Approximating L^2 invariants by finite-dimensional analogues.

 November 22, 2006 talks

\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}}

\newenvironment{question}[1][]{\par\textbf{Question (#1).}}{} \newenvironment{theorem}[1][]{\par\textbf{Theorem (#1).}}{} \newenvironment{lemma}[1][]{\par\textbf{Lemma (#1).}}{} \newenvironment{proof}{\textit{Proof.}}{}

I gave a couple of seminar talks on [1].

Here’s the main result in the paper. Let X be a CW-complex, and filter \Gamma = \pi_1 X as \Gamma = \Gamma_1 \rhd \Gamma_2 \rhd \cdots with [\Gamma_i : \Gamma_{i+1}] < \infty so that \bigcap_i \Gamma_i = \{ 1 \} . Let X_i be the cover of X corresponding to the normal subgroup \Gamma_i .

Then, the limit of the “normalized” Betti numbers \lim_{j \to \infty} b_j( X_i ) / [\Gamma : \Gamma_i] is equal to b^{(2)}_j(X) , the L^2 Betti number of X . In particular, the limit of the normalized Betti numbers is independent of the filtration! In other words, we have “approximated” the L^2 invariant by a limit of finite-dimensional approximations.

The awesome thing about this result is how “easy” the proof is; it’s just some linear algebra (eh, functional analysis), but I don’t claim to have a very conceptual understanding of why it is true. In the big book [2] on this subject, there is a more conceptual explanation of the proof; the book also mentions some basic generalizations.

[1] W. Lück, Approximating L\sp 2 -invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994) 455–481.

[2] W. Lück, L\sp 2 -invariants: Theory and applications to geometry and K -theory, Springer-Verlag, Berlin, 2002.

Algebraic topology and distributed computing.

 November 6, 2006 talks

\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}}

\newenvironment{question}[1][]{\par\textbf{Question (#1).}}{} \newenvironment{theorem}[1][]{\par\textbf{Theorem (#1).}}{} \newenvironment{lemma}[1][]{\par\textbf{Lemma (#1).}}{} \newenvironment{proof}{\textit{Proof.}}{}

I gave a seminar talk on [1].

This paper doesn’t do it (but Rajsbaum’s MSRI talk did), but the result can be reformulated combinatorially, so that the algebraic topology appears as an instance of Sperner’s lemma; this is the sort of thing that should be done at mathcamp.

Here is something that amuses me, but I know that if anyone else said it, I would find it extraordinarily annoying: seeing as these results apply to anything (I mean, the local model of computation is irrelevent), this is an example of how deterministic systems, when combined with each other, yield non-deterministic results (though I have to be careful what I mean by “deterministic”—the system as a whole is determined, but non-deterministic from the perspective of the agents in that they cannot determine the outcome). Clearly I should write a philosophy paper, called “Free will and algebraic topology: a primer,” in which people are vertices in the simplicial complex of all possible worlds.

It will be better for all of us if I stop now.

[1] M. Herlihy, S. Rajsbaum, Algebraic topology and distributed computing—a primer, in: Computer Science Today, Springer, Berlin, 1995: pp. 203–217.