10/20/2016 UIC Workshop Day 1

Manachem Magidor started the morning off with the first tutorial. He is speaking on "Compactness and Incompactness for Chromatic Numbers and Other Cardinal Sins." He phrases compactness in the intuitive way "If every small substructure has a property, then the whole structure does", where small means lower cardinality. He gave quite a few examples of compactness phenomena, stretching across algebra, topology,and set theory. I really liked his intuitive descriptions of weakly compact, strongly compact and supercompact cardinals.

• A cardinal is weakly compact with respect to a certain property if the property passes to small pieces of it,
• A cardinal is strongly compact with respect to a certain property if the property passes to all pieces of it, and
• A cardinal is supercompact it if strongly compact with respect to all second order properties,

He then spoke on chromatic numbers, coloring numbers, stationary reflection, and the relationship between the three. The chromatic number is highly malleable and so hard to study and definitely fixing a bound on the chromatic number is hard property to make an object compact for. Enough stationary reflection on the other hand provides compactness for the property of bounding the coloring number. This is as far as we got today, I'm curious to see where it goes next.

The first talk in the afternoon was by Sherwood Hachtman, He spoke on "Forcing Analytic Determinacy." Essentially, this is related to the reverse math question of how much set theory do you need to get that lightface analytic determinacy is equivalent to 0# existing. More combinatorially, you can ask determinacy with respect to Turing cones. While Woodin conjectured that Z_2 is enough strength (and it certainly is for boldface analytic determinacy), recent work by Cheng and Schinder seems to indicate that Z_3 may be necessary. Hachtman didn't have concrete answers to this yet, but inspired by the technique of genericity iterations, he has forced Turing cones to be inside of some sets who should have them.

This all was pretty interesting to me as I didn't know that anybody was thinking about reverse math questions surrounding determinacy. The classic result about Borel determinacy is certainly cool enough to warrant more attention, so I'm not sure why I had never really thought about it. It would be neat if the lightface analytic determinacy really was force-able, and thus an interesting intermediate stage between Borel determinacy and full boldface analytic determinacy.

Finishing out the first day, Spencer Unger presented the "Poor Man's Tree Property." People in the audience said it was like watching a mini James Cummings present. The idea is to replace the tree property with the weaker assumption that there are no special trees, This is consistency work, o the goal is to find a forcing which can ensure that there are no special trees. Because of some obstructions, such a model will have to fail GCH and SCH everywhere. Combining approaches of Magidor and Gitik, Under was able to create a large interval of cardinals, starting at aleph_2 and passing some singular cardinals, on which there are no special trees. The modified Magidor approach works from the ground up and can ensure that aleph_2 allows for no special trees. This approach can be modified for finitely many cardinals, if you thread the forcings together appropriately. The Gitik approach can be used to ensure that a singular cardinal doesn't allow for special trees. For his result, Unger put these approaches together in supercompact Prikry-like forcing which very carefully collapses certain intervals of cardinals. The details are much too complicated for me to understand with my current knowledge of forcing.

Even though it is outside of my skill set, these super high tech forcings are quite interesting, and its good to see people like Unger, who have intuition and a big picture in mind, present about them.