Day 1 of the CUNY Graduate Students in Set Theory conference:
Embarrassingly, I did not know the math lounge was on the fourth floor. So I sat in the presentation room (which was a nice room) like a square until someone showed up there. I perpetually feel out of sync with the rhythm of the group at math conferences. Everyone is very nice though, so its not a big deal.
Kaethe Minden opened the day up talking about subcomplete forcing and Trees. She gave a nice, intuitive introduction to subcomplete forcing and then showed an interesting application of the concept. Subcomplete forcings do not add branches to omega_1 trees. This proof had a fun flavor of large cardinal embedding/forcing proofs and a bit of diagonalization to it.
Next, I went. I feel the talk went well, but not as well as in Salt Lake City. Still, people had questions, and I didn't make any major errors. I thought I would feel more in the loop with the other combinatorial set theorists, but combinatorics under determinacy is just too different from combinatorics under choice.
We took a break here, The lounge (now that I knew where it was) is nice. Lots of chalkboards. The Berkeley lounge has a better view.
Jeffrey Bergfalk then gave a talk on homological characterizations of small cardinals. Essentially, the finite alephs can be characterized by what kind of simplicial complexes they admit. Particularly omega_n is least with no n-dimensional simplicial complex of a particular form. These complexes also have connections to other combinatoric properties. He provided some nice intuitive explanations of the relevant homology, but there is more choice occuring in these proofs than I can apply to my work.
Maxwell Levine closed the morning session with a talk on Weak squares and very good scales. Scales and good may be the most overused terms in set theory. Although this talk was quite technical, I came out of it with a better understanding of square principles and supercompactness than I had before. (The word compact is used because properties occurring on a large set below the cardinal imply properties about the cardinal) Other than that, I mostly gathered that there are some very complicated relationships between these two notions. They essentially contradict each other, but are also compatible in a lot of ways. Also, some very easily stated question about square are quite open it seems.
We had lunch at an Asian buffet. NYC is very crowded, and NYC cheap is Texas moderate. The food was good though.
After lunch, Matt Foreman gave essentially a plenary talk. He gave a broad overview of the interactions of set theory with dynamics, and introduced a particular problem Von Neumann asked. For a general audience, this talk did a good job of being self-contained, and the final result that the isomorphism problem for ergodic measure preserving diffeomorphisms is analytic is quite interesting. Two lines were quite entertaining from this talk:
"Set theory follows behind the rest of math by 50 years"
"The 2-Torus is a smiley face picture"
We took another break.
To start the afternoon session, Alexander Block spoke on the modal logic of the set theoretic multiverse. This is of course an idea every set theory generally gets after studying some forcing, but probably doesn't formalize. The real formalization is quite nice, and there are some nice subtle problems here about the different kind of multiverses one can form and what kind of modal logic they create. A surprising is known, although for more specific types of forcing it is still open.
David Nichols finished the day with some reverse mathematics. In particular he was interested in variations of the 2-dimensions ramsey problem and how they relate to each other. It was previously that over the minimal second order arithmetic they are equivalent. David, however, was able to show that under a strong notion of computable, they are not computably equivalent, and in fact form a strict hierarchy. I always enjoy the results of reverse math.
I'm glad to have my talk out of the way, and look forward to just sitting back and enjoying the talks tomorrow.
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