# Gracious Living

The Future
December 8, 2012, 17:31
Filed under: Uncategorized | Tags:

I abandoned this little blog for a good period of time, and came back to find pending comments, so I guess people are still reading it.  In case you’re curious or like what you’ve read, I’ll briefly say what’s been going on with me and what will happen to Gracious Living.

Since my last post in 2011, I made some poor/’interesting’ decisions, entered a dark period of my life, recovered from that dark period, wrote a thesis about model categories, graduated college, and started grad school at Northwestern.  I’ve been learning a lot about homotopy theory and algebraic geometry over this first quarter.  I’m still the same O. G., confused, thoughtful, struggling, frustrated, excited, and more interested in math than ever.  Northwestern’s been treating me well so far and I’m looking forward to the waiting future.

What I wanted to do with this blog — discuss math from the ground up — was ultimately a quixotic task.  There is simply too much math, and ultimately the only laypeople capable of reading my posts would be those who are somewhat mathematically-minded already, and accept the mathematician’s abilities to define arbitrary objects, trust in abstract metaphor, and work logically from systems of axioms, and his or her motivations in doing these things.  What’s more, a lot of my writing was, in retrospect, very boring — not just because of my tendency to wordiness, but because of the subject matter itself.  In order to reach math’s fascinating peaks, you’ve got to traverse a lot of valleys of definitions and technical lemmas, and insisting on doing all of this in the ‘right order’ may not be the best way to communicate.  (Not to mention that it takes a hell of a long time.)

This blog’s time has passed, but I haven’t stopped blogging.  I currently have two projects ongoing.  On my Tumblr, I write sporadic essays about film, literature, music, and philosophy, post typically short and weird fiction, and do the requisite tumblry things what with the gifs and the reblogging.  I’ve also just started a new math blog on WordPress with a group of other Northwestern students, intended to be something like the Secret Blogging Seminar.  (As I write this, it has exactly one post up, so we’ll see.)  I’m expecting the things we write about there to cover a wider range of difficulty levels, primarily either things we know a lot about and want to talk about or things outside of our respective fields of concentration we feel like learning in public, so to speak.  As I care a lot about being able to explain my ideas to other people, I’ll probably publish the occasional layperson-level post, and certainly will if someone asks me to, but not with the same Bourbakistic demand for foundations as I had here.

Since people still read this, I’ll check on the comments periodically, but the best way to communicate with me is through either of the other two blogs, or by emailing me at allispaul at gmail dot com.  I’m a weird dude but a friendly one and I like corresponding with strangers.

Thank you all for reading and may our paths cross again!

Paul

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It is 2011 now.
January 2, 2011, 09:37
Filed under: Uncategorized

It’s looking to be a pretty good year.  I’m cooling my heels in the Philippines, where it’s nice and sunny but I don’t know practically anyone.  All the more excuse to sit outside in the sun and do some math.  My parents, who are awesome, gave me Stein and Shakarchi’s Princeton Lectures in Analysis, Volume I: Fourier Analysis.  Volume III (Real Analysis) was the textbook for my analysis class this past semester, and I’d briefly skimmed Volume II (Complex Analysis) for a freshman year class, but I wanted to really sort of start from the beginning and get my hands dirty with some of the basic material of analysis.

The subject itself is pretty cool: Fourier series are a way of decomposing any periodic function (i. e. $f(\theta+2\pi)=f(\theta)$ for all $\theta$) as an infinite sum of sines and cosines, and this sum generalizes to an integral called the “Fourier transform.”  A good comparison is the Taylor series, which decomposes a function as an infinite sum of polynomials.  Oddly enough, besides last semester’s analysis class, the only place I’d seen this was the same freshman class where we used Stein and Shakarchi II: you can give a nice algebraic summary by saying that the set of functions whose squares are integrable (“$L^2$ space”) is a vector space with inner product $\langle f,g\rangle=\sqrt{\int fg}$, and the set of functions $\sin nx,\cos nx$, and $1$ forms an (infinite) orthonormal basis for this vector space.  Regrettably, you have to use the Lebesgue integral to do this, and since S&S want their series to be self-contained, they only use the Riemann integral.  Which makes questions like “when does the Fourier series converge properly?” a whoooooole lot easier, but you kind of feel you’re missing the big picture.  On the other hand, the books are very well written, lucid without being verbose, and the problems are usually pretty interesting.  S&S I is geared on the easy side of things and is probably a good read for even interested high schoolers (and just like high school, there are tons of integrals to be done!  woohoo…).

