Weekly Report: Power tourism

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 fiancée, 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!