Oberseminar Sommersemester 2011
Raum:
MA 114 (SR 1D)
April 6: Grigor Sargsyan (UCLA/Rutgers).
Title: Some open problems and
questions surrounding Kechris-Martin theorems.
Abstract:
We will try to reprove a version of the Kechris-Martin theorem using
inner model theory. The motivation for doing so is that the
methods used can perhaps be generalized to higher pointclasses.
April 11: Andrew Marks (UC-Berkeley).
Title: Universality among some recursion-theoretic countable Borel equivalence relations.
Abstract: We will begin by giving an overview of Martin's conjecture on Turing invariant functions and its relationship to the location of Turing equivalence in the hierarchy of countable Borel equivalence relations. We will then discuss analogues of these questions for some other degree structures.
April 20: Dominique Lecomte (Paris 6).
Title: Baire
class $\xi$ colorings: the first three levels (joint work with
Miroslav Zeleny)
Abstract: The $G_0$ dichotomy due to
Kechris, Solecki and Todorcevic characterizes the analytic relations
having a Borel-measurable countable coloring. We give a version of
the $G_0$ dichotomy for $\Delta^0_\xi$-measurable countable colorings
when $\xi \le 3$. A $\Delta^0_\xi$-measurable countable coloring
gives a covering of the diagonal consisting of countably many
$\Delta^0_\xi$ squares. This leads to the study of countable unions
of $\Delta^0_\xi$ rectangles. We also give a Hurewicz-like dichotomy
for such countable unions when $\xi \le 2$.
April 25: Holiday.
May 2: "Holiday."
May 9: Philipp Lücke (Münster)
Title: Descriptive set theory at uncountable cardinals: supercompact cardinals.
May 16: "Holiday."
May 23: Simon Thomas (Rutgers)
Title:
Universal Borel Actions of Countable Groups
Abstract:
If the countable group $G$ has a nonabelian free subgroup, then there
exists a standard Borel $G$-space such that the corresponding orbit
equivalence relation is countable universal. In this talk, I will
consider the question of whether the converse also holds.
May 30: "Holiday."
June 6: Philipp Lücke (Münster)
Title: Descriptive set theory at uncountable cardinals: huge cardinals.
June 13: Holiday.
June 20: Andrew Marks (UC-Berkeley).
Title: Some
combinatorial questions related to the universality of recursive
isomorphism.
Abstract: We consider the question of
whether recursive isomorphism is a universal countable Borel
equivalence relation. We show that recursive isomorphism on
$3^\omega$ is universal, and we isolate a combinatorial conjecture
that would imply that recursive isomorphism on $2^\omega$ is
universal. The conjecture can be stated in several equivalent forms,
and is related to Borel coloring and matching problems.
June 27: Sean Cox (Münster).
Title: Martin's Maximum and tower forcing.
Abstract: Woodin showed that if there is a Woodin cardinal $\delta$ in $V$, then there is a poset $\mathbb{P} \subset V_\delta$ such that forcing with $\mathbb{P}$ yields a wellfounded generic embedding $j: V \to M$ with small critical point. There are several variations, but this talk will deal with the case where $cr(j) = \omega_2$. Woodin's forcing is called the stationary tower, and is an example of the more general notion of a "tower of ideals".Burke showed that if $\kappa$ is a supercompact cardinal and $\delta$ is any inaccessible $> \kappa$, then there is a tower of ideals of height $\delta$ whose generic ultrapower is never wellfounded (the ideals concentrate on $\wp_\kappa(V_\lambda)$ for $\lambda < \delta$). In this talk, I will present some joint work with Matteo Viale which shows that Martin's Maximum has similar consequences for certain towers on $\wp_{\omega_2}(V_\lambda)$.
July 4: "Holiday."
July 12: Philipp Schlicht (Bonn).
Title: Embedding structure of $\kappa$-trees.
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