Title: Unfriendly colorings
Abstract: Given a graph, a red/blue coloring of its vertices is unfriendly if
every red vertex has at least as many blue neighbors as red neighbors,
and vice-versa. Such colorings always exist for finite graphs, but for
infinite graphs their existence quickly becomes quite subtle. We
investigate certain descriptive set-theoretic analogs of these
colorings in the measure-theoretic and Borel contexts. This is joint
work with Omer Tamuz.
Title: On preserving AD via forcings
Abstract: It is known that many concrete forcings such as Cohen forcing destroy
AD. In this talk, we show that one cannot preserve AD via forcings as
long as the forcing increases \Theta and V satisfies AD^+ and
V=L(P(R)). We also provide an example of forcings which preserve AD
while increasing \Theta when V is not of the form L(P(R)).
This is joint work with Nam Trang.
Title: The complexity of the classification problem in ergodic theory
Abstract: Classical results in ergodic theory due to Dye and Ornstein--Weiss show that, for an arbitrary countable amenable group, any two free ergodic measure-preserving actions on the standard atomless probability space are orbit equivalent, i.e. their orbit equivalence relations are isomorphic. This motivates the question of what happens for nonamenable groups. Works of Ioana and Epstein showed that, for an arbitrary countable amenable group, the relation of orbit equivalence of free ergodic measure-preserving actions on the standard probability space has uncountably many classes. In joint work with Gardella, we strengthen these conclusions by showing that such a relation is in fact not Borel. The proof makes essential use of techniques from operator algebras, including cocycle superrigidity results due to Popa, and answers a question of Kechris.
Title: Condensation for mouse pairs
Abstract: Condensation is a common phenomenon in inner model theory. A typical situation is:
given a sufficiently elementary map \pi: M \rightarrow N where N is a fine-structural model that is sufficiently iterable, we'd like to know whether M is an initial segment of N. The first example of this phenomenon is Godel's condensation lemma, where N is a level of the constructible universe L and the conclusion is N is an initial segment of M. There have been generalizations of this result to (short) extender models. We have proved arguably the most general form of condensation for strategic (short) extender mice, where not only we can conclude the models line up but the strategies do as well, provided the strategy of N normalizes well and has strong hull condensation. We will explain these terms, discuss the ingredients that go into the proof, and future applications. This is joint work with John Steel.
Title: A pointwise ergodic theorem for quasi-pmp graphs
Abstract. We prove a pointwise ergodic theorem for locally countable quasi-pmp graphs, which states that the global condition of ergodicity amounts to locally approximating the means of $L^1$-functions via increasing subgraphs with finite connected components. The pmp version of this theorem was first proven by R. Tucker-Drob using probabilistic methods. Our proof is different: it is constructive and applies more generally to quasi-pmp graphs. It involves introducing a graph invariant, a packedness condition for finite Borel subequivalence relations, and an easy method of exploiting nonamenability. The non-pmp setting additionally requires a new gadget for analyzing the interplay between the underlying cocycle and the graph.
Title: Minutiae on Ramseyness
Abstract: There is a wealth of structure on the Ramsey cardinal hierarchy whose investigation was started by Qi Feng. There has been a variety of papers since building hierarchies of different kinds, either through trees of indiscernibles, or using games, to name but two. We report on some recent work of Holy, Schlicht, and with Nielsen.