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Proposal: Mathematical Exposition July 28, 2009

Posted by Phi. Isett in Essays, polymath proposals.
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Disclaimer:  My last entry discussed how the use of personal profiles, a bookmarking system and selective provision of statistics can, in combination, help polymath deal with many of the challenges it faces, including the issues of how to provide leadership and promote overall efficiency.  But when I wrote it, I considered only the problem of producing research mathematics en masse, so I do not claim that the same principles necessarily apply to the collaborative production of mathematical exposition.

First of all, I am not suggesting we discontinue working on problems — rather I’m suggesting a (necessarily) smaller side-project.  Polymath should try to write a textbook… or something expository.

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What Polymath Needs is Wasted Time July 26, 2009

Posted by Phi. Isett in Essays, Uncategorized.
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The following is an extensive reply to a post on Terence Tao’s blog.

My point of view is based in part on my experience as a moderator of a webforum of up to a dozen active members which devoted a few years to the collaborative production of a complex storyline with several, deeply interwoven subplots.  Time will tell how well large collaborations can produce mathematics –they are certainly an amazing tool for story-writing, and some comparisons can be made upon abstraction so the experience may be relevant.  The Google Groups format had tremendous advantages and shortcomings, but an inability to harness people’s free time ultimately lead to our story’s stagnation.

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At the moment, polymath seems to function in many ways analogously to various forms of entertainment and “time-wasting” (reading blogs and webcomics, participating in forums, watching movies, watching YouTube, etc.) – indeed, this “wasted time” is in some sense exactly the incredible resource which polymath must compete to harness, although within a more restricted audience and for the noble purpose of serving mathematics.  I am sort of joking, but this is only my interpretation of Prof. Tao’s original request that the participation in the latest polymath problem solving experiment be casual.

Now, much work has already been done to procrastinate as efficiently as possible– we read a select few blogs/webcomics/twitters and have efficient means of getting to them (feeds, bookmarks); we look at Rotten Tomatoes to decide which movies to watch.  On YouTube, we often look at the “N Views” and the “x/5.0 (y votes)“ user-rating figure to decide which videos are best worth our wasted time (and many videos on YouTube are “replies” to others, which are linked).  YouTube, for instance, has had to developed mechanisms to help ensure good posts do not get squashed, which is a problem polymath also faces.  Regardless of their shortcomings and their obviously differing objectives from those of Polymath, these “procrastination” activities provide good examples of how to effectively organize a mass of information which is spread among many people so as to save time, and that is the problem I want to address here (as opposed to the closely related problem of optimal technology and formatting).  In other words, I will not propose any particular technologies, but rather make a list of things which I believe a technology has to be capable of doing in order to harness massive collaboration and spare time in a satisfactory way – a task which is crucial to maximizing the long term impact of polymath.

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Hello World, the Lagrange Multiplier Theorem, and Kuhn-Tucker conditions July 26, 2009

Posted by Phi. Isett in Calculus, economics, Uncategorized.
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This is a blog which was started on a whim in order to host a very extensive reply to a blog entry of Terence Tao.  This entry is not the reply — the next entry is.

I may continue this blog to contain pieces of mathematics that I find pretty, important and/or not so commonly known.  In any case, I’d feel silly devoting an entire blog to a single reply.  So here.. I’ll take this opportunity to include my favorite proof of the Lagrange multiplier theorem, and later on maybe I’ll post a nice proof of Stirling’s formula, a group algebraic proof of the Riemann-Lebesgue lemma, or something…

The Lagrange Multiplier Theorem helps us solve constrained maximization / minimization problems, making it (among other things) extremely important in economics.  A (weak form of) the Lagrange Multiplier theorem can be stated as follows:

Let f : \mathbb{R}^n \to \mathbb{R} be a real-valued, continuously differentiable function (called the “objective function” — the thing we want to maximize or minimize), and let g : \mathbb{R}^n \to \mathbb{R}^m be continuously differentiable (called the “constraint function” for reasons which will soon be clear).

If x_0 is an extremizer of the restriction of f to the (codimension m) “constraint manifold” given by M \equiv \{ x : g(x) = 0 \}, then there is a unique linear functional \lambda : \mathbb{R}^m \to \mathbb{R} satisfying the equality of linear maps Df(x_0) = \lambda \circ Dg(x_0).

We assume m \leq n — things tend to be more complicated when the number of constraints exceeds the number of degrees of freedom.

Proof:

We know that f must satisfy some first order condition: namely (by further restricting f to trajectories on M) that Df(x_0)v = 0 for any v \in \mathbb{R}^n which can be realized as a velocity at x_0 by a path within the constraint manifold. An exercise in the implicit function theorem shows that these velocities include all of \mbox{ Ker } Dg(x_0) (these are all such velocities — they are the directions in which one can move without changing g, and together they are called the “tangent space” of M at x_0 under generic circumstances).

Finally, now that \mbox{ Ker } Dg(x_0) \subseteq \mbox{ Ker } Df(x_0), it is a trivial exercise in linear algebra to show that there is a unique linear functional \lambda : \mathbb{R}^m \to \mathbb{R} defined by the identity Df(x_0) = \lambda \circ Dg(x_0).  Basically, we have shown that the zero set of the linear map Dg(x_0) is contained in the zero set of Df(x_0), but because the two maps are linear, their level sets are all translations of the zero level set, and therefore we conclude that every level set of Dg(x_0) is contained in a level set of Df(x_0).

Does anyone know how to turn off the annoying preview thing for hyperlinks..? Never mind, got it.

Edit (29 Aug, 2009):

The same ideas can be used to give a proof of the Kuhn-Tucker conditions for a constrained maximization problem where  we replace the system of equalities \{ g(x) = 0 \} with the system of inequalities \{ g_i(x) \leq 0 | i =1, \ldots, m \}.  We simply replace the role of trajectories within the constraint manifold by trajectories going into the constraint set.

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