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Logic
Seminar |
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Thursday,
October 11
Thursday,
October 25
Abstract: For
a countable Borel equivalence relation E we
consider the weak choice principle ``every
countable sequence of E-classes has a choice
function''. We establish a relationship between
ergodicity of the equivalence relations and the
study of these choice principles.
We
will separate these choice principles as follows:
if E is F-ergodic (with respect to some measure)
then there is a model of set theory in which
``choice for E classes'' fails yet ``choice for F
classes'' holds. For example, ``choice for
E_\infty classes'' is strictly stronger than
``choice for E_0 classes''.
A key
lemma in the proof is the following statement: if
E is F-ergodic with respect to an
E-quasi-invariant measure \mu then the
countable power of E, E^\omega, is F-ergodic with
respect to the product measure \mu^\omega. The
proof relies on ideas from the study of weak
choice principles.
In
this talk we will go over some of the basic ideas
behind forcing and explain how they were used to
construct models of set theory without the axiom
of choice. We will then establish the relationship
with ergodicity and focus on proving the lemma
mentioned above.
Thursday,
November 1
3:00 -- 4:00 pm // Linde, Room 255 Abstract: A
large part of measured group theory studies
structural properties of countable groups
that hold "on average". This is made precise
by studying the orbit equivalence relations
induced by free Borel actions of these
groups on probability spaces. In this vein,
the cyclic (more generally, amenable) groups
correspond to hyperfinite equivalence
relations, and the free groups to the
treeable ones. In joint work with R.
Tucker-Drob, we give a detailed analysis of
the structure of hyperfinite subequivalence
relations of a treed
quasi-measure-preserving equivalence
relation, deriving some of analogues of
structural properties of cyclic subgroups of
a free group. Most importantly, just like
every cyclic subgroup is contained in a
unique maximal one, we show that every
hyperfinite subequivalence relation is
contained in a unique maximal one.
Thursday,
November 8
3:00 -- 4:00 pm // Linde, Room 255 Abstract: A
large part of measured group theory studies
structural properties of countable groups
that hold "on average". This is made precise
by studying the orbit equivalence relations
induced by free Borel actions of these
groups on probability spaces. In this vein,
the cyclic (more generally, amenable) groups
correspond to hyperfinite equivalence
relations, and the free groups to the
treeable ones. In joint work with R.
Tucker-Drob, we give a detailed analysis of
the structure of hyperfinite subequivalence
relations of a treed
quasi-measure-preserving equivalence
relation, deriving some of analogues of
structural properties of cyclic subgroups of
a free group. Most importantly, just like
every cyclic subgroup is contained in a
unique maximal one, we show that every
hyperfinite subequivalence relation is
contained in a unique maximal one.
Wednesday,
January 9
2:00 -- 3:00 pm // Linde, Room 255 Abstract:
We consider the
concept of disjointness for topological dynamical
systems, introduced by Furstenberg. We show that for
every discrete group, every minimal flow is disjoint
from the Bernoulli shift. We apply this to give a
negative answer to the “Ellis problem” for all such
groups. For countable groups, we show in addition
that there exists a continuum-sized family of
mutually disjoint free minimal systems. In the
course of the proof, we also show that every
countable ICC group admits a free minimal proximal
flow, answering a question of Frisch, Tamuz, and
Vahidi Ferdowsi.
