### 19:28:00 - *The defense*

`Today was my defense. It went well. I "broke everybody's legs", as Heidi wished me yesterday. Jeff had me make slides for it a few days ago, so I mostly used the slides rather than the blackboard; fortunately we didn't have much trouble getting the projector set up.`

Committee was Jeff, Andreas Blass, Alexander Barvinok, Martin Strauss, and Kevin Compton (from the CS department). Karen Smith showed up as well. Also Julian (my academic older brother) happened to be in town so he was there. And various other people; this wasn't supposed to be a complete list, so let me head off the possibility before it turns into one. And of course a number of my housemates showed up -- Beatrix, Angus, Seth, Noelle... perhaps one or two more I'm forgetting? Not sure that any of them understood it besides Beatrix.

Still, the topic obviously was pretty accessible to those with a bit of math background (though I did make a point of stopping to explain what ω^{ω} is; that's been a sticking point when I've talked about this before). Which meant that I could actually get into the proofs quite a bit. Most defenses I've gone to, the subject and the statement of the results takes so long to explain that there's barely any time to go into how any of it was proved, but I actually got to do quite a bit of that. I was afraid I would go over time, and I did a little, but only by like 5 minutes. Which is probably by about how much I started late anyway.

Kevin (er, by which I mean Kevin Carde, not Kevin Compton) mentioned to me afterward that he really liked the argument for why, when k is an integer, the order type of D∩[0,k) is exactly ω^{k} and not any larger. (Look at what happens for k-ε and take a limit.) That was a pleasant surprise.

Most unexpected question at the defense (I forget whether this Compton or Strauss): "What if you allow addition, multiplication, and nim-addition?" My response was that, since 3^{k}⊕1=3^{k}-1, well-ordering should fail for the same reason as when you allow subtraction. But up till then, in all the variants of the problem I had considered, it had never occurred to me to consider nim-addition. (Or nim-multiplication, or nim-inversion.) Also Compton asked about ω^{ω} showing up in other contexts, which gave me a chance to say a bit about how it's possible it might show up with addition-multiplication chains as well... as well as another less speculative generalization which I then forgot to mention, but oh well.

Spent a bunch of time talking to Kevin and Julian and Karen afterward. Mentioned the fact that the closure of the defect set is the set of numbers of the form "a defect plus a whole number"; they were surprised that the latter was a closed set. Since the arguments for these facts were largely unloaded from memory at the time, all I could really say was, "You know, when you put it that way, it does seem weird."

In addition to the usual little math department celebration (Jeff got me sparkling grape juice, since I don't drink), when I got home I found Beatrix had gotten me a cake as well! When Noelle mentioned this via IM to Justine (who has moved out for the summer and is down in Texas), she said [something along the lines of] "Is it an ice cream cake? It should be an ice cream cake." I told Noelle she can tell Justine, I'm glad to hear she cares, but the cake is fine.

**EDIT**: OK, actually Justine just asked if it was an ice cream cake; she didn't say it should be. Since it was Justine, I imagined that was the implication, but perhaps that wasn't intended.

Anyway, that's done. I mean, aside from writing up the revisions to my thesis and getting those turned in...

-Harry

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