Oberseminar Modelltheorie, Geometrie und Gruppentheorie
Das Oberseminar findet in diesem Semester donnerstags von 11:00 Uhr bis 12:00 Uhr im Raum SR1d statt.
Bei Fragen wenden Sie sich bitte an Katrin Tent.
Kommende Vorträge im Wintersemester '23/24
12.10.2023
Rosario Mennuni
TBA
Abstract:
TBA
Vergangene Vorträge
13.07.2023
Rob Sullivan (Münster)
Extending automorphisms in ultrahomogeneous structures
Abstract:
Let M be an ultrahomogeneous relational structure (eg the random graph, or Q as a linear order). Let A be a substructure of M. If A is finite, then all automorphisms of A extend to automorphisms of M, immediately by the definition of ultrahomogeneity. What happens if A is infinite? We will discuss several examples and a general theorem for free amalgamation classes. This is ongoing work with Alessandro Codenotti, Aristotelis Panagiotopoulos and Jeroen Winkel.
06.07.2023
Akash Hossein (Université Paris Saclay)
Forking and invariant types in regular ordered Abelian groups.
Abstract:
The non-forking independence relation is not hard to describe
in DLO : one can show that C is independent from B over A if and only if
every point of C avoid those B-definable intervals which are bad, ie those
that are closed, bounded, and disjoint from A. The main result that I will
present in this talk is that this very simple characterization of forking
also holds in DOAG : we have independence if and only if every point from
dcl(AC) avoids the AB-definable intervals that are closed, bounded, and
disjoint from dcl(A).
In order to establish this characterization of forking by singletons, we
will need to manipulate some group valuations which might seem weird at
first, but actually interact very well with the model theory. In
particular, the notion of a separated family (in a valued vector space)
will naturally correspond to orthogonality of types, and thus allow us to
restrict our problem to smaller subfamilies, which you can guess will be
relevant, given the unary nature of our result.
If time allows, we will show that our result naturally extends to every
regular ordered Abelian group, one just has to additionnally check the
divisibility conditons which characterize non-forking in the stable reduct
of torsion-free Abelian groups.
29.06.2023
Elliot Kaplan
O-minimal fields with monotone derivations
Abstract:
In joint work with Nigel Pynn-Coates, we consider o-minimal fields equipped with a compatible valuation and derivation, under the additional assumption that the derivation is monotone (that is, the valuation of any element is no larger than the valuation of its derivative). We develop a version of differential henselianity for these fields, and we use this to prove an Ax-Kochen/Ershov result. In the special case that the underlying o-minimal field is an elementary extension of the real field with restricted analytic functions, we show that any model which satisfies our analog of differential henselianity is elementarily equivalent to a power series model.
22.06.2023
Alexi Block Gorman (McMaster)
Expansions of the reals by Büchi-automatic sets: choose-your-own-adventure
Abstract:
There are compelling and long-established connections between automata theory and model theory, and this talk will explore some of those connections for expansions of the real additive group. Büchi automata are the natural extension of DFAs and NFAs to a model of computation that accepts infinite-length inputs. We say a subset X of the reals is Büchi-automatic if there some natural number r and some Büchi automaton that accepts (one of) the base-r representations of every element of X, and rejects the base-r representations of each element in its complement. We can analogously define Büchi-automatic subsets of higher arities, and these sets exhibit intriguing behavior from the perspectives of both fractal geometry and model theory. In this talk, we will have the opportunity to discuss how expansions of the reals solely by Büchi-automatic sets fit into the framework of tame geometry, as well as connections to notions from metric geometry and neostability.
15.06.2023
Mariana Vicaria (UCLA)
Residue Domination
Abstract:
Haskell, Hrushovski and Macpherson developed the theory of stable domination: a notion that captures when a structure is controlled by its stable part. A prime example is ACVF, where the stable part coincides with all the sets that are internal to the residue field (which is strongly minimal). In this talk we present residue field domination statements for henselian valued fields of equicharacteristic zero, which in essence captures the notion that the structure is controlled by the sorts internal to the residue field. If time allows us, we present residue domination results for sigma-henselian valued fields (this is joint work with D. Haskell).
25.05.2023 11:00
Nicolas Daans (Antwerp)
Local-global principles, function fields, and definable valuations
Abstract:
To study the arithmetic complexity of a field which carries a valuation, it is useful to know whether the associated valuation ring is a(n existentially) definable subset of the field. If the valuation is henselian, then such definability can often be established under mild assumptions on the value group or residue field. Without assuming henselianity, however, no standard approach exists. In this talk, I will discuss how one can establish existential definability of non-henselian valuation rings, in particular in function fields. Special attention will be given to the role of modern local-global principles from quadratic form theory. The presented research was part of my PhD project and will appear as joint work with my former supervisors Karim Johannes Becher and Philip Dittmann.
25.05.2023 10:00!
Bojana Pantić (Novi Sad)
Echeloned spaces - a fresh generalisation of metric spaces
Abstract:
Recently, based on the idea of loosening up metric spaces, we have
produced another fruitful class of structures, namely the class of
/echeloned spaces/. The likes of such consist of an underlying set
of points X and a specific total preorder relation on X^2 named
/echelon/.
In my talk I will be reporting on our most prominent findings regarding
these structures. Among other things, I will discuss the specifics:
regarding their relationship with metric spaces, the universal
homogeneous echeloned space, a 0-1 type of law for echeloned spaces, as
well as the Ramsey property of ordered echeloned spaces.
This is joint work with Maxime Gheysens, Christian & Maja Pech, and
Friedrich Martin Schneider.
11.05.2023
Harry Schmidt (Basel)
Intersecting curves with isogeny orbits
Abstract:
We will discuss a certain ``modular variant'' of the Mordell-Lang theorem. Suppose that we are given a family of abelian varieties parametrised by an algebraic variety $V$. Given a point $p$ on $V$, we are then interested in all points $q$ on $V$ that correspond to an abelian variety that is isogenous to the abelian variety above $p$. We call the set of all such points the isogeny orbit of $p$. Loosely speaking, this might be thought of as an analog of a divisible group. We can then ask to classify all algebraic varieties $V$, that contain a Zariski-dense isogeny orbit. If everything is defined over the complex numbers, this problem is treated by the work of Orr and more recently Richard, Yafaev. I will discuss the history of the problem and talk about joint work with Griffon, in which we begin attacking the problem over fields of positive characteristic.
04.05.2023
Andrea Vaccaro (WWU)
Games on Approximately Finite C*-algebras
Abstract:
By a well-known classification result in operator algebras due to George Elliott, the isomorphism class of an approximately finite C*-algebra (or simply AF-algebra) is completely determined by its dimension group. The latter is a C*-algebraic invariant which (for separable C*-algebras) takes the form of a (countable) ordered abelian group. The main result of my talk is a model theoretic version of Elliott's result in the context of infinitary logic. In particular, Elliott's arguments can be combined with a metric version of the dynamic Ehrenfeucht–Fraïssé game to show that elementary equivalence up to a rank alpha between AF-algebras is verified if elementary equivalence, up to a rank only depending on alpha, between the corresponding dimension groups holds. I will also show how this result can be used to build a class of simple AF-algebras of arbitrarily high Scott Rank.
27.04.2023
Franz-Viktor Kuhlmann (Szczecin)
(Roughly) tame, (roughly) deeply ramified, and perfectoid fields
Abstract:
I will report on recent results obtained in joint work with Anna Rzepka and
Steven Dale Cutkosky, in particular some that have connections with the paper
by Franziska Jahnke and Konstantinos Kartas that has just been published on
arXiv.
Building on our paper "The valuation theory of deeply ramified fields and its
connection with defect extensions" that has just appeared in the Transactions
AMS, we are now working on the following subjects:
1) Galois extensions of prime degree of (roughly) deeply ramified fields can
only admit one type of defect, which we call "independent". In an essentially
completed follow-up paper we give various characterizations of independent
defect. One of them uses Kähler differentials. We also study the special form
of these characterizations when the valued field has rank 1 (as perfectoid
fields do).
2) In their book "Almost ring theory", Gabber and Ramero use Kähler
differentials to characterize deeply ramified fields. In a second paper in
preparation, we give a down-to-earth proof of their result by developing tools
for the explicit computation of Kähler differentials and then computing them
for all possible types of Galois extensions of prime degree of valued fields.
If time permits, I will also comment on some aspects of the model theory of
roughly tame and roughly Kaplansky fields.
20.04.2023
Yifan Jing (Oxford University)
Measure Growth in Compact Semisimple Lie Groups
Abstract:
The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| >> \min \{ |A|^{1+c}, |G| \}. In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset of a compact semisimple Lie group G, then \mu(AA) > \min \{ 2\mu(A) + c\mu(A)|1-2\mu(A)|, 1 \}, where \mu is the normalized Haar measure on G. I will also talk about how to use this result to solve the Kemperman Inverse Problem, and discuss what will happen when G has high dimension or when G is non-compact.
13.04.2023
Tingxiang Zou (WWU)
Number of rational points of difference varieties in finite
difference fields
Abstract: A difference field is a field with a distinguished automorphism.
Automorphisms of a finite field are powers of the Frobenius map. In this
talk, I will discuss how to estimate the number of rational points of a
difference variety, namely a system of difference equations, in a finite
field with a distinguished power of Frobenius. Like algebraic geometry, one
can assign a dimension, called transformal dimension, to a difference
variety. I will present a result which is a difference version of the
Lang-Weil estimate with respect to the transformal dimension. This is joint
work with Martin Hils, Ehud Hrushovski and Jinhe Ye.
03.04.2023 12-13
Chieu-Minh Tran (NUS)
Measure doubling of small sets in SO(3,R)
Abstract:
Let SO(3,R) be the 3D-rotation group equipped
with the real-manifold topology and the normalized Haar measure \mu.
Confirming a conjecture by Emmanuel Breuillard and Ben Green, we show that if
A \subseteq SO(3,R) is open and has sufficiently small
measure, then \mu(A^2) > 3.99 \mu(A).
We also show a more general result for the product of two sets, which can be
seen as a Brunn-Minkowski-type inequality for sets with small measure in
SO(3,R). (Joint with Yifan Jing and Ruixiang Zhang)
26.01.2023
Daniel Hoffmann (TU Dresden)
Interpreting Galois groups of many sorted structures
Abstract:
My plan is to explain a few technical results from my paper with Junguk
Lee ("Co-theory of sorted profinite groups for PAC structures") - but no
worries - I will try to make it enjoyable. The set-up is quite general,
assume that you have a substructure K of some structure M and you want to
do a bit of model theory on the absolute Galois group of K in M. There is
a natural language for profinite groups (i.e. given in a manuscript by
Cherlin-van den Dries-Macintyre), which makes our Galois group a first
order structure living on its own above M. Now, the question is - can we
somehow interpret this structure of a profinite group inside M?
