Abstracts
Tutorials:
Ramsey Theory on trees and applications to infinite graphs
Abstract
Ultrafilters and model theory
These tutorial lectures will outline some recent
interactions of ultrafilters and ultrapowers and model theory
with a focus on open problems.
Descriptive set theory and geometrical paradoxes
The Banach-Tarski paradox states that the unit ball in R^3 can be partitioned in to 5 pieces which can be rearranged by isometries to partition two unit balls. This result was part of an attempt in the early 20th century to understand the relation between measure theory and generalizations of classical ideas such as decomposing polygons in to congruent sets. In the last few years there has been a resurgence of interest in these geometrical paradoxes and related problems in the foundations of measure theory. These results share the theme that the sets used in paradoxes can be nicer than one might expect.
A generalization of the Banach-Tarski paradox states that any two bounded subsets of R^3 with nonempty interior are equidecomposible by isometries. A recent theorem of Grabowski, Mathe and Pikhurko states that if in addition we assume that our two sets have the same Lebesgue measure, then they are equidecomposible with Lebesgue measurable pieces.
These results rely on recent developments in definable combinatorics of definable graphs, in particular on techniques for constructing Borel matchings in graphs. In this series of talks, we survey some recent results and contrast them with their classical counterparts.
Talks:
Ergodic theorems and measurable combinatorics
Descriptive combinatorics is a branch of descriptive set theory concerned with the behavior of classical combinatorial objects, such as colorings and matchings, under additional definability constraints.
For instance, one may be interested in colorings of graphs on probability spaces in which every color class
is measurable. The Lov\'{a}sz Local Lemma (the LLL, for short) is a powerful tool in classical combinatorics;
and it has recently been adapted to various measurable situations. In this talk, I will discuss the LLL and
its measurable versions and illustrate their power with some applications in ergodic theory, specifically to
ergodic theorems for Bernoulli actions.
Erdös-Rado classes
Ramsey classes are used to construct generalized indiscernibles in models of first-order theories.
However, the lack of compactness means that Ramsey's theorem (and its generalizations) are not enough to build
indiscernibles in nonelementary classes. Instead, order indiscernibles are built with the Erdös-Rado Theorem.
We present a framework for building indiscernibles in (many) nonelementary classes: Erdös-Rado classes
(and some variants). The combinatorial definition of these classes is that they satisfy a certain partition
relation for building large homogeneous sets when given many colors. We connect this to building indiscernibles,
present several examples (some independent of ZFC), and provide some applications.
N-dependent theories
A first-order theory is n-dependent if the edge relation of an infinite random n-hypergraph is not definable
in any of its models. N-dependence is a strict hierarchy increasing with n, with the first level corresponding to
the well-studied class of NIP theories. I will give a survey of recent work on n-dependent theories establishing
connections to higher-arity generalizations of VC-dimension and hypergraph regularity (joint with Henry Towsner)
and on understanding which algebraic structures are n-dependent (joint with Nadja Hempel).
The bi-embeddability relation for torsion-free groups
A conjecture of H. Friedman and Stanley states that the isomorphism relation on countable torsion-free abelian
group has maximal complexity under the notion of Borel reducibility. In this talk we will settle the analogue
problem for bi-embeddability by showing that it is universal among analytic equivalence relation.
This extends a theorem of the speaker in generalized descriptive set theory to the classical framework of
Borel reducibility. This is joint work with Simon Thomas.
Tree properties near singular cardinals
Tree properties are a family of combinatorial principles that generalize König's infinity lemma to the
uncountable, and capture the combinatorial essence of various large cardinal properties.
In this talk we focus on the principles ITP and ISP, which characterize supercompactness for inaccessibles,
but can also be forced to hold at successor cardinals such as $\aleph_2$. In joint work with Sinapova we
showed that ISP cannot hold at the successor of a strong limit singular cardinal with countable cofinality.
On the other hand, we force ITP to hold at such a cardinal, starting from a model with infinitely many
supercompacts. We discuss these results, and how they tie in with a number of open questions about
the tree property and singular cardinal combinatorics.
Guessing models and the singular cardinal hypothesis
I will discuss three recent results on guessing models: (1) guessing models are internally unbounded, (2) for any regular cardinal $\kappa \ge \omega_2$,
$\textsf{ISP}(\kappa)$ implies that $\textsf{SCH}$ holds above $\kappa$,
and (3) forcing posets which have the $\omega_1$-approximation property also
have the countable covering property.
The interplay between inner model theory and descriptive set theory in a nutshell
The study of inner models was initiated by Gödel's
analysis of the constructible universe $L$. Later, it became
necessary to study canonical inner models with large cardinals,
e.g. measurable cardinals, strong cardinals or Woodin cardinals,
which were introduced by Jensen, Mitchell, Steel, and others. Around
the same time, the study of infinite two-player games was driven
forward by Martin's proof of analytic determinacy from a measurable
cardinal, Borel determinacy from ZFC, and Martin and Steel's proof of
levels of projective determinacy from Woodin cardinals with a
measurable cardinal on top. First Woodin and later Neeman improved
the result in the projective hierarchy by showing that in fact the
existence of a countable iterable model, a mouse, with Woodin
cardinals and a top measure suffices to prove determinacy in the
projective hierarchy.
This opened up the possibility for an optimal result stating the
equivalence between local determinacy hypotheses and the existence of
mice in the projective hierarchy, just like the equivalence of
analytic determinacy and the existence of $x^\sharp$ for every real
$x$ which was shown by Martin and Harrington in the 70's. The
existence of mice with Woodin cardinals and a top measure from levels
of projective determinacy was shown by Woodin in the 90's. Together
with his earlier and Neeman's results this estabilishes a tight
connection between descriptive set theory in the projective hierarchy
and inner model theory.
In this talk, we will outline the main concepts and results connecting
determinacy hypotheses with the existence of mice with large
cardinals. Neeman's methods mentioned above extend to show determinacy
of projective games of arbitrary countable length from the existence
of inner models with many Woodin cardinals. We will discuss a number
of more recent results, some of which are joint work with Juan
Aguilera, showing that inner models with many Woodin cardinals can be
obtained from the determinacy of countable projective games.
Speeds of hereditary properties an mutual algebricity
A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property H,
the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,...,n}.
Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete ``jumps" in the possible speeds.
Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollob\'{a}s,
and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized.
In contrast to this, many aspects of this problem in the hypergraph setting remained unknown.
In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds.
The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss.
This is joint work with Chris Laskowski.
Bernoulli disjointness
The concept of disjointness of dynamical systems (both topological and
measure-theoretic) was introduced by Furstenberg in the 60s and has
since then become a fundamental tool in dynamics. In this talk, I will
discuss disjointness of topological systems of discrete groups. More
precisely, generalizing a theorem of Furstenberg (who proved the result
for the group of integers), we show that for any discrete group G, the
Bernoulli shift 2^G is disjoint from any minimal dynamical system. This
result, together with techniques of Furstenberg, some tools from the
theory of strongly irreducible subshifts, and Baire category methods,
allows us to answer several open questions in topological dynamics: we
solve the so-called "Ellis problem" for discrete groups and characterize
the underlying topological space for the universal minimal flow of
discrete groups. This is joint work with Eli Glasner, Benjamin Weiss,
and Andy Zucker.
On some new infinitary logics
I will speak about a new class of infinitary logics introduced in 2012 by S. Shelah in his paper "Nice
Infinitary Logics". These logics have remarkable model theoretic properties despite being quite strong.
I introduce some further new logics of the same kind with the goal in mind of understanding Shelah's logic better.