20 September 2010, 21:34
So, the world cup hit. And I spent a month away from Uni and people and everything, with my body mentally in South Africa but physically here in New Zealand. It was a good month…
A while before I left, we realised that we were getting nowhere with Secret Sharing, and that wasn’t likely to change unless I got tenured. Which I’m nowhere near. So, a new topic was needed. Something that could potentially lead to a paper and some actual results.
But I have a knack for picking the wrong topics…
Anyway, the new topic we decided on was “Golden Mean Matroids”. Simply put, a matroid is golden mean (GM) if it is representable over GF(4) and GF(5) (the actual definition uses subdeterminants and partial fields, so I’ll spare you). In particular, we were trying to characterise the maximum sized GM matroids. Some preliminary work was done by Archer in his PhD thesis, using some buggy software called macek. He conjectured that the maximum-sized GM matroids came in three families: GI, GP, and T. He gave matrix representations for the GI and GP families, with the T family coming from Semple’s PhD thesis. We quickly found Dowling representations for both the T family and the GI family, but the GP family eluded us for awhile.
Turns out it’s not golden mean. Which is nice, so now we only have 2 infinite families to play with (and some junk at rank 3).
The basic strategy is to take a maximum-sized GM matroid at rank k+1, contract a point something. What can we say about the original matroid and the contraction?
As it turns out, the answer to that question is “not much in the time we have available”. So, a new problem was needed. Again.
Thankfully, there is a nice trick in Matroid Theory to make problems easier: excluded minors. So we excluded a minor, which I have called Γ. It’s the relaxation of the non-Fano, so I suppose it should be called (F7–)– or something like that.
So that’s where I am now. I’m working on a subclass of the maximum-sized GM matroids that had better be just the T family. And once that is done, I’ll have to write up all the stuff I’ve done nicely, and that’s my thesis.
After that, who knows? A PhD most probably, but where?
Posted by Michael Welsh at 09:34.
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