19.01.2023
Sergei Starchenko (Notre Dame)
On Hausdorff limits of images of o-minimal families in real tori
Abstract:
Let $\{ X_s \colon x\in S\}$ be a family of subsets of ${\mathbb R}^n$
definable in some o-minimal expansion of the real field.
Let $\Gamma \subseteq {\mathbb R}^n$ be a lattice and $\pi \colon {\mathbb
R}^n/\Gamma \to \mathbb T$ be the quotient map.
In a series of papers (published and unpublished) together with Y.Peterzil we
considered Hausdorff limits of the family $\{ \pi(X_s) \colon s\in S\}$ and
provided their description.
In this talk I describe model theoretic tools used in the description.
15.12.2022
Martin Bays (WWU)
Elekes-Szabó for collinearity on cubic surfaces
Abstract:
Motivated by the problem of understanding higher dimensional Elekes-Szabó
phenomena, we consider the spatial orchard problem on a cubic surface S, which
asks for arbitrarily large finite subsets of S with, asymptotically,
quadratically many collinear triples. With a smoothness assumption on S, we
find that such configurations have to be essentially planar. I will aim to
give at least a flavour of the proof, which involves pseudofinite dimension
calculus, approximate subgroups, divisor classes, arithmetic genus, and
incidence bounds. Joint work with Jan Dobrowolski and Tingxiang Zou.
08.12.2022
Franziska Jahnke (WWU)
Defining valuations in ordered fields
Abstract:
We study the definability of valuation rings in ordered fields (in the language of ordered rings). We show that any henselian valuation ring that is definable in the language of ordered rings is already definable in the language of rings. However, this does not hold when we drop the assumption of henselianity. This is joint work with Philip Dittmann, Sebastian Krapp and Salma Kuhlmann.
01.12.2022
Sylvy Anscombe (Université de Paris)
Existential theories of classes of fields --- henselian and otherwise
Abstract:
I have spoken before about joint work with Fehm on the existential theory of power series fields F((t)), and more recently on work with Dittmann and Fehm on the analogous theories in a language with an additional parameter for the uniformizer. In the former case we found a transfer of decidability: the existential theory is decidable if and only if the existential theory of F, as a field, is decidable. In the latter case, we obtain the same transfer, but in positive characteristic this is conditional on a consequence of local uniformization, a major open conjecture. In an ongoing project (again with Fehm), we broaden these transfer results to deal with classes of residue fields. One surprising result gives Turing equivalences between the existential theory of Q and a number of existential theories of henselian and large fields.
24.11.2022
Christian d'Elbée
The generic theory of fields expanded by a multiplicative endomorphism
Abstract:
I will present the theory ACFH which axiomatises existentially closed
fields expanded by a multiplicative homomorphism. This theory is NSOP_1 and
not simple, and I will explain how to get to those conclusions. I will
mention results about higher amalgamation and imaginaries.
17.11.2022
Rob Sullivan (WWU)
The externally definable Ramsey property
Abstract:
In this talk, I will introduce a weakened version of the Ramsey property:
the "externally definable Ramsey property", where the colourings considered
are restricted to those that are externally definable. We will explore
several examples (and non-examples) of ultrahomogeneous structures with
this property, and we will also discuss how to characterise
ultrahomogeneous structures with the externally definable Ramsey property
in terms of their topological dynamics. This is joint work with Nadav Meir.
10.11.2022
Konstantinos Kartas (IMJ-PRG/Sorbonne Université)
A model-theoretic Fontaine-Wintenberger theorem, Part II
Abstract:
We first review the statement of the model-theoretic Fontaine-Wintenberger
theorem from last time. We then continue with our list of examples of
elementary properties which are transferable between a perfectoid field
and its tilt. Some of those will be of arithmetic interest. Finally, we
explain the key ideas in the proof of the main theorem.
03.11.2022
Konstantinos Kartas (IMJ-PRG/Sorbonne Université)
A model-theoretic Fontaine-Wintenberger theorem, Part I
Abstract:
This is a series of two talks based on upcoming joint work with F. Jahnke,
where we establish certain connections between perfectoid geometry and
model theory of henselian fields. In the first part, we present a
model-theoretic generalization of the Fontaine-Wintenberger theorem. This
reveals an abundance of elementary statements whose truth value is
transferable between a perfectoid field and its tilt. A key ingredient in
the proof is an Ax-Kochen/Ershov principle for perfectoid fields (and
generalizations thereof).
27.10.2022
Aleksandra Kwiatkowska (WWU)
Projective Fraisse limits of graphs with confluent epimorphisms
Abstract:
We show that the class of finite connected graphs with confluent
epimorphism is a projective Fraisse class and we investigate the continuum
(compact and connected space) obtained as the topological realization of
its projective Fraisse limit. This continuum was unknown before. We show
that it is indecomposable, but not hereditarily indecomposable,
one-dimensional, pointwise self-homeomorphic, but not homogeneous. It is
hereditarily unicoherent, in particular, the circle does not embed in it.
However, the universal solenoid, the pseudo-arc, and the Cantor fan do
embed in the continuum.
This is joint work with W. Charatonik and R. Roe.
20.10.2022
Nicolas Daans (Antwerpen)
A universal definition of Z in Q
Abstract:
It is a long-standing open problem whether the ring of integers Z has an
existential first-order definition in Q, the field of rational numbers. A
few years ago, Jochen Koenigsmann proved that Z has a universal
first-order definition in Q, building on earlier work by Bjorn Poonen.
This result was later generalised to number fields and to global function
fields, using classical machinery from number theory and class field
theory related to the behaviour of quaternion algebras over global and
local fields.
In this talk, I will sketch a variation on the techniques used to obtain
the aforementioned results. It allows for a relatively short and uniform
treatment of global fields of all characteristics that is less dependent
on class field theory. Instead, a central role is played by Hilbert's
Reciprocity Law for quadratic forms. Finally, I will touch on quantitative
aspects of the method, and, if time allows, discuss other instances in
which similar techniques can be applied.
14.07.2022
Alessandro Codenotti (WWU)
Generalized Ważewski dendrites as projective Fraïssé limits
Abstract:
In joint work with Aleksandra Kwiatkowska we continue the study of projective Fraïssé limits of trees initiated by Charatonik and Roe in a recent preprint. In particular by introducing new classes of maps between trees, called (weakly) coherent, we construct many generalized Ważewski dendrites as the topological realization of a projective Fraïssé limit of trees. By moving to the more general context of Fraïssé categories developed by Kubiś we are able to obtain all generalized Ważewski dendrites in a similar manner. As an application we recover a countable dense homogeneity result for the endpoints of those dendrites.
30.06.2022
Martino Lupini (Newcastle)
On the heart of abelian Polish groups and related categories
Abstract:
I will explain how methods from logic can be used to provide a
description as a concrete category of the canonical "completion" to an
abelian category (called the left heart) of several categories of
algebraic structures endowed with a topology. These categories include
abelian Polish groups, abelian locally compact Polish groups, abelian
non-Archimedean Polish groups, abelian locally compact totally
disconnected Polish groups, Polish topological vector spaces, separable
Fréchet spaces, and separable Banach spaces over the real or complex
numbers or over a non-Archimedean valued field. I will then discuss how
these descriptions can be used to construct "Borel-definable" refinements
of classical invariants from algebraic topology and homological algebra
that keep track of additional topological and
complexity-theoretic information. In conclusion, I will discuss how these
refinements are, indeed, finer, richer, and more rigid than their purely
algebraic counterparts. This is joint work with Bergfalk and
Panagiotopoulos.
15.06.2022 (Wednesday!) 10:00 (!) SRZ 213 (!)
Manuel Bodirsky (TU Dresden)
The Almost-Sure Theories of Classes Defined by Forbidden Homomorphisms
Abstract:
This talk is about the almost-sure theories for classes of finite
structures that are specified by homomorphically forbidding a finite set
F of finite structures. If F consists of undirected graphs, a full
description of these theories can be derived from the
Kolaitis-Proemel-Rothschild theorem, which treats the special case where
F = {K_n}. The corresponding question for finite sets F of directed
graphs is wide open. We present a description of the almost-sure
theories of classes described by homomorphically forbidding finite sets
F of oriented trees; all of them are countably categorical.
Joint work with Colin Jahel
02.06.2022
Andre Nies (Auckland)
Meet groupoids and computable presentations of totally disconnected, locally compact groups
Abstract:
Recently there has been a lot of interest in computation in totally disconnected, locally compact groups (for instance by G. Willis and his co-workers in Newcastle). We give various notions of computable presentation of such a group and show their equivalence. One of them relies on the (countable) meet groupoid given by the compact open cosets. The class of computably t.d.l.c. groups turns out to be robust and have good closure properties. The notion of computable presentation allows us to rigorously formulate the question whether the modular function, or the scale function, is computable. This is joint work with Alexander Melnikov, arxiv.org/abs/2204.09878.
19.05.2022
Andre Nies (Auckland)
Meet groupoids and the Borel complexity of the isomorphism relation between oligomorphic groups
Abstract:
A meet groupoid is an algebraic structure that is a groupoid and at the same time a meet semilattice with least element. The cosets of open subgroups of a topological group, together with the empty set, form a meet groupoid in a natural way, given by set multiplication in case the associated subgroups match, and the intersection operation. Meet groupoids, in the equivalent form of coarse groups, were introduced by Tent in a 2018 paper with Kechris and the speaker.
In joint work with Schlicht and Tent, we use meet groupoids to show that the isomorphism relation between oligomorphic closed subgroups of Sym(N) is Borel reducible to a Borel equivalence relation with all classes countable.
12.05.2022
Philipp Hieronymi (Bonn)
A strong version of Cobham's theorem
Abstract:
Let k,l>1 be two multiplicatively independent integers. A subset X of N^n is k-recognizable if the set of k-ary representations of X is recognized by some finite automaton. Cobham’s famous theorem states that a subset of the natural numbers is both k-recognizable and l-recognizable if and only if it is Presburger-definable (or equivalently: semilinear). We show the following strengthening. Let X be k-recognizable, let Y be l-recognizable such that both X and Y are not Presburger-definable. Then the first-order logical theory of (N,+,X,Y) is undecidable. This is in contrast to a well-known theorem of Büchi that the first-order logical theory of (N,+,X) is decidable. Our work strengthens and depends on earlier work of Villemaire and Bès.
The essence of Cobham's theorem is that recognizability depends strongly on the choice of the base k. Our results strengthens this: two non-Presburger definable sets that are recognizable in multiplicatively independent bases, are not only distinct, but together computationally intractable over Presburger arithmetic. This is joint work with Christian Schulz.
