Mathematische Logik
Sommersemester 2014

Mathematisches Institut Abteilung für Math. Logik

Dies ist die Homepage des Oberseminars "Mathematische Logik" im Sommersemester 2014.

Enrique Casanovas,
Heike Mildenberger (Forschungssemester),
Martin Ziegler.

Zeit und Ort

Mi 16:30-18:00, SR 404 in der Eckerstr. 1., vorher ab 16 Uhr Tee in Zimmer 313 


  • 30.4.2014
    Martin Ziegler:
    Scharf 2-transitive Gruppen
  • 7.5.2014, schon um 16 Uhr
    Heike Mildenberger:
  • 21.5.2014
    Andreas Baudisch:
    Free amalgamation and automorphism groups
  • 4.6.2014
    Itay Kaplan:
    Strict non-forking in resilient theories
    Strict non-forking is a forking notion defined by Shelah for NIP theories and proved very useful in analysing dividing in NIP and NTP2 theories. In joint work with Alex Usvyatsov, we proved symmetry of this notion in NIP. Recently we generalized this to a more general setting, that of resilient theories. Resilient theories were defined by Chernikov and Ben-Yaacov as a class of theories containing both NIP and simple theories and contained in NTP2 theories. I will give all the definitions and try to prove the theorem. 
  • 11.6.2014

  • 18.6.2014
    Daniel Palacìn:
    Supersimplicity and the Fitting subgroup
    The Fitting subgroup of a given group G is the subgroup generated by all nilpotent normal subgroups of G. While it is always normal, it may not be nilpotent. Wagner proved that the Fitting subgroup of a stable is always nilpotent. However, this is not known for the wider class of groups with a simple theory. 
    In this talk we present some of the main tools and notions of groups in simple theories, and focus on those which have ordinal Lascar rank. Our aim is to sketch the proof that the Fitting subgroup of a type-definable supersimple group is again nilpotent. This generalizes a proof of Milliet in the finite rank case.

  • 25.6.2014, Tee schon 15:50 Uhr
    Rafel Farré:
    dp-minimality, strong dependence and dp-rank in Ordered Abelian Groups
    In recent years there has been great interest in NIP theories, largely due to the work of Shelah. Trying to solve the equation x / dependent = superstable / stable
    Shelah has developed the notion of strongly dependent theory.Also, as a generalization of other notions of minimality, dp-minimality has been introduced. Due to work of Hrushovski and others, valued fields play an important role in the development of NIP theories. There are also interesting conjectures (due to Shelah among others) saying that NIP fields (maybe with extra conditions) are certain Henselian valued fields.
    Prior to the characterisation of valued fields which are dp-minimal or strongly dependent, which are interesting open questions, it is necessary to know which Ordered Abelian Groups (OAG for short) are dp-minimal or strongly dependent. (It is well known that any OAG is NIP)
    We provide an answer to those questions about OAGs. We develop tools to compute dp-rank, and apply them to OAGs, characterising dp-minimality and strong dependence in OAGs. We are able to compute the exact value of the dp-rank of any OAG.

  • 1.7.2014, Sondertermin am Dienstag, 14:00 -15:30 Uhr, SR 119, Eckerstr.1
    Vassilis Gregoriades:
    Uniformity functions in descriptive set theory
    It is a typical application of effective descriptive set theory to witness a given property about sets in a good uniform way, where the term “good”usually means
    (1) continuous or (2) Borel.
    An example about (1) is the Suslin-Lusin Separation Theorem: two disjoint analytic sets in Polish spaces are separated by a Borel set, and the Suslin-Kleene Theorem: there exists a continuous function u : ωω × ωω ωω such that whenever (α,β) encodes a pair of disjoint analytic sets then u(α,β) encodes a Borel separating set.
    An example about (2) is the following result of Louveau: if P X×Y is a Borel set, where X and Y are Polish spaces, such that every section Px is a Σ^ξ0 set then there exists a Borel-measurable function u : Xωω and a Σ^ξ0 set Gξ ωω ×Y such that P is the set of all (x,y) for which the pair (u(x),y) belongs to Gξ. (The latter solves the so-called Section Problem.)

