Day 2 of Best.
Erin Carmody was the special speaker for the morning. Her talk was entitles "Killing Them Softly." After describing degrees of inaccessibility with words to a ridiculous level (inaccessible, hyper inaccessible, richly inaccessible, utterly inaccessible, deeply, truly, eternally,...), she introduced a better way of describing these degrees using meta-ordinals and something like Cantor normal form. She was able to use variants of this approach to describe degrees of Mahlo cardinals and degrees of measurable cardinals. The real work here though is the creation of clever forcing which can bump large cardinals down in degree very sensitively (basically one degree at a time).
Next Douglas Ulrich talked about a new notion of cardinality for countable first order theories. I believe I saw his co-author present this in Utah. It had been a while, and I still find the idea of considering Scott sentences for models outside of your current universe pretty exciting. It's also interesting how looking at the number of models up to back and forth equivalence doesn't suffice to distinguish different first order theories.
After that I spoke. I think it went okay. People had questions. The only thing is I should have a waited a bit longer to prepare my slides as I had already kind of moved beyond them with what I have actually proved. People did like the little graphics I put in.
Up next was Monroe Eskew. He spoke on rigid ideals. This is one of those curious problems where the difference between MA and CH really shows up. In MA it seems that the results on rigid ideals are uninteresting. Under just CH (really GCH) however, all kinds of consistency questions show up. Monroe is able to employ some clever forcing and large cardinal conditions to get the existence of rigid ideals.
The last talk before lunch was Daniel Soukoup. He talked about orientations of graphs with large chromatic number. Basically, the chromatic number of a graph can determine some structure of its subgraphs. When you orient a graph, that number might change, and the structure of the oriented subgraphs is different than the structure of the subgraphs. This problem is intractable when the chromatic number is finite, but Soukoup is able to gets results when the chromatic number is uncountable.
The next special speaker was Martin Zeman. His talk was on master conditions from huge embeddings. Although the title wouldn't suggest it, the talk was mostly about the non-saturated ideal, the properties of precipitousness, pre-saturation, saturation, and other analogous properties. So this really took me back on a nostalgia trip to the stationary tower. The huge embeddings and master conditions come into play because the problem reduces to a problem of the closure properies of a certain forcing (which is actually a different forcing modded out by the master condition).
Paul Ellis then spoke on a Borel amalgamation property. The sentence FAP implies BAP implies SAP was in this talk. It's great. This material is Ellis' attempt to generalize out some properties that certain Fraisse structures have. Basically is the complexity of the isomoprhism problem of the associated group of automorphisms sufficiently hard. The generalization is natural and the results are interesting: it's not clear what conditions guarantee the hardness.
The second to last speaker was John Clemens. His talk was on the relative primeness of equivalence relations. He is essentially studying a sort of pigeonhole property for quotients of the reals. I think his work and mine will eventually have some intersection. Mostly his work right now is a repackaging of previous results, but as we all know, having the right words to describe a problem is half the battle.
The last speaker was Lijiana Babinkostova. Her talk was titles "On a property of Corson." Her talk was basically on games which can describe topological properties. In particular, she liked to use this context to look at modified versions of the standard games to get a hold on some more technical topological notions. For instance, instead of density, she was interesting in theta-density, which only requires that the set meet the closure of every open set.
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