05.05.2022
Simon André (WWU)
Infinite finitely generated simple sharply 2-transitive groups
Abstract:
A group G is said to be sharply 2-transitive if it acts on a set X containing at least two elements in such a way that any pair of distinct elements of X can be mapped to any other such pair by a unique element of G. A typical example is the affine group GA(K) for its natural action on the field K. The first examples of sharply 2-transitive groups other than GA(K) were constructed a few years ago by Rips, Segev and Tent. In my talk, I will describe a construction of infinite finitely generated simple sharply 2-transitive groups. Based on works with Katrin Tent and Vincent Guirardel.
28.04.2022
Sylvy Anscombe (Université de Paris)
Existential theories of henselian fields, parameters welcome
Abstract:
The first-order theories of local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are far less well understood than their characteristic zero analogues: the fields of real, complex and p-adic numbers. On the other hand, the existential theory of an equicharacteristic henselian valued field in the language of valued fields is controlled by the existential theory of its residue field. One is decidable if and only if the other is decidable. When we add a parameter to the language, things get more complicated. Denef and Schoutens gave an algorithm, assuming resolution of singularities, to decide the existential theory of rings like Fp[[t]], with the parameter t in the language. I will discuss their algorithm and present a new result (from ongoing work, with Dittmann and Fehm) that weakens the hypothesis to a form of local uniformization, and which works in greater generality.
21.04.2022
Juan Pablo Acosta
One dimensional definable groups in some valued fields and definable groups in regular groups
Abstract:
I present a complete description of one dimensional commutative groups in
algebraically closed valued fields up to a finite index subgroup and quotient
by a finite group of order a power of the residue characteristic. I also
present a similar description in the case of ultraproducts of p-adic numbers,
the pseudo-local fields.
In addition I give a description of all groups definable in a regular ordered
group $R$ with finite quotients $R/nR$, up to a finite index group. This
generalizes previous results for rational numbers, where the proof is new, and
for Presburger arithmetic.
14.04.2022
Martin Hils (WWU)
Beautiful pairs of valued fields and spaces of definable types
Abstract:
By classical results of Poizat, the theory of beautiful pairs of models of a stable theory T is "meaningful" precisely when the set of all definable types in T is strict pro-definable. We transfer the notion of beautiful pairs to unstable theories and study them in particular in valued fields, establishing Ax-Kochen-Ershov principles for various questions in this context. Using this, we show that the theory of beautiful pairs of models of ACVF is "meaningful" and infer the strict pro-definability of various spaces of definable types in ACVF, e.g., the stable completion introduced by Hrushovski-Loeser, and a model theoretic analogue of the Huber analytification of an algebraic variety.
This is joint with Pablo Cubides Kovacsics and Jinhe Ye.
07.04.2022
Martin Bays (WWU)
Incidence bounds in positive characteristic via valuations and distality
Abstract:
Joint work with Jean-François Martin. The Szemerédi-Trotter theorem bounds the number of incidences between finite sets of points and straight lines in the real plane, and generalisations to other algebraic binary relations play an important role in understanding the interaction between (pseudo)finite sets and field structure in characteristic zero. In positive characteristic, these incidence bounds fail drastically without further restrictions, but may be expected to hold under certain conditions. We confirm this in the case of fields admitting a valuation with finite residue field, e.g. finitely generated extensions of F_p, by seeing that the restricted distality provided by the valuation suffices to trigger a result of Chernikov-Galvin-Starchenko.
03.02.2022 11:00 SRZ216/217
Jinhe Ye (IMJ-PRG)
Curve-excluding fields
Abstract:
Consider the class of fields with Char(K)=0 and x^4+y^4=1 has only 4 solutions in K, we show that this class has a model companion, which we denote by CXF, curve-excluding fields. Curve-excluding fields provide examples to various questions. Model theoretically, they are model complete and algebraically bounded. Field theoretically, they are not large. This answers a question of Junker and Macintyre negatively. Joint work with Will Johnson and Erik Walsberg.
03.02.2022 9:30 SRZ216/217
Gianluca Basso (Lyon 1)
The dynamics of kaleidoscopic groups
Abstract:
Duchesne, Monod and Wesolek described how to each permutation group
of countable degree a group acting on a certain one-dimensional tree-like
continuum. This is called its kaleidoscopic group. We reframe the construction
in terms of countable structures and determine which dynamical properties are
preserved when passing to the kaleidoscopic group. This requires a novel
structural Ramsey theorem and produces a new class of examples exhibiting
a poorly understood phenomenon. This is joint work with Todor Tsankov.
27.01.2022
Leon Pernak (WWU)
Introduction to hyperbolic towers and generalized Fraïssé limits II
Abstract:
We will study a generalized version of what is known as Fraïssé's method in model theory to elementary embeddings. The method allows us to obtain for certain classes of structures a limit structure which is universal (i.e. all members of the class embed elementarily in the limit), saturated and homogeneous. A quick discussion of theorems solving Tarski's problem together with the preparations in the previous talk will then allow us to prove that groups elementarily equivalent to a fixed torsion-free hyperbolic group form such a strong elementary Fraïssé class, and consequently prove the existence of a universal, saturated and homogenous elementary free group.
20.01.2022
Ulla Karhumäki (Helsinki)
Recognising $PGL_2(K)$ and tight configurations in finite Morley rank context
Abstract:
It is expected that a `small’ odd type infinite simple group of finite Morley rank is isomorphic to the Chevalley group $PGL_2(K)$ for $K$ algebraically closed. We discuss different set-ups with the common feature that a group of finite Morley rank $G$ is a definable closure of a group $PGL_2(F)$ (or $PSL_2(F)$) for some infinite field $F$. I will explain what kind of assumptions are needed for the identification of $G$ with $PGL_2(K)$ in such set-ups. I will also present a tool, particularly suitable for our set-up, for recognising a connected odd type group of finite Morley rank with $PGL_2(K)$. Some parts of the talk are joint work with P. Ugurlu and some work in progress with A. Deloro.
13.01.2022
Leon Pernak (WWU)
Introduction to hyperbolic towers and generalized Fraïssé limits
Abstract:
A long standing question in both model theory and group theory has been
whether all non-abelian free groups are elementarily equivalent. The
question was famously asked by Tarski around 1945 and answered
positively by Z. Sela and, independently, by O. Kharlampovich and A.
Myasnikov in 2006. In pursuit of the question, Sela studied what he
called (hyperbolic) towers. This concept was further investigated by C.
Perin and R. Sklinos, among others, who worked out multiple details and
further applications of the concept.
The central justification for studying hyperbolic towers is the
following result due to Sela:
If G is a non-abelian torsion-free hyperbolic group and a hyperbolic
tower over some non-abelian subgroup H, then H is elementarily embedded
in G.
And its converse, due to Perin:
If a torsion-free hyperbolic group H is elementarily embedded into some
torsion-free hyperbolic group G, then G is a hyperbolic tower over H.
The definition of hyperbolic towers relies heavily on concepts in
geometric group theory. Therefore we will quickly discuss fundamental
groups of complexes and graphs of groups. The latter provide a tool to
decompose groups into amalgamated products and HNN-extensions, known as
Bass-Serre theory. Towers then consist of multiple layers of such
decompositions with certain additional properties. We will work our way
through the definitions along multiple examples and state the main
results.
If time permits, we will also discuss a generalization of classical
Fraïssé limits, which was used by Kharlampovich-Myasnikov and later by
Guirardel-Levitt-Sklinos to provide a homogeneous group in which all
non-abelian free groups (Kharlampovich-Myasnikov) or more generally,
all elementarily equivalent torsion-free groups
(Guirardel-Levitt-Sklinos) embed elementarily.
16.12.2021
Blaise Boissonneau (WWU)
Artin-Schreier extensions & combinatorial complexity
Abstract:
Model-theoretic combinatorial complexity and algebraic properties are
interlinked in somewhat surprising ways. An example of that is a well-known
result by Kaplan, Scanlon and Wagner, which states that infinite NIP fields of
characteristic p have no Artin-Schreier extension. This result has since then
being proven to also hold for n-dependent fields, and a weaker version holds
for NTP2 fields. In all these settings, we can reduce the argument down to an
explicit formula, and use it to lift complexity from the residue field; most
notably, this allows us to obtain a classification of n-dependent henselian
valued fields.
09.12.2021
Rob Sullivan (Imperial College London)
Hrushovski constructions in the context of the KPT correspondence
Abstract:
Hrushovski constructions - specifically, particular classes of 2-sparse graphs - behave differently to previous classes studied in the context of the KPT correspondence, as first shown in papers of Evans, Hubička and Nešetřil (2019, 2021). We will discuss results from these papers, including how to find Ramsey expansions of classes of sparse graphs via a Ramsey theorem for classes of structures with set-valued functions. We will also discuss some new results from the speaker's PhD thesis (supervised by Evans), extending results from the above papers.
02.12.2021
Francesco Gallinaro (Leeds) ONLINE
Exponential sums equations and tropical geometry
Abstract:
Zilber’s quasiminimality conjecture predicts that all subsets of the complex numbers that are definable using the language of rings and the exponential function are either countable or cocountable. Building on Zilber’s work, Bays and Kirby have proved that the quasiminimality conjecture would follow from the exponential-algebraic closedness conjecture, also due to Zilber, which predicts sufficient conditions for systems of equations in polynomials and exponentials to have complex solutions.
In this talk, I will give an introduction to this topic before presenting some recent work which solves the conjecture for a class of algebraic varieties which corresponds to systems of exponential sums. This turns out to be closely related to tropical geometry, a "combinatorial shadow" of algebraic geometry which reduces some questions about algebraic varieties to questions about polyhedral objects.
25.11.2021
Arturo Rodriguez Fanlo (Oxford) ONLINE
Piecewise hyperdefinable groups and rough approximate subgroups
Abstract:
Piecewise hyperdefinable sets are natural generalizations of interpretable sets. A standard example is the quotient of a subgroup generated by a definable set over a type-definable normal subgroup. On the other hand, approximate subgroups are subsets of a group similar to subgroups up to a finite discrete-like error. Rough approximate subgroups generalise approximate subgroups by allowing also a no-discrete-like error. The most relevant case of rough approximate subgroups occurs in metric groups when the no-discrete error is given by the metric.
Firstly, we will discuss the general structure of piecewise hyperdefinable groups. Then, we will see an application to rough approximate subgroups and some combinatorial consequences in the particular case of metric groups. All this corresponds to my Ph.D. thesis which is divided into two papers: \emph{On piecewise hyperdefinable groups} (arXiv:2011.11669) and \emph{On metric approximate subgroups} (joint with Hrushovski, soon in arxiv).
