In my book *Elliptic operators, topology and asymptotic methods* (both the first and the second editions) I give a discussion of the representation theory of the groups Spin and Pin which was based (as far as I can now remember) on some notes that I took when I attended Adams’ famous course on the exceptional Lie groups, as a Part III student in 1981. I no longer seem to have those, unfortunately (although meanwhile a version of Adams’ own notes on his approach has been published by University of Chicago Press). Meanwhile, in 2010 Darij Grinberg pointed out on Math Overflow that the argument I gave was garbled: see this link. In this post I want to explain what is garbled and how the useful part of the argument can be recovered. Continue reading

# Tag Archives: incompetence

# Thinking about “maximal Roe algebras”

One of the things that has happened in coarse geometry while I was busy being department chair is a bunch of papers about “maximal Roe algebras” (some references at the end). Of course these are objects that I feel I ought to understand, so I spent some time trying to figure out the basics.

Let \(X\) be a bounded geometry uniformly discrete metric space. (Something like bounded geometry seems to be necessary, for a reason that I’ll explain below.) We know how to form the *translation algebra* \( {\mathbb C}[X] \) (the *-algebra of finite-propagation matrices on \(X\) ), and this has an obvious representation (the *regular representation*) on \(\ell^2(X)\). Then the usual version of the (uniform) *Roe algebra* is just the C*-algebra obtained by completing \({\mathbb C}[X]\) in this representation. Because it involves only the regular representation we may call this the *reduced* Roe algebra (in analogy to the group case). Continue reading

# The Peter Principle Revisited

Not really “coarse mathematics” in the sense I initially intended, but feels rather relevant to the work of a department head. This paper uses a simulation to demonstrate that, under certain hypotheses, a “hierarchical” organization will function more efficiently by promoting people at random than by always promoting those who are most competent in their current position.

# Rescue on the Prow

This tells the tale of my abortive attempt (May 2010) to climb the Prow on Washington Column (V 5.7 C2) with Aaron McMillan. I have wanted to climb this thing ever since I started aid climbing, but despite going up there twice, haven’t yet succeeded. Continue reading