Abstracts
Clinton
Conley
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.
Daisuke Ikegami
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.
Martino Lupini
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.
Nam Trang
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.
Anush Tserunyan
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.
Philip Welch
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.