6/15/2016

Day 1 of the 23rd Best Conference (insert joke here)

This is one of the more social conferences I've been to. It probably helps that there is a nationally renowned brewery down the road.

The morning started off with a special 1 hour talk by Natasha Dobrinen on Ramsey Spaces coding universal triangle-free graphs and applications to Ramsey degrees. She really gave quite a good talk, and it was full of things I sort of understood. You start off with a Fraisse structure, you then want to know things about colorings of it, and the methodology involves a rather clever coding of an infinite graph into the full binary tree. She used the "choose" notation for the coloring. It's interesting how pure set theory and topological Ramsey theory can have such different notation for the same idea. I think I like the "choose" notation more. She half apologized for just scanning in hand drawn pictures of her trees, but I think I actuallu prefer those to a latexed up version. Plus Latexing all of that up might be the very definition of a waste of time.

Shezad Ahmed went next on Jonsson cardinals and pcf theory. This was close to the same talk as at CUNY, but I definitely got more out of it this time around. pcf theory is tough to talk about because there is so much background, The "tension", as he describes it, that a successor of a singular cardinal being Jonsson introduces into the objects of pcf theory is interesting. It seems like its one of those situations where it makes your life complicated because its allowing you to study phenomena at a level of detail previously unavailable.

Then William Chan gave a talk on how every analytic equivalence relation with all Borel classes is Borel somewhere. The work is actually quite general, and ends up essentially being a problem in determinacy theory, although aspects of the problem are resolvable in ZFC(and a little more). This work might entail one of the more creative uses of the Martin Solovay tree construction that I've seen. For the biggest results he has, he uses very strong forms of determinacy. I wonder how much of that is truly necessary (I bet he does too).

Next, Kameryn Williams spoken on minimal models of Kelley-Morse set theory. Again, this was essentially the same talk as at CUNY. It's still funny that the result is actually that there is no least model of KM, This talk hadn't stuck with me as much as Shezad's had, so on this second time hearing it, I actually picked up quite a bit more of what was going on. The clever trick that allows him to even get off the ground is this idea of looking at a slight strengthening of KM which then has a nice first order characterization in terms of an inaccessible cardinal. I'm intrigued about the possible interplay between large cardinal notions and second order set theories,

Kaethe Minden finished out the morning with a talk on subcomplete forcing and trees. This was also essentially the same talk as at CUNY. I'm still really impressed with pictures and with her ability to explain a very complicated forcing concept. I've said it before, I have a hard time with forcing talks in general. But this one is nice.

After lunch, Simon Thomas gave a special one hour talk. He spoke on the isomorphism and bi-embeddability relations for finitely generated group. It was great; he is consistently a very good speaker. I think  my favorite line this time around was "A Kazhdan group is punchline." That there are very hard problems with just finitely generated groups is fascinating. I guess I'll just remember some more lines from the talk. "The old work was great, but you can see the sweat. You shouldn't be able to see the sweat." And so he comes up with a much better topological space to approach his problem. Finally, "Some people think you should believe in your conjectures. I strongly disagree with this/" One more. "No presentation should be perfect. You should always introduce an intentional error. This is the intentional error for this talk. Some people don't need to introduce intentional errors." Although related to his talk at Utah, this was obviously quite different as he had some more time.

Next up was Cody Dance. He talked about indiscernibles for L[T_2,x]. I think I like this work better than his work on the embedding generated by the club filter in L(R). Of course, I'm probably just being biased because I like determinacy theory more than inner model theory. Basically, because Jackson recently proved some new things about the Martin Solovay tree construction, it seems that we might be able to use less description theory and more indiscernible theory to get the weak and strong partition property. If the methods hold out, this would probably finally allow Jackson to push the analysis of these properties past the first inductive pointclass.

After Cody was Kyle Douglas Beserra. He spoke on the conjugacy problem for automorphisms of countable regular trees. Kyle is coming to UNT in the fall, having just finished his masters. Instead of looking at the complexity of classifying trees up to isomorphism, he as looking at the complexity of classifying isomorphisms of trees up to conjugacy. It seems one can mostly lift the methods used to study the trees to study their automorphisms as well. The really clever bit though is when the trees are unrooted and finitely branching, and the automorphism fixes a tree, Then he gets to do some invariant descriptive set theory and find some nice universal countable equivalence relations to Borel reduce to and be  reduced to from.

At this point my yellow notepad ran out of pages. It's okay, I had a backup.

Dan Hathaway then talked about disjoint Borel functions. He reminded the audience of the notion of a conflict response relation, which other people seemed familiar with, but I was not. Later I found out that the notion has connections to cardinal invariants. He used it to try get a handle on the minimum size of a set of Borel functions you would need to always be able to respond to someone else's Borel function with a disjoint one. Interestingly enough, this problem reduces to looking at just real numbers and any ordering that sort of looks like the ordering L puts on the reals. To show this reduction he has to create his own variant of Hechler forcing (which looks itself like a variant of Mathias forcing to me). The great part is that this analysis is quite flexible, and the functions don't have to be Borel. If you look into the projective hierarchy you just start using the mouse orders for the mouse with the correct number of Wooding cardinals. It's cool to see someone exploit the interplay between the mice and the projective classes.

Almost finishing out the day was Paul Mckenney. He was looking at the classic problem of whether or not the power sets of \omega and \omega_1 are isomorphic mod finite. In particular, he was studying some consequences of this. He had a few different results on this. The first was that if there is a cardinal preserving automorphism of the power set of omega_1 mod finite, then the continuum function collapses at omega_1. The next was about Q_B sets. This is frustrating because this talk is very hard to write about on here, but it really was very good.

The last talk of the day(this was quite a bit of talks) was given by Paul Corazza. He talked about the axiom of infinity, QFT, and large cardinals. This was basically basically just an alternate way to talk about the strength of elementary embedding, stuffed inside a somewhat goofy package. There is no real connection here to QFT. He does get some new large cardinal notions out of it though.