# Weekly Report: Power tourism

Published: Sun 18 June 2006

It's been a while since I managed to write one of these. The reason is simple enough - my weekends have been packed; and I don't get around to it during the weeks.

During the last three weekends, first my parents and my brother, and then for the last two and the week inbetween my fiancĂ©e, have been visiting me in Jena. Thus, I have covered more ground in these three weeks when it comes to tourism than I probably will be able to do in several months. I have seen the Blue Man Group in Berlin (WOW!), I have seen the Dornburg, the Feengrotten and Weimar. I have eaten at the expensive luxurious restaurant at the top of the old university tower in Jena (it's bloody scary, but quite cool - the restaurant is on the 29th floor; in a city where only one single house goes above 10 floors).

The future is taking a more determined shape as well: I'm going to Sweden for a two-week vacation in the beginning of August, and then I'm going to Leeds for a workshop on triangulated categories. The Leeds-trip is paid for by the university, since my advisor still thinks that it's bloody important that I end up being fluent in the categorifications of homological algebra; and because the participant list was both cool and good to get connections in. I will be meeting a few old friends there, as well as quite a few people I've only read about (Avramov - the 'cause' of my master's thesis is coming!) . All in all, I'm very much looking forward to this. It'll be a blast.

It's still not very clear what I'll actually be doing for my PhD. So far, reading up on the bloody stuff and get a solid grip on it is work enough, so to speak. But I may end up trying to find a result along the lines of "For almost all groups [tex]G[/tex], the depth of [tex]H^*(G,\mathbb F_p)[/tex] is equal to the p-rank of [tex]Z(G)[/tex]" for some sensible meaning to 'almost all'. Another possible direction was suggested to me at a department dinner last tuesday: The algorithms in place for efficient calculation of modular group cohomology can probably be "lifted" to algorithms to work with path algebras; and this may actually end up being relevant for some ring-theoretical statements. This may also be a good direction to go off to - I don't know.

For a completely different tune: John Baez has an absolutely marvelous last week's find. He talks about mathematics and music; more specifically about the application of group theory to modern (i.e. post-tonal) music theory. The idea is that tunes live in [tex]\mathbb Z/12\mathbb Z[/tex]; and more specifically chords live in [tex](\mathbb Z/12\mathbb Z)^3[/tex]. So for instance C corresponds to 0, C# to 1 and D to 2. C-major is the triple (0,4,7) and C-minor is (0,3,7). Thus, we have an action of the dihedral group [tex]D_{12}[/tex] with 24 elements on chords; generated by the transposition [tex](a,b,c)\mapsto(a+1,b+1,c+1)[/tex] and the reflection [tex](a,b,c)\mapsto (c,b,a)[/tex]. Transposition preserves major/minor chords; whereas reflection swaps them. And then funky stuff starts happening when you start considering whether there are any other interesting group actions of 24-element groups on chords - turns out that there is one: the so-called PLR-group. Go read it!