18.11.2021
Chieu Minh Tran (Notre Dame) ONLINE
The inverse Kemperman problem
Abstract:
Let $G$ be a locally compact group with a left Haar measure $\mu$, and let
$A,B \subseteq G$ be nonempty and compact. In 1964, Kemperman showed that
if $G$ is unimodular (i.e., $\mu$ is also the right Haar measure, e.g.,
when $G$ is $\mathbb{R}/\mathbb{Z}$, $\mathrm{SL}_2(\mathbb{R})$, or
$\mathrm{SO}_3(\mathbb{R})$), then $$ \mu(AB) \geq \min \{\mu(A)+\mu(B),
\mu(G)\} .$$ The inverse Kemperman problem (proposed by Griesmer,
Kemperman, and Tao) asks when the equality happens or nearly happens. I
will discuss the recent solution of this problem by Jinpeng An, Yifan
Jing, Ruixiang Zhang, and myself highlighting some ideas from
model-theoretic group theory.
11.11.2021 ONLINE
Konstantinos Kartas (Oxford)
Decidability via the tilting correspondence
Abstract:
We will discuss new decidability results for mixed characteristic
henselian fields, whose proof goes via reduction to positive
characteristic. In order to achieve the reduction, we will use
extensively the theory of perfectoid fields/p-adic Hodge theory and
also the earlier Krasner-Kazhdan-Deligne principle. Our main results
are the following:
(1) A relative decidability result for perfectoid fields. This says
that, under a certain natural assumption, a perfectoid field K is
decidable relative to its tilt K^♭. As an application, we obtain
several decidability results for tame fields of mixed characteristic,
transposing a recent result of Lisinski (building on earlier work of
Kuhlmann). We also obtain a different application by transposing work
of Anscombe-Fehm to mixed characteristic.
(2) An undecidability result for the asymptotic theory of p-adic fields
(fixed p). This says that the set of sentences in the language of
valued fields with cross-section which are true in all but finitely
many finite extensions of Qp is undecidable. This should be contrasted
with the Ax-Kochen/Ershov Theorem, saying that each individual p-adic
field is decidable in said language.
4.11.2021
Anna-Maria Ammer (WWU)
First-order theory of free generalized n-gons.
Abstract:
We extend the results of Hyttinen and Paolini on free projective planes to
free generalized n-gons. We show that the theory of free generalized
n-gons corresponds to the theory of open generalized n-gons and is
complete. We further characterize the elementary substructure relation and
show that the theory of open generalized n-gons does not have a prime
model.
28.10.2021
Simon André (WWU)
First-order theory of free groups and hyperbolic groups
Abstract:
I will explain why non-abelian free groups have the same
$\forall\exists$-theory, a result first proved by Sacerdote in 1973 using
combinatorial group theory, then proved in a more geometrical way by Sela
using the theory of groups acting on trees (developed by Rips, Sela and
others). I will also discuss some generalisations of this result to larger
classes of groups, such as virtually free, hyperbolic or acylindrically
hyperbolic groups.
21.10.2021
Jan Dobrowolski (WWU)
Kim-independence in first-order logic and beyond
Abstract:
The class of NSOP1 theories, originally introduced by Džamonja and Shelah in
2004, has been studied very intensively in the last few years since the
striking discovery of an independence relation called Kim-independence by
Ramsey and Kaplan (based on earlier ideas of Kim and a paper by Chernikov and
Ramsey), which generalises forking independence in simple theories, and
retains all its nice properties except base monotonicity in the class of NSOP1
theories (over models). Algebraic examples of non-simple NSOP1 structures
include Frobenius fields (e.g. omega-free pseudo-algebraically closed fields),
infinite-dimensional vector spaces with a generic bilinear form, algebraically
closed fields with a predicate for a Frobenius subfield, and, in positive
logic, existentially closed exponential fields and algebraically closed fields
with a generic submodule.
14.10.2021
Martina Liccardo (Napoli Federico II)
Elimination of imaginaries in lexicographic products of ordered abelian groups
Abstract:
We will investigate the property of elimination of imaginaries for some
special cases of ordered abelian groups. As main result, we will show that the
lexicographically ordered groups Z^n and Z^n \times Q eliminate imaginaries
once we add finitely many constants to the language of ordered abelian groups.
08.07.2021
Victor Lisinski (Oxford University)
Decidability of equal characteristic tame Hahn fields in the language L_t
Abstract:
The model theory of tame fields in the language of valued fields has been
extensively studied by Kuhlman. In particular, the theory of an equal
characteristic tame field in this language is given by the theory of the
residue field and the theory of the value group. Building on Kuhlman's
results, we give an AKE-principle for tame valued fields of equal
characteristic in L_t, the language of valued fields with a distinguished
constant symbol t. Furthermore, we use this principle together with
Kedlaya's work on the connection between generalised power series and
finite automata to show that a tame Hahn field of equal characteristic is
decidable in L_t if it has decidable residue field and decidable value
group. In particular, we obtain decidability of F_p((t^{1/p^\infty})) and
F_p((t^Q )) in L_t.
Time permitting, we will also see how approximation methods used in this
work reveal a condition on algebraicity for generalised power series in
terms of the order type of the support.
01.07.2021 SLZ117 + online
Shirly Geffen (WWU)
Rokhlin type properties for finite group partial actions on C*-algebras
Abstract:
The classical Rokhlin lemma from ergodic theory has motivated the
concept of Rokhlin type properties for group actions on C*-algebras. They are
mainly considered in order to classify C*-algebras arising from dynamical
systems.
In this talk, I'll review these ideas and extend them to the setting of
partial actions by finite groups.
This is joint work with Fernando Abadie and Eusebio Gardella.
24.06.2021
Tomas Ibarlucia (Université de Paris)
«Stably free» actions of the free group
Abstract:
I will present a construction of a special kind of action of a free group
on any countable or separable saturated structure, which enjoys a strong
form of freeness given by (local) stability theory. This is one of the main
ingredients of a proof that automorphism groups of separably categorical
structures have Property (T) of Kazhdan.
22.06.2021 10:30 SRZ 216/217(!) + online
Maciej Malicki (Polish Academy of Sciences)
Non-locally compact Polish groups and non-essentially countable orbit equivalence relations
Abstract:
It is a long-standing open question whether every Polish group that is not
locally compact admits a Borel action on a standard Borel space whose
associated orbit equivalence relation is not essentially countable. In the
talk, I will answer this question positively for the class of all Polish
groups that embed in the isometry group of a locally compact metric space.
This class contains all non-Archimedean Polish groups, for which there is
an alternative, game-theoretic proof giving rise to a new criterion for
non-essential countability. I will also discuss the following variant of a
theorem of Solecki: every infinite-dimensional Banach space has a
continuous action whose orbit equivalence relation is Borel but not
essentially countable. This is joint work with A. Kechris, A.
Panagiotopoulos, and J. Zielinski.
17.06.2021
Ziemowit Kostana
When is the automorphism group of an uncountable Fraisse limit (sort of) universal?
Abstract:
In many typical situations, the countable universal homogeneous model of some
first-order theory has the property that its automorphism group contains a
homeomorphic copy of the automorphism group of each countable model of the
theory.
In the uncountable case, the analogous statement is typically false. However,
there are some situations when the automorphism group of an uncountable
homogeneous structure is universal (as a topological group) for the class of
automorphism groups of some carefully chosen structures. I would like to
present three such examples -- two from the realm of linear orders, and one
from the theory of Boolean algebras.
10.06.2021
Aristotelis Panagiotopoulos (WWU)
The definable content of (co)homological invariants: Cech cohomology
Abstract:
In this talk we will develop a framework for enriching various
classical invariants of homological algebra and algebraic topology with
additional descriptive set-theoretic information. The resulting "definable
invariants" can be used for much finer classification than their purely
algebraic counterparts. We will illustrate how these ideas apply to the
classical Cech cohomology invariants to produce a new "definable cohomology
theory" which, unlike its classical counterpart, provides a complete
classification to homotopy classes of mapping telescopes of d-tori, and for
homotopy classes of maps from mapping telescopes of d-tori to spheres. In
the process, we will prove Ulam stability for quotients of Polish abelian
non-archimedean groups G by Polishable subgroups H. A special case of our
Ulam stability theorem answers a question of Kanovei and Reeken regarding
quotients of the $p$-adic groups.
20.05.2021
Colin Jahel
Some progress on the unique ergodicity problem
Abstract:
In 2005, Kechris, Pestov and Todorcevic exhibited a correspondence between combinatorial properties of structures and dynamical properties of their automorphism groups. In 2012, Angel, Kechris and Lyons used this correspondence to show the unique ergodicity of all the minimal actions of some subgroups of $S_\infty$. In this talk, I will give an overview of the aforementioned results and discuss recent work generalizing results of Angel, Kechris and Lyons in several directions.
06.05.2021
Simone Ramello (Università di Torino)
Étale methods in the model theory of fields
Abstract:
Introduced by Johnson, Tran, Walsberg and Ye, the so-called étale-open
topology effectively acts as a dictionary between the topological and the
algebraic worlds; in particular, algebraic properties of a field an be
characterized as topological properties of the étale-open topology on the
affine line, and moreover the étale-open topology turns out to be the
"usual suspects" when considered over certain classes of fields -- the
Zariski topology on separably closed fields, the order topology on real
closed fields... The étale-open topology also allows to characterize large
fields, as introduced by Pop in 1996, precisely as the class of fields over
which it is not the discrete topology. This allows to turn a statement like
the stable fields conjecture into an almost purely topological question, at
least in the large case, and eventually leads to proving the conjecture in
this scenario.
29.04.2021
Lasse Vogel (HHU Düsseldorf)
Invariants of definable sets in pseudo-finite structures
Abstract:
Since many of the main tools in model theory, for example the switching to
an elementary extension, do not work for finite structures, one can take a
particular interest in pseudo-finite structures, which are infinite
structures that behave sufficiently similar to finite ones. One also might
be interested in an overview of the definable sets in a given
pseudo-finite structure up to isomorphism. For this goal invariants of
definable sets are a useful tool, especially one can define a stronger,
non-standard version of cardinality for definable sets that is invariant
under definable bijections. We will break this invariant down into
components which are easier to understand and review their relations. We
will also ask ourselves, how conclusively this version of cardinality
describes the definable sets in certain types of pseudo-finite
structures.
22.04.2021
Tim Clausen (WWU)
Mock hyperbolic reflection spaces and Frobenius groups of finite Morley rank
Abstract:
Joint work with Katrin Tent.
A Frobenius group is a group G together with a proper nontrivial malnormal
subgroup H. A classical result due to Frobenius states that finite Frobenius
groups split, i.e. they can be written as a semidirect product of a normal
subgroup and the subgroup H. It is an open question if this holds true for
groups of finite Morley rank, and the existence of a non-split Frobenius group
of finite Morley rank would contradict the Algebraicity Conjecture. We use
mock hyperbolic reflection spaces, a generalization of real hyperbolic spaces,
to study Frobenius groups of finite Morley rank.
We show that the involutions in a connected Frobenius group of finite Morley
rank and odd type form a mock hyperbolic reflection space. These spaces
satisfy certain rank inequalities and we conclude that connected Frobenius
groups of odd type and Morley rank at most 6 split.