    In this talk we will state the basic notions of effective descriptive set theory, give further examples and present new results of this type. These results deal with diverse topics including: the Baire property of analytic sets, refinements of Polish topologies, and the hierarchy of Borel-measurable functions. The work on the latter topic is joint with T. Kihara and has applications to the Decomposability Conjecture, cf. the following talk by T. Kihara.

  • 2.7.2014 , schon um 16 Uhr , Tee ab 15:25 in Raum 313
    Takayuki Kihara:
    Decomposition, dimension, and degrees
    Nikolai Luzin asked whether every Borel function on the real line can
    be decomposed into countably many continuous functions. Although the
    full Luzin problem was negatively answered in the 1930s, this problem
    and related results prompted many researchers to develop the theory of
    decomposability, and the long-running line of this research recently
    led us a beautiful conjecture on decomposability of the hierarchy of
    Borel functions. Recently, by using deep methods from computability
    theory, V. Gregoriades and the speaker obtained a new decomposability
    theorem in the hierarchy of Borel functions. In this talk, we see how
    computability theory gives a new insight into this descriptive set
    theoretic problem.
    Moreover, in computability theory, this result led to the discovery of
    the importance of the notion of topological dimension in computability
    theory. Motto Ros-Schlicht-Selivanov asked whether there are at least
    two piecewise homeomorphism types of Polish spaces having no
    transfinite inductive dimension. A. Pauly and the speaker pointed out
    that R. Pol's counterexample to P. Alexandrov's problem (a strongly
    infinite dimensional compact metrizable space which is not
    decomposable into countably many finite-dimensional subspaces) is not
    piecewise homeomorphic to Hilbert cube. We will also discuss
    computability-theoretic aspects of this space.

  • 16.7.2014, Tee schon ab 15:50 Uhr
    Pierre Simon:
    Definably amenable NIP groups
    I will talk about definably amenable groups in NIP theories, that is NIP groups admitting a translation-invariant measure on the class of definable sets. Examples include stable groups, compact Lie groups and solvable NIP groups. In a joint work with Artem Chernikov, we characterize "generic types" in this context, describe the space of invariant measures and prove "generic compact domination". I will present this work and, if time permits, mention some more recent developments.

  • 23.7.2014, Tee schon ab 15:50 Uhr
    Amador Martín-Pizarro:
    Schöne Gruppen
    In Zusammenarbeit mit Thomas Blossier wird gezeigt, daß, gegeben ein schönes Paar (M,E) von Modellen einer stabilen Theorie mit nfcp und EI jede definierbare Gruppe sich projiziert, bis auf Isogenie, auf die E-rationalen Punkte einer T-definierbaren Gruppe H über E mit einer T-definierbaren Gruppe als Kern.
    Wir geben auch eine vollständige Beschreibung interpretierbarer Gruppen für Paare algebraisch abgeschlossener Körper.

  • 30.7.2014,Tee schon ab 15:50 Uhr
    Martin Hils:
    Bewertete Körper mit Automorphismus und NTP2
    Im Vortrag werde ich folgenden allgemeinen Erhaltungssatz für NTP2 diskutieren. Ein bewerteter Körper mit Automorphismus in Restklassencharakteristik 0, der Körper-Quantoren in der dreisortigen Sprache von Pas eliminiert, ist NTP2, sofern sowohl der Restklassenkörper als auch die Wertegruppe (jeweils mit induziertem Automorphismus) NTP2 sind.
    Hieraus folgt zum Beispiel, dass algebraisch abgeschlossene bewertete Körper mitNichtstandard-Frobenius-Automorphismus in Restklassencharakteristik 0 NTP2 sind.
    (Es handelt sich um eine gemeinsame Arbeit mit Artem Chernikov.)

Last update on July 28, 2014, R.S.