oh great. fall term soon begins.

woefully unproductive: that's how the last few weeks have been. i've been stuck on this or that little lemma, and they are not terribly interesting. sometimes i feel like i merely play with numbers.

unsurprisingly and unfortunately, this unproductivity is my own fault; having promised too many people that i'd do too many things, there's little time (and peaceful time, at that) to do what i'm supposed to be doing:

mathematical research, and enough of it to make a thesis.

i'm also supposed to be teaching soon (after Labor Day), but that involves teaching calculus to children who look like fast-forwarded adults, and who don't care about calculus anyway. so that hardly counts.

too many seminars and talks to give, too many students already and more returning to town, and too much impatience on my part.

metric co-tangent bundles: a happy rant.

man, this co-tangent bundle stuff is so damned cool. right now i'd say that it's cooler than elvis.

to work in this theory, one has to use a little of everything:

the weak-* topology from functional analysis, and worrisome issues of metrizability and nets vs. sequences;

metric derivations which are reminiscient of differential geometry, but with Lipschitz and not smooth flavour;

and the bread and butter of the analysis on metric spaces: the boogeymen fractals, like Cantor sets, Sierpenski carpets, Laakso constructions, and other such things;

it is like seeing old friends, meeting faces that were always familiar but to whom you were never introduced, and then having friendly arm-wrestling contests and laughing after somebody wins.

i mean, it's hard, and sooner or later i'll post here again about how annoying and frustrating all of this is, but i like to think that it's interesting work.

in life, there are plenty of ways to practice drudgery and misery, but to do so in an interesting way .. that is rare, indeed!

we are creatures, driven by conflict.

this crackpottery [1] must end. if it doesn't, i might become obsessive and brooding and moody, and that serves no one.

i'm beginning to think that the tukia-sullivan tiling technique is simply incompatible with the notion of (weak) second derivatives, and to solve the problem, one needs an entirely new technique.

[see here for my reason(s)]

maybe i should pull the plug on this last thesis problem, and let it die. i've already been given this metric co-tangent theory on my plate, and maybe i should start thinking about what can be done there, and what might make a good thesis.

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i hate giving up.

i've given up on too many things over these few years, whether they be research problems or not, and i don't see it ever reducing in number. what hope does it give, when all the things i've set out to accomplish have resulted in failure?

some people have told me that even good, even great mathematicians can't solve all problems they want, and that's true. i don't expect to be great or good or even mediocre, but what does that say about us?

that we settle for less, and dare not dream of mountains higher than those we've seen before and scaled? that we stop hoping for lofty heights and enslave ourselves in heavy chains, for our own protection against setback and failure?

what is the point, then, of ever attempting anything?

f*ck. i hate it when i argue my way into something intractable, mathematical or philosophical. this is exactly what i mean by becoming "brooding and moody."

education and knowledge have not made my life any better, but then again, was there any reason to expect them to do so?

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[1] as you might guess, yes: 'crackpottery' is the state or quality of being a crackpot, and no: it's notin the dictionary. like the word 'carefreedom,' i think i made it up.

never trust groups; functional analysis and metric geometry, oh my!

my crackpot ideas (for reviving the dead thesis problem back into life) aren't working. that should come as no surprise, of course, but these days i can't help but take such things personally ..

.. which can't possibly be healthy. then again, how healthy can it be, to think about mathematics for most of the day? q:

so, after having thought a little about group actions in the contexts of the conformally-natural extension of circle homeomorphisms (cf Douady and Earle) and the hyperbolic tiling extension (cf Kirby, Siebenmann, Sullivan), i think i'm entitled to this opinion:

i hate group-equivariant mappings, at least when it comes to second derivatives.

you'd think that actions by isometries would be reasonably nice, but no. there is an inherent problem between equivariance via group conjugations F = g F g^-1 (which are, heuristically speaking, rescalings of space) and second derivatives (which are quantities describing curvature).

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meanwhile, the metric co-tangent bundle theory does look interesting .. if that's what it's called.

between the work of N. Weaver and that of J. Cheeger, there are function algebras and abstract constructions running amok and amidst the analysis on metric spaces and measure-theoretic geometry. it feels like i'm walking around a space station and gawking at the various alien races, all the while asking myself,

"what am i supposed to be doing, here?"

and somehow, i suppose that when i answer that question, i'll know what my thesis problem is.

is it all fun and games?

upon the death of my thesis problem i devoted myself to being (mathematically) unproductive and avoided entangling alliances of all sorts.

it worked well until this afternoon.

as you may have guessed, i started entertaining crackpot theories, mostly about how to raise the problem from the dead. i'll mention something the moment that i see something rigorous in the works, but don't hold your breath; i won't.

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as for the title, it seems like the p-harmonic folks are spending more time at games, especially tug-of-war!

• a friend of mine told me once about a game-theoretic formulation of mean curvature flow via pde,
  
  
• earlier this year, Peres, Schramm, Sheffield, and Wilson gave a formulation of the infinity-Laplacian in terms of "tug of war,"
  
  
• then earlier this summer, i heard a talk of Jan Malý about a game-theoretic version of the eikonal equation,
  

and now it seems that Peres and Sheffield have gone further and interpolated some of their earlier joint work:


Tug of war with noise: a game theoretic view of the p-Laplacian

Fix a bounded domain Ω in R^d, a continuous function F on the boundary of Ω, and constants ε > 0, p > 1, and q > 1 with p^-1 + q^-1 = 1. For each x in Ω, let u^ε(x) be the value for player I of the following two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed and the player who wins the toss chooses a vector v of length at most epsilon to add to the game position, after which a random "noise vector" with mean zero and variance (q/p)|v|² in each orthogonal direction is also added. The game ends when the game position reaches some y on the boundary of Ω, and player I's payoff is F(y).

We show that (for sufficiently regular Ω) as ε tends to zero the functions u^ε converge uniformly to the unique p-harmonic extension of F. Using a modified game (in which ε gets smaller as the game position approaches the boundary), we prove similar statements for general bounded domains Ω and resolutive functions F.

[stunned]

in my meeting with the advisor today, i think we killed the thesis problem ..

.. and no, 'killed' doesn't mean 'solved;'
it means 'unsolvable' or at least 'intractable.'
maybe it means 'impossible.'

i prefer the term 'insoluble,' myself.

i mean, i got a corollary out of the deal,
and if i write it up, maybe that will be a paper.

but for now, i think i'll crawl under a rock, and stay there for a while.

a personal library update.

today in the second floor atrium, there was a huge pile of mathematical books, many of them classics, and there was a sign which read:

free books

now i am eight books richer, though it's hard to say if i will ever seriously use them ..

.. but no matter. though there may be no such thing as a free lunch, there is such a thing as a free book!

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i finally updated my LibraryThing mathematical book catalog, and again, you can view it here:

http://www.librarything.com/catalog.php?view=grey_ghost

i'm up to ninety-five (95) books, though not all of them are that "mathematical." judge them as you like.

anyways, back to work. the theorem won't prove itself, after all.