Moreover, by using a construction from the theory of K-loops we show that if G
is a connected Frobenius group of degenerate type with abelian complement,
then G can be expanded to a group whose involutiuons almost form a mock
hyperbolic reflection space. The rank inequalities allow us to show structural
results for such groups. As a special case we get Frecon's theorem: There is
no bad group of Morley rank 3.
15.04.2021
Franziska Jahnke (WWU)
Model theory of henselian valued fields
Abstract:
We study the class of sets definable by first-order formulae in henselian
valued fields. The guiding principle, building on classical work by Ax-Kochen
and Ershov, is that definable sets in well-behaved henselian valued fields are
governed by those of the residue field and the value group. We present a new
theorem generalizing these classical ideas.
We then discuss when a valuation is so intrinsic to the field that its
valuation ring is definable using just the arithmetic of the field. This is
closely linked to a conjecture by Shelah that considers fields for which the
class of definable sets has restricted combinatorial complexity, i.e., no
formula has the independence property. The conjecture predicts that any
infinite such field is separably closed, real closed, or admits a nontrivial
henselian valuation.
We present the state of the art regarding the conjecture, including a theorem
that any NIP henselian valued field satisfies an Ax-Kochen/Ershov principle.
Arithmetic definability of valuations turns out to be a key tool in this
context.
04.02.2021
Ben Castle (UC Berkeley)
A Partial Result on Zilber's Restricted Trichotomy Conjecture
Abstract:
Zilber's Restricted Trichotomy Conjecture predicts that every sufficiently rich strongly minimal structure which can be interpreted from an algebraically closed field K, must itself interpret K. Progress toward this conjecture began in 1993 with the work of Rabinovich, and recently Hasson and Sustretov gave a full proof for structures with universe of dimension 1. In this talk I will discuss a partial result in characteristic zero for universes of dimension greater than 1: namely, the conjecture holds in this case under certain geometric restrictions on definable sets. Time permitting, I will discuss how this result implies the full conjecture for expansions of abelian varieties.
28.01.2021
Stefan Ludwig (WWU)
Metric valued fields in continuous logic
Abstract:
By work of Ben-Yaacov complete valued fields with value groups embedded in the real numbers can be viewed
as metric structures in continuous logic. For technical reasons one has to consider the projective line ov
er such a field rather than the field itself.
In this talk we introduce the above setting and give a classification of the elementary classes of metric
valued fields in equicharacteristic 0 in terms of their residue field and value group. This can also be se
en as an approximate AKE principle. As a second result we give a negative answer to a question of Ben-Yaac
ov on the existence of a model companion for metric valued fields enriched with an isometric isomorphism.
21.01.2021
Rosario Mennuni (WWU)
Model theory of double-membership
Abstract:
It is an old result that the "membership graph" of any countable
model of set theory, obtained by joining x and y if x is in y *or*
y is in x, is isomorphic to the random graph. This is true for
extremely weak set theories but, crucially, they have to satisfy
the Axiom of Foundation.
In joint work with Bea Adam-Day and John Howe we study the class of
"double-membership graphs", obtained by joining x and y if x is in
y *and* y is in x, in the case of set theory with the Anti-
Foundation Axiom. In contrast with the omega-categorical,
supersimple class of "traditional" membership graphs, we show that
double-membership graphs are way less well-behaved: their theory is
incomplete and each of its completions has the maximum number of
countable models and is wild in the sense of neostability theory.
By using ideas from finite model theory, we characterise the
aforementioned completions, and show that the class of countable
double-edge graphs of Anti-Foundation is not even closed under
elementary equivalence among countable structures. This answers
some questions of Adam-Day and Cameron.
14.01.2021
Allison Wang (Cambridge University)
Hyperfiniteness and Ramsey notions of largeness
Abstract:
The lowest non-trivial complexity class in the theory of
Countable Borel Equivalence Relations (CBERs) is the class of
hyperfinite CBERs. One difficulty that arises in studying this class is
determining which CBERs are hyperfinite. Measure theory can
be used to answer this question, but not many techniques can. For instance,
a Baire category approach cannot distinguish hyperfinite
CBERS: a result of Hjorth and Kechris states that every CBER on a Polish
space is hyperfinite when restricted to some comeager set.
We will discuss a classical proof of Mathias's theorem that every CBER on
the Ellentuck Ramsey space is hyperfinite when restricted
to some pure Ellentuck cube. Mathias's theorem implies that a
Ramsey-theoretic approach also cannot distinguish hyperfinite CBERs.
This is joint work with A. Panagiotopoulos.
07.01.2021
Florian Felix (WWU)
A New Approach to Ershov's Wonderful Fields
Abstract:
A "wonderful field" is a field together with embeddings into the p-adic
numbers and the reals that fulfils a full local-global principle with regard
to all absolutely irreducible varieties. They were introduced by Yuri Ershov
in 2002 in a number of papers and became relevant again in 2019 in the PhD
thesis of Kesavan Thanagopal as they give rise to an example for a finite
field extension L/K where K is decidable, but L is not. In this talk we will
introduce and motivate wonderful fields, give an alternative proof for their
existence and briefly talk about their model-theoretic properties.
17.12.2020
Aleksandra Kwiatkowska (WWU)
Simplicity of the automorphism groups of homogeneous structures
Abstract:
The program of understanding the normal subgroup structure of groups that
arise as automorphism groups of countable structures dates back at least to
the ’50s, when Higman described all proper normal subgroups of the
automorphism group of rationals (Q,<). In recent several years
Tent-Ziegler, following the work of Macpherson-Tent, proved simplicity for
many automorphism groups of countable graphs and metric spaces. In the talk,
we prove simplicity for the automorphism groups of order and tournament
expansions of homogeneous structures such as the bounded Urysohn metric space
and the random graph. In particular, we show that the automorphism group of
the linearly ordered random graph is a simple group. This is joint work with
Filippo Calderoni and Katrin Tent.
10.12.2020
Katrin Tent (WWU)
Defining R and G(R)
Abstract:
In joint work with Segal we use the fact that for Chevalley groups G(R) of
rank at least 2 over a ring R the root subgroups are (nearly always) the
double centralizer of a corresponding root element to show under mild
restrictions on the ring R that R and G(R) are bi-interpretable. (This holds
in particular for any field k.) For such groups it then follows that the group
G(R) is finitely axiomatizable in the appropriate class of groups provided R
is finitely axiomatizable in the corresponding class of rings.
03.12.2020
Sahana Balasubramanya (WWU)
Quasi-parabolic group actions on hyperbolic spaces
Abstract:
Gromov's classification of groups acting on hyperbolic spaces exactly classifies the 5 different possibilities that can arise in this situation. Of these, quasi- parabolic actions are the least understood and several questions remain unanswered in this regard. In this talk, I will talk about the motivation behind our work, and describe some known structural results related to quasi-parabolic actions.
I will also talk about a new construction, developed by me, C.Abbott and A.Rasmussen, that allows us to build quasi-parabolic actions for new groups. This construction is a generalization of the work of Caprace -Cornulier-Monod-Tessera from their paper 'Amenable hyperbolic groups'. As a result, we are able to completely classify the quasi-parabolic actions of certain groups on hyperbolic spaces.
26.11.2020
Franziska Jahnke (WWU)
Modelltheorie bewerteter Körper
Abstract:
Wir betrachten arithmetische Definierbarkeit in henselsch bewerteten Körpern.
Einerseits beschäftigen wir uns mit der Frage, wann eine henselsche Bewertung
so intrinsisch ist, dass die Arithmetik des Körpers sie bereits vollständig
beschreibt. Andererseits sagt die Shelah Vermutung voraus, dass unendliche
Körper mit (aus kombinatorischen Sicht) wenig komplexer Arithmetik bereits
separabel abgeschlossen, reell abgeschlossen oder henselsch bewertet sind. Wir
stellen den aktuellen Stand der Arbeit an dieser Vermutung vor. Arithmetische
Definierbarkeit von Bewertungen spielt auch hier eine zentrale Rolle.
19.11.2020
Rosario Mennuni (WWU)
The domination monoid
Abstract:
This talk is concerned with the interaction between the semigroup
of invariant types and the preorder of domination, i.e. small-type
semi-isolation. In the superstable case, the induced quotient
semigroup, which goes under the name of "domination monoid",
parameterises "finitely generated saturated extensions of U" and
how they can be amalgamated independently. In general, the
situation is much wilder, and the domination monoid need not even
be well-defined. Nevertheless, this object has been used to
formulate AKE-type results, can be computed in various natural
examples, and there is heuristic evidence that well-definedness may
hold under NIP. I will give an overview of the subject and present
some results on these objects from my thesis.
12.11.2020
Tingxiang Zou (WWU)
Around pseudofinite difference fields
Abstract:
Ultraproducts of finite fields (pseudofinite fields) have been
studied by model theory extensively. Lots of asymptotic qualitative
behaviors of finite fields have been discovered through this way. On the
other hand, the model companion of difference fields (fields with a
distinguished automorphism) has also been well understood. The theory is
called ACFA. It is widely believed that non-trivial ultraproducts of
\bar{F_p} with frobenius are models of ACFA by an unpublished paper of
Hrushovski. However, (non-trivial) ultraproducts of finite fields with
Frobenius remain model-theoretically mysterious. In this talk, I will
present some attempts and conjectures towards understanding them.
05.11.2020
Silke Meißner (TU Darmstadt)
Simplicity of the Automorphism Groups of Some n-Order Expansions
Abstract:
In [1] it was shown that the automorphism groups of (1-)order
expansions of several homogeneous structures with a stationary independence
relation are simple. We follow the basic approach from [1] in order to show
that the same holds true for n-order expansions, 1<n. However, the passage
from 1-orders to n-orders causes some extra challenge that requires a
different proof idea. My proof involves a reduction of the problem to
1-orders that comes, in return, with significant constraints we have to deal
with.
[1] F. Calderoni, A. Kwiatkowska, K. Tent. Simplicity of the
Automorphism Groups of Order and Tournament Expansions of Homogeneous Structures.
29.10.2020
Jan Dobrowolski (WWU)
Sets, groups, and fields definable in vector spaces with a bilinear form
Abstract:
We study dimension, definable groups, and definable fields in vector
spaces over algebraically closed [respectively, real closed] fields
equipped with a non-degenerate alternating bilinear form or
a non-degenerate [positive-definite] symmetric bilinear form. After
a brief overview of the background, I will discuss a notion of dimension
and some other ingredients of the proof of the main result, which states
that, in the above context, every definable group is
(algebraic-by-abelian)-by-algebraic
[(semialgebraic-by-abelian)-by-semialgebraic]. It follows from this
result that every definable field is definable in the field of scalars,
hence either finite or definably isomorphic to it [finite or
algebraically closed or real closed].
