Saturday, July 14, 2012

regrets: i really should have gone to this colloquium.

maybe i should quit following the internet so closely. often it makes you feel connected, in tune with events happening right now. other times, it just reminds you of what you've missed out.

take, for example, this excerpt from a title/abstract for peter jοnes's uni. helsinki maths colloquium from this past may (2013):
...The new result in topic 3 concerns Lebesgue measurable sets $E$ of small Lebesgue measure (in any dimension). The set $E$ can be decomposed into a bounded number of sets with the property that each (sub)set has a nice "tangent cone". Roughly speaking each subset has very small intersection with any Lιpschitz curve whose tangent vector (to that curve) always lies inside a fixed cone. This had been proven in dimension two by Albertι, Csörnyeι, and Preιss by using special, two dimensional combinatorial arguments.

The main technical result needed in our work is a $d$-dimensional, measure theoretic version of (a geometric form of) the Erdös-Sζekeres theorem. (The discrete form of E-S is known only in $d = 2$.) In what is perhaps a small surprise, certain ideas from random measures can be used effectively in the deterministic setting. Our result yields strong results on Lιpschitz functions: For any Lebesgue null set $E$ in $d$ dimensions, there is a Lιpshitz [sic] mapping of Euclιdean $d$-space to itself, that is nowhere differentiable on $E$. (Thus Rademacher's theorem, which states that such a map is a.e. differentιable, is sharp. Any Lebesgue null set can be in the set of non-differentιability for such a Lιpschitz function.)

[jaw drops]
why didn't i attend this colloquium?
i could have learned so much!
why didn't anyone *tell* me?


this topic of measurable differentiable structures on metric spaces is small, but right now it has the potential to advance very quickly. as a general principle, techniques towards lebesgue null sets in euclidean spaces have interesting consequences for analysis on metric spaces ..

now i'm absolutely curious:
how do random measures fit into the picture?
what does "random" mean, here? [1]

if they lead to a higher-dimensional erdös-sζekeres theorem, then what replaces the ordering for curves?
curiouser and curiouser ..

[1] if they mean this in the probabilistic sense, then wouldn't this correspond to a measure on the space of measures? or does this refer to baιre categοry, with the class of measures on $\mathbb{R}^d$ treated as a metric space under an appropriate weak topοlogy?

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