Probably the awesomest Christmas present I got was the sadly-out-of-print Mathematics Made Difficult by Carl E. Linderholm, written in the 70’s.  As the title would suggest, the book is fed up with the never-ending attempts to make math seem easy or fun, and instead aims to present basic arithmetic in as difficult a way as possible.  It’s an odd book, and you see why it’s out of print — it’s geared towards mathematicians and math students, defining the set of natural numbers, for example, as an initial element in the category of pointed monoids, but at the same time it’s possessed by a Groucho Marx-style wit and wordplay, and ends up making all of these wry observations about the philosophical issues of thinking about math that are usually consigned to pop-sci books.

“[One might object, to the proposition that $2\times 2=4$, that] 2 units multiplied by 2 units is not 4 units but 4 squared units.  But 4 squared is 16.  Hence, $2\times 2=16$.

“[To which I reply that] it is not at all obvious a priori that 16 is not exactly 4.”

And in the introduction, he starts a paragraph with “Consider a category…” and proceeds to give a long, rambling list of properties that category should be assumed to hold.  Then he ends with “Then it is possible to think, for we have just defined a Boolean algebra.”  Hilarious.  (A Boolean algebra is a mathematical structure that looks like the class of sets with the operations of union, intersection, and complementation.  Deductive logic itself basically has an underlying Boolean algebra structure, so the existence of one allows us to conclude that logic, and so thought, exist too.)

I’m also reading Infinite Jest for a book club-type thing with my friends, and helping my girlfriend put together a website.  As someone who’s always been sort of interested in techy stuff like programming and web design, but never had any real reason to go out and do it, I cannot emphasize more the coolness of actually having somewhere to put one’s skills to the test and force oneself to learn.

But I guess the reason why I’m starting this post is that I need to pick another math book to read.  I’ve fortunately got a pretty nice electronic collection, and I’d like to find a subject that’s been somewhat out of my path so far — algebraic geometry, number theory, and category theory are all looking good.  Lethe Ar has a review of J. P. “Duh Bear” Serre’s Arithmetic here — I share his disdain and confusion for the subject but it looks like it might be an interesting read… Or before tackling those, should I brush up on my commutative algebra?  I’m fine with groups and fields, pretty good with homological stuff, but always a little cautious with rings and modules.  I’ve got Dummit/Foote and Atiyah/Macdonald, both of which I believe are pretty standard texts (Holomorphusion‘s been going through Atiyah/Macdonald, FWIW).

If any math person reading this has any recommendations, I’d love to hear them.  And, as usual, if anybody’s curious about some topic, I always welcome suggestions.  Happy New Year!

Symmetric Groups
December 18, 2010, 11:36
Filed under: Algebra, Math, Uncategorized | Tags: , , , ,

We’ve seen symmetric groups before.  The symmetric group on an arbitrary set, $S_X$ or ${\rm Sym}(X)$, is the group of bijections from the set to itself.  As usual, we’re only interested in the finite case $S_n$, which we call the symmetric group on $n$ symbols.  These are pretty important finite groups, and so I hope you’ll accept my apology for writing a post just about their internal structure.  The language we use to talk about symmetric groups ends up popping up all the time.

Differential Geometry and the Sphere Theorem
December 16, 2010, 20:38
Filed under: Topology, Uncategorized | Tags: , , , ,

I just wrote a paper about the Sphere Theorem for my differential geometry class.  Since I can’t really get it out of my head, I thought it’d be fun to use to give a brief overview of differential/Riemannian geometry, in more or less layman’s terms.  This is somewhat of a departure from my normal posting style: though you’ll get more out of this if you’ve understood the topology posts so far, I’ll try to write in broad enough strokes that calculus is the only prerequisite.  Really, the most complicated idea behind this is the idea of a map from $\mathbb{R}^m$ to $\mathbb{R}^n$ being differentiable.  I also refer to vectors, but I think that that’s a pretty intuitive concept, in general.