22.10.2020
Aristotelis Panagiotopoulos
Dynamical obstructions to classification II
Abstract:
One of the leading questions in many mathematical research
programs is whether a certain classification problem admits a
“satisfactory” solution. What constitutes a satisfactory solution depends,
of course, on the context and it is often subject to change when the
original goals are deemed hopeless. Indeed, in recent years several
negative anti-classification results have been attained. For example: by
the work of Hjorth and Foreman-Weiss we know that one cannot classify all
ergodic measure-preserving transformations using isomorphism types of
countable structures as invariants; and by the work of Thomas we know that
higher rank torsion-free abelian groups do not admit a simple
classification using Baer-invariants such as in the rank-1 case.
In this talk I will provide a gentle introduction to *Invariant Descriptive
Set Theory*: a formal framework for measuring the complexity of such
classification problems and for showing which types of invariants are
inadequate for a complete classification. In the process, I will present
several anti-classification criteria which come from topological dynamics.
In particular, I will discuss my recent joint work with Shaun Allison, in
which we provide a new obstruction to classification by (co)homological
invariants and use it to attain anti-classification results for Morita
equivalence of continuous-trace C∗-algebras and for the isomorphism problem
of Hermitian line bundles.
15.10.2020
Aristotelis Panagiotopoulos
Dynamical obstructions to classification
Abstract:
One of the leading questions in many mathematical research
programs is whether a certain classification problem admits a
“satisfactory” solution. What constitutes a satisfactory solution depends,
of course, on the context and it is often subject to change when the
original goals are deemed hopeless. Indeed, in recent years several
negative anti-classification results have been attained. For example: by
the work of Hjorth and Foreman-Weiss we know that one cannot classify all
ergodic measure-preserving transformations using isomorphism types of
countable structures as invariants; and by the work of Thomas we know that
higher rank torsion-free abelian groups do not admit a simple
classification using Baer-invariants such as in the rank-1 case.
In this talk I will provide a gentle introduction to *Invariant Descriptive
Set Theory*: a formal framework for measuring the complexity of such
classification problems and for showing which types of invariants are
inadequate for a complete classification. In the process, I will present
several anti-classification criteria which come from topological dynamics.
In particular, I will discuss my recent joint work with Shaun Allison, in
which we provide a new obstruction to classification by (co)homological
invariants and use it to attain anti-classification results for Morita
equivalence of continuous-trace C∗-algebras and for the isomorphism problem
of Hermitian line bundles.
09.07.2020
Paul Wang (ENS)
Groupoids and covers
Abstract:
Hrushovski established a correspondence between definable groupoids and
internal covers, using the notion of binding groupoids. It was later shown to
be an equivalence of categories by Haykazyan and Moosa. They also managed to
extend it to the case of 1-analysable covers with independent fibers. There
was a suggestion to treat the more general case of 1-analysable covers using
simplicial groupoids. In this talk, I will present the internal case, describe
how the suggestion can (hopefully) be realised, and give a few examples.
02.07.2020
Juliane Päßler (WWU)
Forking and JSJ decompositions in the free group
Abstract:
Sela proved that the first order theory of torsion free hyperbolic groups, and therefore that of free groups, is stable. This result implies that there is a notion of independence in the free group. Perin and Sklinos succeeded to characterize forking independence in the free group in a purely group theoretic way by introducing a special kind of JSJ decomposition of a free group relative to a set of parameters.
In my talk I will present this result as well as some special cases where it is easier to understand forking independence.
25.06.2020
Christoph Kesting (WWU)
NTP2 Transfer for multiplicative valued difference fields with lift and section
Abstract:
For multiplicative valued difference fields in equicharacteristic 0 there is an analogous notion to henselianity called $\sigma$-henselianity. This gives rise to a lift of the residue field into the valued field, which we will add to our language, alongside a section of the valuation. We will discuss some of the properies of these fields and a NTP2 transfer result applicable to them.
18.06.2020
Timo Krisam (WWU)
Distal Theories and the Type Decomposition Theorem
Abstract:
A distal theory is a NIP-theory that can be viewed as being order-like. First
we will discuss some characterizations and related concepts of distality.
After that we will take a look at Pierre Simon's type decomposition theorem
for NIP, seeing how a NIP-theory can be viewed as a mix of something stable
and something ordered. This will be done in the context of a basic example.
28.05.2020
Juan Pablo Acosta (WWU)
One dimensional groups definable in the p-adic numbers and groups definable in Presburger Arithmetic
Abstract:
I will present two results.
A list of all one dimensional groups definable in the
p-adic numbers up to a finite index subgroup and a
quotient by a finite subgroup.
And a list of all groups definable in Presburger arithmetic up to a
finite index subgroup.
14.05.2020
Martin Hils (WWU)
Classification of imaginaries in valued fields with automorphism
Abstract:
The imaginaries in the theory ACVF of non-trivially valued algebraically
closed valued fields are classified by the so-called 'geometric' sorts. This
is a fundamental result due to Haskell-Hrushovski-Macpherson. We show that the
imaginaries in henselian equicharacteristic 0 valued fields may be reduced,
under rather general assumptions, to the geometric sorts and to imaginaries in
RV together with sorts for certain finite-dimensional vector spaces over the
residue field.
In the talk, I will mainly speak about an application of this reduction, which
has been the initial motivation of our work: The imaginaries in the theory VFA
of algebraically closed non-trivially valued fields of equicharacteristic 0,
endowed with a non-standard Frobenius automorphism, are classified by the
geometric sorts. This requires an understanding of imaginaries in pure short
exact sequences, and a key result from Hrushovski's groupoids paper.
07.05.2020
Tim Clausen (WWU)
On the geometry of sharply 2-transitive groups
Abstract:
We show that the geometry associated to certain non-split sharply 2-transitive
groups does not contain a proper projective plane. For a sharply 2-transitive
group of finite Morley rank we improve known rank inequalities for this
geometry and conclude that a sharply 2-transitive group of Morley rank 6 must
be of the form AGL_1(K) for some algebraically closed field K.
30.04.2020
Martin Bays (WWU)
Local uniform honest definitions
Abstract:
I will present a result from some recent work-in-progress with Itay Kaplan and
Pierre Simon: using the density of compressible types in NIP theories (which I
talked about in a previous Oberseminar, but will recall) and some consequences
of the (p,q)-theorem, we give an explicit uniform honest definition for
externally definable sets \phi(A,b) where \phi is NIP, improving on previous
results of Chernikov-Simon which gave the existence of such uniform honest
definitions only under a global NIP hypothesis.
23.01.2020
David Bradley-Williams (DHHU Düsseldorf)
Spherically complete models of hensel minimal fields
Abstract:
A valued field is called maximal if it admits no proper immediate extentions
(having the same residue field and value group). Krull observed that every
valued field must have some maximal immediate extension; Kaplansky established
sufficient conditions for uniqueness. In doing so, Kaplansky proved that a
field is maximal if and only if it is spherically complete: that the
intersection of any chain of closed (valuative) balls is non-empty. As can be
expected, spherical completeness can be convenient for analytic/geometric
arguments. Model theoretically, it can be helpful to transfer to a spherically
complete model, if at least one exists.
But, while every valued field has a spherically complete extension, this need
not be an elementary extension (even as a valued field). Furthermore, it
might be important to preserve extra (algebraic/analytic) structure on the
field.
Cluckers, Halupczok and Rideau have recently introduced hensel minimality for
expansions of valued fields. In this talk, we discuss the existence of
spherically complete models of hensel minimal expansions of valued fields;
joint work with Immanuel Halupczok.
09.01.2020
Daniel Palacin (Albert-Ludwigs-Universitaet Freiburg)
Stable sets and additive combinatorics
Abstract:
Given a subset A of a finite abelian group G, we denote by A+A the
subset of elements of G which are the sum of two elements of A. A fundamental
question in additive combinatorics is to determine the structure of subsets A
satisfying that A+A has size at most K times the size of A, where K is a fixed
parameter. It is easy to verify that these subsets are translates of subgroups
when K=1. Furthermore, for arbitrary K and for abelian groups of bounded
exponent, a celebrated theorem of Ruzsa asserts that A is covered by a finite
union of translates of subgroups, whose sizes are commensurable to the size of
A. Improvements of this result have been subsequently obtained by many authors
such as Green, Tao and Sanders, as well as Hrushovski who obtained an
analogous result for non-abelian groups using model theoretic tools.
In this talk I shall present a model theoretic version of Ruzsa's theorem for
subsets A satisfying suitable model theoretic conditions, such as stability.
This is joint work with Amador Martin-Pizarro and Julia Wolf.
19.12.2019
Philip Dittmann (TU Dresden)
Around model completeness properties for fields
Abstract:
Many of the fields studied in model theory are model complete or become so in
natural expansions of the language of rings. I will discuss recent joint work
with D. Leijnse concerning a weakening of model completeness which is
satisfied by fields such as the rational numbers or more general global
fields, for which a full model-theoretic understanding is unattainable.
12.12.2019
Javier de la Nuez (University of the Basque Country)
Minimality results for automorphism groups of homogeneous structures
Abstract:
Any group G of permutations can be endowed with the so called standard
topology, the group topology in which a system of neighbourhoods of the
identity consists of the collection of all fix-point stabilizers of finite
sets and in case G is the automorphism group of a countable structure M it
is a Polish group topology. For certain classes of highly homogeneous M
such as the random graph or the dense linear order this yields interesting
examples of Polish groups with remarkable dynamical properties. Here we
take a look at the question of minimality, i.e. of whether there are no
Hausdorff group topologies on G strictly coarser than the standard
topology. We present a couple of new minimality results in case M is
the Fraïssé limit of a class with free amalgamation, as well as for
the isometry group of the Urysohn space with the point-wise convergence topology.
(Joint work with Zaniar Ghadernezhad)
05.12.2019
Sebastian Krapp (Üniversität Konstanz)
Valuations Definable in the Language of Ordered Rings
Abstract:
Let L_r := {+,-,*,0,1} be the language of rings and let L_or :=
{+,-,*,0,1,<} be the language of ordered rings.
The study of definable valuations (i.e. valuations whose corresponding
valuation ring is a definable set) in certain fields is motivated by the
general analysis of definable subsets of fields as well as by recent
conjectures on the classification of NIP fields. There is a vast collection of
results giving conditions on L_r-definability of henselian valuations in a
given field, many of which are from recent years (cf. [Fehm-Jahnke19]). So
far, not much seems to be known about L_or-definable valuations in ordered
fields. Since L_or is a richer language than L_r, it is natural to expect
further definability results in the language of ordered rings.
