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
Vorträge
- 30.4.2014
Martin Ziegler:
Scharf 2-transitive Gruppen
- 7.5.2014, schon um 16 Uhr
Heike Mildenberger:
tba
- 21.5.2014
Andreas Baudisch:
Free amalgamation and
automorphism groups
- 4.6.2014
Itay Kaplan:
Strict non-forking in resilient theories
Abstract:
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
Pfingstpause
- 18.6.2014
Daniel Palacìn:
Supersimplicity and the
Fitting subgroup
Abstract:
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
Abstract:
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
Abstract:
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 ⊆ × is a Borel
set, where and are Polish
spaces, such that
every section Px
is a  ξ0
set then there exists a Borel-measurable function
u : → ωω and a
 ξ0
set Gξ
⊆ ωω × 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
Abstract:
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
Abstract:
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
Abstract:
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.)
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