Gracious Living

The Future
December 8, 2012, 17:31
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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!


Ideals: Numbers But Better
March 8, 2011, 14:27
Filed under: Algebra, Math | Tags: , , ,

I left you with a bit of a teaser.  We’d defined rings, integral domains, and fields, and even seen a few examples, but in such a short exposition, there wasn’t very much time to give you the tools to work with them.  There turn out to be ideas that make better sense in a ring, like primality and divisibility.  But to understand them, we need to develop a little machinery, which in this case is the theory of ideals.  As I show below, ideals are like better-behaved numbers, and help us understand the structure of, among other things, the integers.

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The Construction of the Reals: Metric Completion and Dedekind Cuts
March 6, 2011, 21:31
Filed under: Algebra, Math, Set Theory, Topology | Tags: , , , ,

It looks like I’m getting views now, which is surprising.  I’ve been pretty busy with schoolwork, but I really want to get this blog up to speed, particularly because I’d like to start discussing things as I’m learning about them.  I’d also like to make more non-mathematical posts, but maybe these are best left to a separate blog?  Thoughts?

Our first example of a field was the field of rationals, \mathbb{Q}.  Recall that this was the field of fractions of the integers, which were in turn the free abelian group on one generator with their natural multiplication.  But now it appears that we’re stuck.  While we intuitively know what \mathbb{R} should be — it’s a line, for crying out loud — there seems to be no algebraic way of “deriving” it from \mathbb{Q}.  A first guess might be to add in solutions of polynomials, like \sqrt{2} as the solution of f(x)=x^2-2, but not only does this include some complex numbers, it also misses some real numbers like e and \pi.  (We call such numbers — those that aren’t solutions of polynomials with rational coefficients — transcendental.  It’s actually quite difficult to prove that transcendental numbers even exist.)

Instead, we turn to topology.  Below, I give two ways of canonically defining \mathbb{R}, one using the metric properties of \mathbb{Q}, one using its order properties.  I found this really interesting when I first saw it, but I can’t see it interesting everyone, so be warned if you’re not a fan of set theory or canonical constructions.  One of the topological techniques we’ll see will be useful later, but at that point it’ll be treated in its own right.

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Rings, Integral Domains, and Fields
February 2, 2011, 19:36
Filed under: Algebra, Math | Tags: , , , , , ,

In which I sort of breeze through a couple of really awesome and really important concepts.  Last time, we classified abelian groups — now we’ll see what happens if we require additional structure on the groups.  In particular, I’m going to construct \mathbb{Z} and \mathbb{Q} similarly to how the Peano axioms constructed \mathbb{N}.

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The Classification Theorem for Finitely Generated Abelian Groups
January 29, 2011, 16:23
Filed under: Algebra, Math | Tags: , , , ,

Wow, it’s been a long time since I’ve written anything on this blog.  I’m taking algebraic topology and an algebraic number theory course this semester, and I started reading through Atiyah and MacDonald’s Commutative Algebra over the winter.  So I thought I’d continue with a little algebra.  The algebra we’ve done thus far has been highly noncommutative, for the most part — we investigated groups like free groups, symmetric groups, matrix groups, and dihedral groups in which the order of operations mattered.  As you might expect, with abelian groups, the theory becomes much simpler, and the subject called “commutative algebra” is just the study of abelian groups with extra structure — something like a scalar multiplication, as in the case of vector spaces, or some other operation.  But first, we need to understand abelian groups.

When talking about abelian groups specifically, we usually write them additively: the group operation applied to a and b is a+b, and then we can build expressions like 3a+2b.  The proof I give below is due to J. S. Milne, who in turn says it’s similar to Kronecker’s original proof.  Of course, I’ve added more detail in places where I thought it was necessary, and taken it out where I thought it wasn’t.  There are other, more common proofs, typically using matrices, but I find them unwieldy and inelegant.

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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!

Countability Axioms
December 31, 2010, 01:55
Filed under: Math, Topology | Tags: , , , ,

So far we’ve seen two basic families of properties of topological spaces.  The connectedness axioms tell us in what ways it is possible to break our space apart into pieces.  The compactness axioms tell us how bounded the space is.  What we’re going to look at today is a set of axioms that deal with cardinality.  It should be mentioned that topology, for the most part, doesn’t really care about large cardinals — at most, we’re dealing with \aleph_1, the cardinality of our favorite counterexample \omega_1, and c, the cardinality of the reals.  These are equal if we accept the continuum hypothesis, and in either case we often talk about them in terms of countable subsets — sequences and the like.  The reason that countability is so important is that the properties we’re about to study are typical of metric spaces, and metrizability is a central question of point-set topology.

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Limit Point and Sequential Compactness
December 30, 2010, 02:51
Filed under: Math, Topology | Tags: , , ,

Back to topology.  The interesting thing about compactness, as I see it, is that its definition isn’t very intuitive.  We want to talk about what are basically “closed and bounded” sets without really using closedness, which doesn’t behave well with subspaces, or boundedness, which doesn’t behave well with anything.  At the time this idea came about in the early part of this century, the mathematicians ssewho invented it (Borel, Weierstrass, and Bolzano, mainly, I think) messed around with a couple other definitions but they turned out to rely on intuition from Euclidean space, and so not be nearly as useful.  In this post, we’ll study those other definitions.

I debated with myself for a long time about what to do first: this, the separation axioms (which measure how good the topology is at distinguishing disjoint sets), or the countability axioms (which measure, well, how countable the topology is).  In a sense, each one depends on the other two for examples and theorems, and in this case, though it’s easy to show that all the properties I’m listing are different, we’ll need to refer to the other properties to discover when they are the same.  So I expect this post to be brief; after we have all the other properties defined, it’ll be easier to talk about what implies what.

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Subgroups of Free Groups are Free
December 28, 2010, 07:32
Filed under: Algebra, Math | Tags: , , , , ,

Okay, first post for a while.  As I promised quite a while back, let’s prove together that subgroups of free groups are free.  It’s surprising that this is nontrivial to prove: just try to come up with some subgroups of F_2 and you’ll see what I mean.  In fact, using only basic algebraic topology and a bit of graph theory, we can come up with a really simple argument that replaces this one.  Perhaps that’s an argument in favor of algebraic topology.  But I think this angle is sort of interesting, and it should be a fresh experience for me, at least.

The proof is due to Jean-Pierre “Duh Bear” Serre in his book Trees.  A heads up if you track this down — Serre has a really weird way of defining graphs.  Fortunately, for this proof at least, a little bit of work translates things into the same language of graphs and digraphs that we saw when talking about Cayley graphs.  I review that below the fold.  It takes a while to set up the machinery, though the proof itself isn’t too long.  To recompense, I’ve left out a couple minor details, which you’re probably able to fill in.  If some step doesn’t make sense, work it out — or try to disprove it!

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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.

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