedia and another source to make sure they agree, but I don't ...


80

$\newcommand{\alnul}{\aleph_0}\newcommand{\mfr}[1]{\mathfrak{#1}}\newcommand{\Ra}{\Rightarrow}\newcommand{\card}[1]{\left|#1\right|}\newcommand{\powerset}[1]{\mathcal P(#1)}\newcommand{\Lra}{\Leftrightarrow}\newcommand{\Zobr}[3]{#1\colon#2\to#3}$I have no doubt that you there are many useful online resources for these, but many such identities are available ...


57

In no particular order: Algebraic number theory notes by Sharifi: http://math.arizona.edu/~sharifi/algnum.pdf Dalawat's first course in local arithmetic: http://arxiv.org/abs/0903.2615 Intro to top grps: http://www.mat.ucm.es/imi/documents/20062007_Dikran.pdf Representation theory resources: http://www.math.columbia.edu/~khovanov/resources/ Classical ...


39

English language Of course, there is mathoverflow, many users of this site are active there too. google: mathoverflow site:math.stackexchange.com sci.math, Usenet group, google: "sci.math" site:stackexchange.com AoPS - Art of Problem Solving, phpBB, supports TeX using LatexRender, google: artofproblemsolving site:stackexchange.com MHF - mathhelpforum, ...


39

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 ...


27

To give a personal anecdote. I was thinking about the complexity of a particular algorithm for a particular type of graph (I'm a Computer Scientist rather than a Mathematician). I pop onto Wikipedia to find that a particular subproblem of the algorithm is equivalent to a known problem in mathematics. I dig a little more and find that this problem is ...


24

I find that the impressiveness of a Khan Academy video for me is negatively related to how much I know of the subject. As a math graduate student and calculus teacher, I find Khan's math/calculus videos the least impressive of the lot, his physics/chemistry/biology videos mildly impressive, and his history videos the most impressive. What this suggests to ...


24

This big list is included in Appendix A of Introduction to Topological Manifolds by John M. Lee: Let $f:X\to Y$ and $g:W\to X$ be maps, and suppose $R\subseteq W$, $S,S'\subseteq X$, and $T,T'\subseteq Y$. $T\supseteq f(f^{-1}(T))$. $T\subseteq T' \Rightarrow f^{-1}(T)\subseteq f^{-1}(T')$. $f^{-1}(T\cup T')=f^{-1}(T)\cup f^{-1}(T')$. $f^{-1}(T\cap T')=f^{-...


23

Some old qualifying exams from Harvard: http://www.math.harvard.edu/quals/index.html


23

Gathmann's notes on algebraic geometry. I think they are one of the best places from where one can start learning algebraic geometry. http://www.mathematik.uni-kl.de/~gathmann/alggeom.php Lecture notes on complex representation theory of finite groups, character theory by A Bartel http://homepages.warwick.ac.uk/staff/A.Bartel/docs/reptheory.pdf Lecture ...


23

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! ...


22

LaTeX plugin in Pidgin: http://sourceforge.net/projects/pidgin-latex/ This screenshot is taken from http://sourceforge.net/projects/pidgin-latex/screenshots/124729


22

For Spanish speaking users, I can link to Carlos Ivorra's website. He has the following material: Logic and set theory Consistency tests Set theory Descriptive set theory Non-standard analysis Algebra Geometry Analysis Functions of complex variable Number Theory Class Field Theory Algebraic Topology Algebraic Geometry Algebraic Curves Homological Algebra ...


19

Zev Chonoles, a graduate student at the University of Chicago and sometime poster here, has several wonderful sets of lecture notes of the first year graduate courses at U of Chicago. They're terrific and strongly recommended. E. Kowalski of ETH Zurich in Germany has some very good,substantial notes at his webpage on analysis, representation theory and ...


18

http://illuminations.nctm.org/ActivityDetail.aspx?ID=20 In the above link you can have a tool where you can draw graphs, check degree, find eulerian path, hamiltonian path.


18

yEd is a free cross-platform application that lets you interactively create nodes and edges via drag and drop, format them with different shapes and styles, and apply various graph layout algorithms to arrange the graph neatly.


18

I think the "How to become a pure mathematician" website is what you're looking for. It offers a well structured approach from very basic mathematics all the way up to graduate level, with links to useful resources and books. Also, any university curriculum can be useful to see in what order university students study different subjects. An example is ...


18

I recommend 3Blue1Brown's Channel. For an example, see this video. Also The Mathologer EDIT2: Unfortunately they already stopped making videos anymore: EDIT: I'd also like to add a newer channel PBS Infinite Series


17

Mathim - online chat with the possibility to use of LaTeX syntax (the first result that google returned for latex online chat or latex online chat math) http://mathim.com Screenshot from my short experiment with this tool: Mathim was also discussed here: http://www.physicsforums.com/showthread.php?t=193510 http://www.physicsforums.com/showthread.php?t=...


17

I'm quite partial to Apostol's books, and although I haven't read them (yet) his analytic number theory books have an excellent reputation. Introduction to Analytic Number Theory (Difficult undergraduate level) Modular Functions and Dirichlet Series in Number Theory (can be considered a continuation of the book above) I absolutely plan to read them in the ...


16

Harvard has one, Cambridge Tripos used to be harder (so I heard), Purdue also has some posted, and University of Florida's Math Department essentially answered your question (by copying from Dave Renfro) before you even asked by listing qual exams of other universities (UF's page seems to have disappeared completely; anyone who knows the replacement is free ...


16

Note: I will update this list as addition resources come to my attention. Lecture Notes: [Lecture Notes on Geometric Graph Theory by Janos Pach] ps [Princeton Lecture Notes] http://web.math.princeton.edu/math_alive/5/Notes1.pdf http://web.math.princeton.edu/math_alive/5/Notes2.pdf [PSU Lecture Notes by Christopher Griffin] http://www.personal.psu.edu/...


16

If you haven't read the chapter on Dirichlet's theorem on primes in arithmetic proression in Serre's Course in arithmetic, I highly recommend that you do. You can read it independently of what came before. I liked the book of Ayoub when I was a student. My memory is that it is somewhere between a textbook and a monograph, and that it covers lots of ...


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