In my talk, I will outline some progress in the study of L_or-definable
valuations from my joint work with S. Kuhlmann and G. Lehéricy. In this
regard, I will present sufficient topological conditions on the value group
and the residue field of a henselian valuation v on an ordered field such that
v is L_or-definable. Moreover, I will show how the study of L_or-definable
valuations connects to ordered fields dense in their real closure as well as
above mentioned conjectures regarding NIP fields.
All valuation and model theoretic notions will be introduced.
28.11.2019
Silvain Rideau-Kikuchi (IMJ Paris Diderot)
A model theoretic theoretic account of the tilting equivalence
Abstract:
Generalizing work of Krasner and Fontaine-Wintenberger on the isomorphism of
absolute Galois groups between a mixed characteristic perfectoid field K and
its charactersitic p tilt K^flat = prolim_{x -> x^p} K, Scholze introduced a
notion of perfectoid adic space and proved an equivalence of category between
perfectoid adic spaces over K and over K^flat.
The goal of this talk will be to give a model theoretic translation of these
results. We will show that, in a well chosen continuous structure, K and
K^flat are bi-interpretable and that this immediately yields an equivalence of
categories between type spaces over K and K^flat. We then explain how these
result relate to (and differ from) Scholze's results in adic geometry.
14.11.2019, 21.11.2019
Blaise Boissonneau (WWU Münster)
NIPity in Algebraic Extensions of Q_p
Abstract:
A very important open question in the study of NIP fields is the
following: do they always admit a non-trivial henselian valuation? Qp is a
good example, since it is NIP and its valuation is definable. In this talk, I
will explore algebraic extensions of Qp and classify which ones are NIP. On
the way, it will give some insight about NIP fields in the general case.
07.11.2019
Sahana Balasubramanya (WWU Münster)
Acylindrical actions on products of hyperbolic spaces
Abstract:
I will give a background about acylindrical actions, hyperbolic spaces and
acylindrically hyperbolic groups. I will especially highlight some of the
existing theory as it relates to the behavior of "random" elements and
"random" subgroups of such groups (i.e. those obtained by random walks on
the group). The question I am interested in (in joint ongoing work with T.
Fernos) is how much of this behavior generalizes to acylindrical actions on
products of hyperbolic spaces. The motivation behind this question is the
behavior of Higher rank lattices, Mapping class groups and decomposable
RAAGs. I will talk about some of the result we have obtained in this
direction.
24.10.2019
Pierre Touchard (WWU Münster)
Burden in exact sequences of abelian groups
Abstract:
In model theory, the burden is a notion of dimension for NTP2 structures,
which form a relatively tame class of first order structures. Chernikov and
Simon (2016) consider exact sequences of abelian groups A -> B -> C, and
modulo the hypothesis that B/nB is finite for all integers n, they prove a
quantifier elimination result. Then, they compute the burden in a particular
case: if the burden of A and burden of C are equal to 1, so is the burden of
A->B->C. However, the condition of finite classes modulo n is restrictive. For
instance, it is known that abelian groups with finite classes modulo n are
exactly the ones of burden 1 (Jahnke, Simon, Walsberg). Using a new quantifier
elimination result of Aschenbrenner, Chernikov, Gehret and Ziegler, one can
show in general that the burden of a pure exact sequence of abelian groups is
given by the following formula: bdn(A->B->C) = max_n (bdn(A/nA) + bdn(nC)). I
will present some examples and a sketch of the proof. I will also present some
applications of this result.
17.10.2019
Amador Martin-Pizarro (Albert-Ludwigs-Universität Freiburg)
Equational and non-equational theories
Abstract:
A first-order theory is equational if every definable set is a Boolean
combination of instances of equations, that is, of formulae such that the
family of finite intersections of instances has the descending chain
condition. Equationality is a strengthening of stability yet so far only
two examples of non-equational stable theories are known. We construct
non-equational stable theories by a suitable colouring of the free
pseudospace, based on Hrushovski and Srour's original example.
27.06.2019
Benjamin Brück (Universität Bielefeld)
Parabolic subgroups and spherical complexes for Out(RAAGs)
Abstract:
The class of right-angled Artin groups (RAAGs) is often see as interpolating
between free abelian and free groups. I will present a complex interpolating
between two objects associated to these classes of groups: The Tits building
of GL_n(Q) and the free factor complex. The key objects for the definition are
parabolic subgroups in the outer automorphism group Out(A_\Gamma) of the
corresponding RAAG. The resulting complex has the homotopy type of a wedge of
spheres and its dimension is determined by the rank of a Weyl group group
associated to Out(A_\Gamma).
(see https://arxiv.org/abs/1906.05606)
18.06.2019 15:00-16:00 M2
Allen Gehret (UCLA)
Towards a Model Theory of Logarithmic Transseries
Abstract:
In this talk I will first define and describe the mathematical
object $\mathbb{T}_{\log}$: the ordered valued differential field of
logarithmic transseries. I will then discuss a strategy I have developed for
proving $\mathbb{T}_{\log}$ is model complete in a certain language that I
will introduce. I reduce the problem of model completeness down to two
precise conjectures concerning the nature of logarithmic derivatives,
solutions of linear differential equations, and differential-transcendence.
07.06.2019 11:00 217
Luca Motto Ros (University of Turin)
A descriptive main gap theorem
Abstract:
Answering a question of S. Friedman, Hyttinen and Kulikov, we
show that there is a tight connection between the depth of a classifiable
shallow theory $T$ and the Borel rank of the isomorphism relation
$\cong^\kappa_T$ on its models of size $\kappa$, for $\kappa$ any cardinal
satisfying $\kappa^{< \kappa} = \kappa > 2^{\aleph_0}$. This yields a
descriptive set-theoretical analogue of Shelah’s Main Gap Theorem. We also
discuss some limitations to the possible (Borel) complexities of
$\cong^\kappa_T$, and provide a characterization of categoricity of $T$ in
terms of the descriptive set-theoretical complexity of $\cong^\kappa_T$.
Joint work with F. Mangraviti.
06.06.2019
Pierre Simon (University of California, Berkeley)
On omega-categorical linear orders
Abstract:
What can be said about the closed definable subsets of an omega-categorical
linear order? This question arose in the investigation of NIP
omega-categorical structures and might also be relevant to understanding
Ramsey expansions of homogeneous structures. I will discuss a conjectural
answer and partial results towards it.
05.06.2019 M6 10:30
Gianluca Paolini (University of Turin)
First-Order Model Theory of Free Projective Planes
Abstract:
We prove that the theory of open projective planes is complete and strictly
stable, and infer from this that Marshall Hall's free projective planes
$(\pi^n : 4 \leq n \leq \omega)$ are all elementary equivalent and that
their common theory is strictly stable and decidable, being in fact the
theory of open projective planes. We further characterize the elementary
substructure relation in the class of open projective planes, and show that
$(\pi^n : 4 \leq n \leq \omega)$ is an elementary chain. We then prove that
for every infinite cardinality $\kappa$ there are $2^\kappa$ non-isomorphic
open projective planes of power $\kappa$, improving known results on the
number of open projective planes. Finally, we characterize the forking
independence relation in models of the theory and prove that $\pi^\omega$
is strongly type-homogeneous.
09.05.2019, 16.05.2019, 23.05.2019 SR1d 11-12
29.05.2019 M6 11-12
Matthias Aschenbrenner (UCLA)
A tour of asymptotic differential algebra
Abstract:
Hardy fields and transseries are two among several approaches for enriching
the real continuum by infinitesimal and infinite quantities, with
applications to dynamical systems and model theory. Both come with natural
notions of ordering and differentiation. In joint work with van den Dries and
van der Hoeven we developed a framework of “asymptotic differential algebra”
to unify these approaches. In these lectures I will introduce the main
objects of interest, explain the basic algebraic framework, sketch the
important results obtained so far, and time permitting, outline current
challenges and further possibilities.
25.04
Nadia Hempel (UCLA)
Gruppen und Körper in neostabilen Theorien
Abstract:
In diesem Vortrag geben wir zuerst, anhand von Beispielen fundamentaler
Begriffe wie Sprache und Struktur, eine kurze Einführung in die Grundlagen der
mathematischen Logik.
Anschließend, stellen wir verschieden elementare Klassen
von Strukturen vor und betrachten innerhalb dieser algebraischen Objekte
(Gruppen, Ringe, Körper). Kurz gefasst sind diese Klassen durch das Verbot von
gewissen Kombinatorischen Mustern in den definierbaren Mengen charakterisiert.
Dies hat erstaunliche Konsequenzen für Gruppen als auch Körpern. Außerdem gibt
es Verbindungen zur Theorie von Gruppen mit Kettenbedingungen auf
Zentralisatioren die wir aufzeigen werden.
Abschließend, stellen wir die
Hierarchie der n-abhängigen Theorien vor (für jede natürliche Zahl n).
Der n-Zufallsgraph ist das kanonische Objekt welches n-abhängig aber nicht
(n-1)-abhängig ist. Folglich ist dies eine strikte Hierarchie. Jedoch liegt
die Frage ob algebraische Objekte in jeder dieser Klassen existiert nahe.
Motiviert durch diese Frage stellen wir alle bisher bekannten Resultate von
n-abhängigen Gruppen und Körpern vor.
18.04.2019
Pierre Touchard (Universitaet Muenster)
Stably embedded elementary submodels and Henselian valued fields
Abstract:
We will discuss when an elementary submodel is stably embedded. This
question has been studied by Marker and Steinhorn for o-minimal theories.
In 2015, Kovacsics and Delon characterised pairs of algebraically closed
valued fields $M \prec N$ with $M$ stably embedded in $N$ by the
corresponding property of the value groups. In this talk, we will see how
one can generalise this result for all Henselian valued fields.
We will also discuss the following question: when can stable embeddedness
of elementary pairs be expressed in first order logic, when we add a
predicate $P$ for the smaller model.
11.04.2019
Filippo Calderoni (Universitaet Muenster)
On the complexity of ordered abelian groups
Abstract:
After a gentle introduction to Borel classification theory, we shall discuss
the complexity of the bi-embeddability relation in the case of various
countable structures. In particular, we shall show that the bi-embeddability
relation on ordered divisible abelian groups is as complicated as possible
among analytic equivalence relations. As we will see, our results are
connected with the model theory of o-minimal theories. In the end, we shall
discuss some open questions that we hope to settle in the near future. This
work is joint with D. Marker and L. Motto Ros and still in progress.
14.02.2019
Kobi Peterzil (University of Haifa)
Locally definable and approximate subgroups of semialgebraic groups
Abstract:
(w. E. Baro and P. Eleftheriou)
A symmetric subset X of a group G is called a k-approximate subgroup of G
if k-many group-translates of X cover XX.
Given a natural number n, we let X(n)=XX^{-1}.....XX^{-1} (n times), and ask:
are there n and k such that X(n) is a k-approximate subgroup of G? We do not
know the answer when G=(R^n,+) and X is an *arbitrary* smooth curve.
