Continuing this series (earlier posts here, here and here) on the paper of Weinberger and Yu, I’m expecting to make two more posts: this one, which will say something about the class of groups for which they can prove their Finite Part Conjecture, and one more, which will say something about what can be done with the conjecture once one knows it. Continue reading

# Tag Archives: exhaustion

# “Finite part of operator K-theory” III – repeat

(I posted this yesterday but it seems to have vanished into the ether – I am trying again.)

This series of posts addresses the preprint “Finite Part of Operator K-theory for Groups Finitely Embeddable into Hilbert Space and the Degree of Non-rigidity of Manifolds” (ArXiv e-print 1308.4744. http://arxiv.org/abs/1308.4744) by Guoliang Yu and Shmuel Weinberger. In my previous post I gave the description of their main conjecture (let’s call it the Finite Part Conjecture) and showed how it would follow from the Baum-Connes conjecture (or, simply, from the statement that the Baum-Connes assembly map was an injection). Continue reading

# Rope solo experiment

Well, the weather has been too good not to go climbing, so I headed down to Donation yesterday afternoon. Much to my surprise, I had the whole place to myself.

The picture (not very good) shows the rope solo system that I used. This is my first time using the Peztl MiniTraxion for this purpose. (One advantage of having climbed a couple of walls is that one has a lot of interesting gear to play with on an occasion like this.)

Steph Davis has a good post on rope solo systems (and the follow-up comments are helpful too). What I did was tie the climbing rope in to the anchor at its midpoint, with a figure-8 knot on a steel biner, so I had two independent anchored strands. One strand went through the mini-traxion, which rode on a full strength chest harness (also clipped to the sit-harness by a short sling). The other strand went through a Gri-Gri.

When climbing, the mini-traxion side was weighted to my gear bag and then fed automatically. the Gri-Gri side I pulled rope through when convenient, and also tied off with a separate backup knot every now and again. To transition to rappel, just open the cam on the mini-trax and rappel on the Gri-Gri.

At first I was pretty nervous trusting the system and moved very slowly and inelegantly. But it soon became clear that it would work fine. The pulley moves up with you and gives a good catch. Still, I should probably have started with something a bit easier. Things feel different when you are the only one around!

If working something steep, one should carry prusiks so as to be able to unweight the mini-trax and transition to rappel when hanging in space. Of course, one could bring ascenders; but that seems to be taking the idea of raiding the aid box to an unnecessary extreme!

# Connes Embeddings and von Neumann Regular Closures of Group Algebras

This is an interesting paper of Gabor Elek’s which touches on some things I’ve posted about recently – especially (i) the Atiyah conjecture and (ii) the idea (which shows up in the work of Ara et al) that one can use some kind of “asymptotic rank” instead of “asymptotic trace” in some contexts where you want to build “continuous dimension” type invariants.