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Oberseminar Wintersemester 2011
Raum:  N2 (on the ground floor of the new building)

The schedule remains tentative.

We head to the seminar room from the eighth floor around 2:10pm.

  • Thursday, October 20:  Kostya Slutsky (Münster/UIUC).

    Title:  Graev metrics on free products and HNN extensions of groups with two-sided invariant metrics.
    Abstract:  M. Graev in the 40s constructed special two-sided invariant metrics on free groups $F(X)$ starting with a metric on the basis $X$. We will discuss similar constructions for free products (possibly with amalgamation) of groups with two-sided invariant metrics and for HNN extensions of such groups.

  • Thursday, October 27:  Sean Cox (Münster).

    Title:  Ideal projections as forcing projections I.
    Abstract:  The notion of catching antichains for an ideal first appeared in the Foreman-Magidor-Shelah paper on Martin's Maximum, and was used extensively by Woodin in stationary tower arguments. Roughly, an ideal $J$ catches antichains for another ideal $I$ iff the support of $J$ is sufficiently large and there is a canonical ideal projection $\pi$ from the $J$-positive sets onto the $I$-positive sets, such that $\pi$ also transfers generic objects downward. Some instances of antichain catching are equivalent to saturation of $I$ (namely, when $J$ is the canonical club filter for $I$). However this is not the case in general. Martin Zeman and I have shown that the property ``$(\exists J)(J$ catches antichains for $I)$'' is strictly weaker than saturation of $I$ and strictly stronger than precipitousness of $I$.

  • Thursday, November 3:  Sean Cox (Münster).

    Title:  Ideal projections as forcing projections II.

  • Thursday, November 10:  There will be no seminar this week.

  • Thursday, November 17:  Maciej Malicki (Polish Academy of Sciences).

    Title:  Trees, ultrametric spaces and their automorphism groups.
    Abstract:  I will talk about automorphism groups of rooted trees, and, more generally, isometry groups of ultrametric spaces. I will be particularly interested in properties such as property (FA'), property (FA), uncountable strong cofinality and ample generics. First, I will present results relating the last two properties to the behaviour of the algebraic closures of finite sets in the case of countable rooted trees. Then I will focus on the structure of isometry groups of Polish ultrametric spaces, giving an explicit description of ultrametric spaces called W-spaces in terms of wreath products. Building on the this, and the results for trees, I will characterize W-spaces having uncountable strong cofinality.

  • Wednesday, November 23 (4pm, Raum SR7):  David Aspero.

    Title:  Measuring club-sequences, together with CH.
    Abstract:  I will focus on a new method for building forcing notions giving rise to models of CH. The type of construction I will describe is not a forcing iteration in the classical sense. It can be roughly described as a finite support iteration with systems of models as side conditions and with "global" symmetry constraints. As an application of this method I will present the construction of a model of Moore's 'measuring' together with CH. This is joint work with M.A. Mota.

  • Thursday, November 24:  Howard Becker (Münster).

    Title:   Choosing a weakly Cauchy subsequence.
    Abstract:  Consider a separable Banach space. Given a sequence in the space which has a weakly Cauchy subsequence, how difficult is it to select a weakly Cauchy subsequence? Is it always possible to do the selection in a Borel way? In a universally measurable way? The answer depends on the Banach space, and in the measurable case also depends on the model of set theory that the Banach space lives in.

  • Thursday, December 1:  Stefan Geschke (Bonn).

    Title:  Structural results about continuous $n$-colorings.
    Abstract:  We consider continuous colorings of the $n$-element subsets of a Polish space, which we call $n$-colorings for short, and their so-called homogeneity numbers. It turns out that there is a finite list of $n$-colorings on $2^\omega$ such that an $n$-coloring on a Polish space $X$ has uncountable homogeneity number iff it contains a coloring from the list. The proof is based on a generalization of a Ramsey-style theorem of Blass. If time permits, I will also say something about the existence of universal $2$-colorings.

  • Thursday, December 8:  Vassilis Gregoriades (Darmstadt).

    Title:  The complexity of the set of points of continuity of a multi-valued function.
    Abstract:  We consider a notion of continuity concerning multi-valued functions between metric spaces. We focus on multi-valued functions assigning points to nonempty closed sets and we answer the question asked by M. Ziegler (Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Effective Linear Algebra, to appear in Ann. Pure Appl. Logic) of determining the complexity of the set of points of continuity of such a multi-valued function. We will see that, under appropriate compactness assumptions, it is $\mathbf{\Pi^0_2}$. Under a natural weakening of these assumptions it can be a true $\mathbf{\Sigma^0_3}$ set, while in the general case it might not even be Borel. In particular, this notion of continuity is not metrizable, in the sense that if we view a multi-valued function $F$ from $X$ to $Y$ as a single-valued function from $X$ to the family $\mathcal{F}(Y)$ of closed subsets of $Y$, it is not necessarily the case that there is a metrizable topology on $\mathcal{F}(Y)$ such that $F$ is continuous at an arbitrary $x \in X$ in the usual sense (of single-valued functions) exactly when it is continuous at $x$ in the sense of multi-valued functions. We will also present some similar results about a stronger notion of continuity, which is closely related to the Fell topology and thus in some cases turns out to be metrizable.

  • Thursday, December 15:  There will be no seminar this week.

  • Thursday, December 22:  Holiday.

  • Monday, December 29:  Holiday.

  • Thursday, January 5:  Holiday.

  • Thursday, January 12:  TBA.

  • Thursday, January 19:  John Clemens (Münster).

    Title:  Combinatorial structure for analytic equivalence relations.
    Abstract:  Orbit equivalence relations of Polish group actions form a proper subclass of the collection of analytic equivalence relations, and it is a long-standing problem to find a complete characterization of when an arbitrary equivalence relation is induced by some action of a Polish group. Such equivalence relations inherit combinatorial structure from the group actions inducing them, which can be used to analyze their complexity and other properties. In this talk I will discuss work in progress on trying to generalize some of the combinatorics of orbit equivalence relations to general analytic equivalence relations, and how a number of questions about the Borel reducibility hierarchy can be reformulated in this framework.

  • Thursday, January 26 and Thursday, February 2:  Dominik Adolf (Münster).

    Title:  Determinacy from $\mathtt{PFA}(\aleph_2)$ and a suitable ideal on $\omega_1$.
    Abstract:  Schindler and Claverie showed that $\mathtt{PD}$ follows from $\mathtt{PFA}$ for antichains of size at most $\aleph_2$ and the existence of a precipitous ideal on $\omega_1$. We discuss the prospects for deriving $\mathtt{AD}$ in $L(\mathbb{R})$ from the same hypotheses.

  • Thursday, May 31:  Mona Rahn (Bonn).

    Title:  TBA.
    Abstract:  TBA.

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