M Theory Lesson 88
The blogosphere is wonderful! In a comment at the Everything Seminar the remarkable Terence Tao points to a short article on the mathematics of Gian-Carlo Rota. On page 9 there is a brief comment on profinite combinatorics, a subject that Rota dreamed of developing. Unfortunately there is only a single obscure reference to Rota's own writing on the subject.
Given a finite field of order $q$, a continuous geometry in the sense of von Neumann is a profinite limit of (lattices of subspaces of) projective geometries
$P(1,q) \rightarrow P(2,q) \rightarrow P(4,q) \rightarrow \cdots \rightarrow P(2^{n}, q) \cdots$
and this limit contains subspaces of any dimension $d \in [0,1]$. This led Rota to consider the Riemann zeta function, and for those with library access the (two and a half page) paper to look at is:
K. S. Alexander, K. Baclawski, G-C. Rota
A stochastic interpretation of the Riemann zeta function
Proc. Nat. Acad. Sci. USA 90 (1993) 697-699
Wait, I found a reprint! A stochastic process $Z_s$ indexed by $s \in \mathbb{N}^{+}$ is found such that a probability distribution for $Z_{s} = n$ is given by $n^{-s} \zeta (s)^{-1}$. Section 3 discusses in general Mobius inversion for an infinite lattice (recall that inversion was crucial to Leinster's concept of Euler characteristic for a finite category). By specialising to the sequence of cyclic groups $\mathbb{Z}_n$ and the profinite integers (mentioned in the last lesson) one obtains the Riemann zeta correspondence for $Z_s$ a suitable random variable on the $s$-th power of the profinite integers.
Given a finite field of order $q$, a continuous geometry in the sense of von Neumann is a profinite limit of (lattices of subspaces of) projective geometries
$P(1,q) \rightarrow P(2,q) \rightarrow P(4,q) \rightarrow \cdots \rightarrow P(2^{n}, q) \cdots$
and this limit contains subspaces of any dimension $d \in [0,1]$. This led Rota to consider the Riemann zeta function, and for those with library access the (two and a half page) paper to look at is:
K. S. Alexander, K. Baclawski, G-C. Rota
A stochastic interpretation of the Riemann zeta function
Proc. Nat. Acad. Sci. USA 90 (1993) 697-699
Wait, I found a reprint! A stochastic process $Z_s$ indexed by $s \in \mathbb{N}^{+}$ is found such that a probability distribution for $Z_{s} = n$ is given by $n^{-s} \zeta (s)^{-1}$. Section 3 discusses in general Mobius inversion for an infinite lattice (recall that inversion was crucial to Leinster's concept of Euler characteristic for a finite category). By specialising to the sequence of cyclic groups $\mathbb{Z}_n$ and the profinite integers (mentioned in the last lesson) one obtains the Riemann zeta correspondence for $Z_s$ a suitable random variable on the $s$-th power of the profinite integers.
posted by Kea | 1:21 PM
2 Comments:
Hi Kea,
I returned home from a little conference in Röros, Norway about various anomalous phenomena organized by Society of Scientific Exploration. Participants were science professionals and the atmosphere very very warm. Lectures were absolutely excellent and I got for the first time in my life through the entire lecture of mine;-)! Very enjoyable event.
I should check what the continuous geometry of von Neumann really means before saying anything but what you say brings strongly in mind TGD related notions which also suggest generalization of the notion of continuous geometry in various directions. I wrote long comments but was stuck with a question about how to generalize from a finite field to p-adic numbers and decided it to be better to think thoroughly before I say anything!
...and I got for the first time in my life through the entire lecture of mine
That's great news, Matti! Glad to hear you enjoyed yourself.
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