UPDATE: I have not kept this blog up-to-date for over two years now. I have started a new blog at ayoucis.wordpress.com
This blog is something that I devote a fair amount of my time to (hopefully that is apparent). So, why do I do it? That question is best answered in two parts: the selfish and unselfish reasons.
The act of writing math, of discussing it is the didactic equivalent of ambrosia for me. Phrasing theorems or ideas in my own language, my own notation and then proving them with either a reinterpreted [from a book] or completely new [to me] proof is how I best learn. So, this blog is how I learn. As my math knowledge base has grown when I find a new book that interests me I rarely read the chapter and then do the problems. I find another approach much more helpful. I first like to skim through a chapter and see what the ‘big picture’ is. Then I think about it, doing some examples to get the intuitive/conceptual idea down. I then search far and wide, internet and book(s) to find every conceivably interesting theorem relating to this topic. I write these theorem statements all down and attempt to prove them on my own, without the assistance of a book. When I am unable to complete a proof I look at a book for an idea and then try to proceed, but this is a final effort. I subscribe to the school of mathematical learning which can be best summed up by Paul Halmos’s famous quote: “Don’t just read it [mathematics]; fight it!” So, this blog is, taking Halmos’s metaphor one step further, my war journal.
The other selfish reason for this blog is one of hopefully fruitful forethought. Eventually everyone forgets theorem statements, proofs, ideas, etc. Consequently, I will undoubtedly need to one day look back at my notes on ____ and ‘refresh my memory’. Because I wrote these notes in my own words, my own notation, and with my own personal slant they will hopefully be much more conducive to this learning/remembering. So ideally this will be my personal account of all the math I’ve learned from basic undergraduate to postdoctorate, and beyond.
The unselfish reason I maintain this blog is much easier to convey–I want to make material concerning all kinds of mathematics more widely available to anyone who has interest in math. This is not to say that I believe that my writing style or insight is greater than any of the hundreds of other online mathematical expositors, but hopefully at least one person’s take on mathematics is similar enough to my own that this will help. This is one of the key motivations for why I blog about, in particular, what I blog about. For example, at the time of this posts writing I am studying commutative algebra and differential geometry. Yet, if you looked at my blog at this current moment you would see that I am ostensibly blogging about multivariable analysis a la Spivak’s Calculus on Manifolds and general ring theory. It does help me to continually revisit these topics to better understand them, but that is only a small part of my intentions (although it does make my ‘war journal’ more complete/coherent). The main reason is that I would like this blog to be a self-contained account of most subjects. Since you can’t do commutative algebra without general ring theory, here I am plodding through isomorphism theorems and decomposition of modules. In this way, my aim is very similar to that of John Armstrong’s blog Unapologetic Mathematician.
So that is the long-winded reason I make this blog, but I hope that it can, in some way, be useful to you. If you have any questions, feel free to contact me through this blog or at my e-mail email@example.com!