(Joint work with Eli Glasner, Todor Tsankov, and Benjamin Weiss) Wednesday,
January 16
2:00 -- 3:00 pm // Linde, Room 255 Abstract: Given a measure-preserving equivalence relation E, there is a Polish space S(E) of subequivalence relations, which admits a natural action of the full group [E]. Does S(E) have a dense orbit? We will present results due to François Le Maître which show that the answer is yes when E is the hyperfinite ergodic equivalence relation, and that the answer is no when E is induced by a measure-preserving action of a property (T) group. Wednesday,
January 23
2:00 -- 3:00 pm // Linde, Room 255 Abstract: Given a measure-preserving equivalence relation E, there is a Polish space S(E) of subequivalence relations, which admits a natural action of the full group [E]. Does S(E) have a dense orbit? We will present results due to François Le Maître which show that the answer is yes when E is the hyperfinite ergodic equivalence relation, and that the answer is no when E is induced by a measure-preserving action of a property (T) group. Wednesday,
February 6
2:00 -- 3:00 pm // Linde, Room 255 Wednesday,
March 13
2:00 -- 3:00 pm // Linde, Room 255 Abstract: We generalize the theorem of Þórisson, characterizing when two measures agree on invariant sets, to the setting of cardinal algebras. Thursday,
April 4
3:00 -- 4:00 pm // Linde, Room 255 Abstract: We discuss the notion of weak containment and weak equivalence for pmp actions of countable groups and its relation with invariant random subgroups. Thursday,
April 11
3:00 -- 4:00 pm // Linde, Room 255 Abstract: 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. Thursday,
May 16
3:00 -- 4:00 pm // Linde, Room 255 Abstract: We give a Baire category characterization of when a subset of a Polish space is \Sigma^0_{n+2}-hard for n > 0. Our proof uses a priority argument, and Antonio Montalban's true stages machinery. We apply this characterization to the decomposability conjecture; the problem of describing when a function is a union of countably many continuous functions defined on \Pi^0_n sets. Tuesday,
May 28
3:00 -- 4:00 pm // Linde, Room 255 Abstract: I will give an overview of the application of nonstandard methods to the study of partition regularity of Diophantine equations. I will the explain how these methods can be used to generalize the classical Rado criterion for linear equations to obtain natural necessary conditions for arbitrary Diophantine equations, which are also sufficient for certain degree 2 equations. This is joint work with Jordan M. Barrett and Joel Moreira. Tuesday,
June 4
3:00 -- 4:00 pm // Linde, Room 255 Abstract Thursday,
June 13
3:00 -- 4:00 pm // Linde, Room 255 Abstract: I will define a class of equivalence relations called Polishable equivalence relations that lies between the class of orbit equivalence relations of Polish group actions and the class of idealistic equivalence relations of Kechris and Louveau. I will present a Scott analysis for such equivalence relations. I will compare this analysis with the Scott analysis for isomorphism equivalence relations from continuous model theory and with versions of the Scott analysis for (certain) orbit equivalence relations of Polish group actions. As a tool in the proofs, I will introduce transfinite filtrations from one topology to another, a new notion of interpolation between topologies that may be of independent interest. Tuesday,
August 13
2:00 -- 3:00 pm // Linde, Room 255 Abstract: Given a discrete group G, two G-flows X and Y are said to be disjoint if their only joining is the trivial joining 2^X. Earlier this year, it was shown by Glasner, Tsankov, Weiss and Zucker that the Bernoulli shift 2^G is disjoint from every minimal G-flow. Their proof is by cases depending on the group, and relies heavily on difficult machinery developed for ICC groups due to Frisch, Tamuz and Vahidi Ferdowsi. Recently, Bernshteyn has found a much shorter proof eliminating all casework, which reduces the problem to an application of the Lovász Local Lemma from combinatorics. We will present this proof, which proceeds via showing a result interesting in its own right, namely that if U is a nonempty open set in 2^G, then then there is some n such that any union of n translates of U always contains an orbit. Tuesday,
August 20
2:00 -- 3:00 pm // Linde, Room 255 Abstract: Given two graphs G and H, a trivial upper bound for the chromatic number of G\times H is the chromatic number of G (and also the chromatic number of H). Hedetniemi's conjecture, dating back to 1966, states that this upper bound is always an equality, that is to say that \chi(G\times H) = \min(\chi(G),\chi(H)) for any finite simple graphs G and H. Hajnal showed in 1985 that the generalization to infinite graphs is false, but the conjecture for finite graphs, and even the generalization to Borel chromatic numbers of analytic graphs on Polish spaces, remained unresolved. Recently, Shitov has shown that Hedetniemi's conjecture is false, using the exponential graphs of El-Zahar and Sauer. We will present an outline of this proof.
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information: A. Kechris, kechris@caltech.edu |
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