We give a positive answer to the above question when G is an abelian
semialgebraic group over some real closed field and X is a semialgebraic subset
of G (more generally when X is definable in certain o-minimal expansions of R).
In these cases, we obtain uniform bounds for k and n in terms of X.
I will describe the above result and the connection to a still-open problem
about definably generated abelian groups in o-minimal structures.
24.01.2019
Pablo Cubides Kovacsics (TU Dresden)
Definable subsets of Berkovich curves
Abstract:
Let K be an algebraically closed complete rank 1 non-trivially valued
field. Let X be an algebraic curve over K and let X^an be its
analytification in the sense of Berkovich. We functorially associate to
X^an a definable set X^S in a natural language. As a corollary, we
obtain an alternative proof of a result of Hrushovski-Loeser about the
iso-definability of curves. Our association being explicit allows us to
provide a concrete description of the definable subsets of X^S. This is
a joint work with Jérôme Poineau.
17.01.2019
Daniel Hoffmann (Uniwersytet Warszawski)
From PAC to NSOP
Abstract:
There are several attempts to describe theories by Galois groups, and new
notions of Galois group have been defined for this purpose (Shelah Galois
group, Kim-Pillay Galois group, Lascar Galois group). My project goes in the
other direction: instead of introducing new Galois groups, finding theories
which are controlled by the "classical" Galois groups.
In the case of the theory of fields, there is a special class of fields,
pseudoalgebraically closed fields (PAC fields). PAC fields were the core of
research in field theory in the second half of the 20th century. Why? Because
the theory of a PAC field is controlled by its absolute Galois group, so all
the machinery from Galois theory can be invoked and used with success; e.g.
Nick Ramsey showed that a PAC field is NSOP1 if and only if its absolute
Galois group is NSOP1. Therefore it makes sense to develop model-theoretic
Galois theory in the case of PAC structures, a generalization of PAC fields.
With my co-authors, I obtained recently a generalization of the Elementary
Equivalence Theorem for PAC structures: two PAC structures share the same
first order theory provided they have isomorphic absolute Galois groups. We
hope that also Ramsey’s result might be generalized, and this is a work in
progress. In my talk, I will summarize the situation and explain connections
between some results from my preprints, because combining them together gives
us an algorithm for obtaining PAC structures with an absolute Galois group
which can be "calculated", and so there is a prospective way to generate new
examples of NSOP1 structures.
10.01.2019
Philip Dittmann (Oxford)
First-order logic in finitely generated fields
Abstract:
The expressive power of first-order logic in the class of finitely
generated fields, as structures in the language of rings, is relatively
poorly understood. For instance, Pop asked in 2002 whether elementarily
equivalent finitely generated fields are necessarily isomorphic, and
this is still not known in the general case.
Building on work of Pop and Poonen, and using cohomological techniques
based on tools due to Kerz-Saito and Gabber, I shall show that every
infinite finitely generated field of characteristic not two admits a
definable subring which is a finitely generated algebra over a global
field. This implies that any such finitely generated field is
biinterpretable with arithmetic, and gives a positive answer to the
question above in characteristic not two.
20.12.2018
Martin Bays (Universitaet Muenster)
Density of compressibility
Abstract:
Joint with Itay Kaplan and Pierre Simon.
Distal theories are NIP theories which are "wholly unstable". Chernikov and
Simon's "strong honest definitions" characterise distal theories as those in
which every type is _compressible_. Adapting recent work in machine learning
of Chen, Cheng, and Tang on bounds on the "recursive teaching dimension" of a
finite concept class, we find that compressibility is dense in NIP structures,
i.e. any formula can be completed to a compressible type in S(A). Considering
compressibility as an isolation notion (which specialises to l-isolation in
stable theories), we obtain consequences on the existence of models with
certain properties.
13.12.2018
Tingxiang Zou (Lyon 1)
Counting in pseudofinite structures
Abstract:
In pseudofinite structures, the non-standard size of definable
sets often reveals important algebraic or model theoretic properties of the
corresponding theories. In this talk, we will give two new examples of this
correlation. One is between coarse dimension and the transformal
transcendence degree in certain class of pseudofinite difference fields.
The other example is that in pseudofinite H-strucures built from
one-dimensional asymptotic classes, the coarse dimension of a tuple
corresponds to the leading coefficient of the SU-rank of this tuple. This
is the first step to show that they are examples of multidimensional
asymptotic classes (mac).
06.12.2018
Gregory Cherlin (Rutgers)
Splitting twisted automorphism groups
Abstract:
The twisted automorphism group of a structure is the normalizer
in the full symmetric group of its automorphism group. Following Cameron
and Tarzi, we ask when the group of twisted automorphisms splits over
the group of automorphisms. After explaining their result for the case of
randomly edge colored infinite complete graphs, we discuss joint work with
Rebecca Coulson which answers this question in the case of metrically
homogeneous graphs.
29.11.2018
Zoé Chatzidakis (CNRS, ENS)
Non-existence of prime models of pseudo-finite fields
Abstract:
Let F be a non-principal ultraproduct of finite fields, T its
theory, and A a subfield of F which is relatively algebraically
closed in F. It is known that any two models of T which contain A
as a relatively algebraically closed subfield are elementarily
equivalent over A.
Theorem: If A is not pseudo-finite, then T has no prime model over
A.
The proof of this result is fairly easy when A is countable: one just
uses that prime models over A are atomic. The proof when A is
uncountable is more involved, as it involves constructing 2^{|A|}
non-isomorphic models of T which are of transcendence degree 1 over
A. We also discuss the existence or non-existence of \kappa-prime models
of T (i.e., \kappa-saturated models of T containing A and which
A-embed into any \kappa-saturated model of T containing A), for regular
uncountable \kappa such that \kappa =\kappa^{<\kappa}.
15.11.2018
Florian Felix (Muenster)
Quantifier elimination and NIP in separably closed valued fields
Abstract:
Jizhan Hong has shown that the theory of separably closed valued fields
of infinite imperfection degree has quantifier elimination in the
language of valued rings together with certain lambda p-coordinate
functions. We will elaborate on his proof, and using this result we will
prove that the same theory has NIP by counting coheirs.
08.11.2018
Simon Thomas (Rutgers)
The characters and invariant random subgroups of the finitary symmetric group
Abstract:
In this talk, I will discuss the relationship between the characters and the
invariant random subgroups of the finitary symmetric group. (I will not assume
any prior knowledge of character theory or the theory of invariant random
subgroups.)
25.10.2018
Sylvy Anscombe (University of Central Lancashire)
Cohen Rings: their structure, complete theories, and NIP transfer
Abstract:
The story of complete local rings is an old one. For us: a Cohen ring (of
residue characteristic p) is a complete local ring A with maximal ideal pA.
Under the extra hypothesis of regularity, these are complete discrete
valuation rings. The structural results are due to many people, including
famous names like Hasse, Schmidt, Witt, Teichmuller, and Mac Lane, in the
1930s. If the residue field is imperfect, there are some extra
difficulties, worked out by Teichmuller and Mac Lane. In the 1940s, Cohen
extended the structure theory by removing the hypothesis of regularity. We
elaborate on his theory, supplementing the structural results with an
`embedding lemma'. Turning to model theory, we apply the embedding lemma to
give a relative quantifier elimination, and to describe the complete
theories. Finally, if there is time, I will explain how we apply this to
extend Belair's transfer of NIP from residue field to valued field in
henselian mixed characteristic valued fields.
(This is work in progress with Franziska Jahnke)
18.10.2018
Timo Krisam (Universität Münster)
Generische Prädikate und bdn(T_P)
Abstract:
Wir betrachten eine vollständige Theorie T in der Sprache L,
die Quantorenelimination hat. Zu L fügen wir ein unäres Prädikat P hinzu
und erhalten die Sprache L_P. Chatzidakis und Pillay haben gezeigt, dass
unter bestimmten Annahmen T als eine L_P-Theorie einen Modellbegleiter T_P
besitzt, welcher einige Eigenschaften von T, unter anderem einfach zu
sein, erhält. Wir werden sehen, dass auch die Bürde von T erhalten bleibt,
also bdn(T) = bdn(T_P) gilt. Als direktes Korollar ergibt sich, dass falls
T NTP2 ist, so auch T_P.
12.07.2018
Ayhan Gunaydin (Bosphorus University)
Tame Expansions of o-minimal Structures
Abstract:
Expanding a model theoretically “tame” structure in a way that it
stays “tame” has been a theme in the recent years. In the first part of this
talk, we present a history of work done in that frame. Then we focus on the
case of expansions of o-minimal structures by a unary predicate. There is a
dividing line according to whether the predicate is dense or discrete; even
though the results obtained are similar, there is an enormous difference in
the techniques used. We shall present some of the results obtained in the
dense case. Starting from a set of abstract axioms, we obtain a decomposition
theorem for definable sets and a local structure theorem for definable
groups.
The abstract axioms mentioned above are “smallness”, “o-minimal open core” and
“quantifier elimination up to existential formulas”. We shall illustrate a
proof of the fact that the first two imply “quantifier elimination up to
bounded formulas”, which is a weak form of the last axiom and we give reasons
why it is really weaker than that axiom.
(Joint work with P. Eleftheriou and P. Hieronymi)
05.07.2018
Yatir Halevi (Hebrew University Jerusalem)
Infinite stable division rings of finite dp-rank are algebraically closed fields
Abstract:
As the title suggests, we will show that every infinite stable
division ring of finite dp-rank is an algebraically closed field.
Joint work with Daniel Palacín.
28.06.2018
Esther Elbaz (Paris 7)
Grothendick ring of pairing function with no cycles
Abstract:
If $p$ is a bijection between a set $M$ and $M^2$, we say that it is a
pairing function with no cycles if for any term $t(x_1,..., x_n)$
formed with $p$, and that is not a variable, we have, for every
$a_1,\ldots, a_n \in M$, $t(a1,\ldots, a_n) \neq a_1$.
The theory of pairing function with no cycles is the simplest example
of a stable theory that is not a limit of super stable theories. It has been
studied by several authors and, in particular, it has been shown that it is
complete and admits quantifier elimintation in a natural language.
Grothendieck rings of a structure have been introduced in Model Theory
in 2000. Its construction relies on indentifying definable sets that
are in definable bijection and generalizes the definition of
Grothendieck ring already known in algebraic geometry.
We will first give a brief survey of this model theoretic notion. Then
we'll compute the Grothendieck ring of pairing function with no
cycles, and show that it is isomorphic to $\mathbb{Z}[x]/(X-X^2)$.
14.06.2018
Silvain Rideau (Paris Diderot)
Groups and fields in ACVF
Abstract:
(joint with Ehud Hrushovski)
We will give a structure theory for groups interpretable in ACVF, decomposing
them in term of groups internal to the residue field, groups internal to the
value group and group schemes over the valuation ring. Groups with a stably
dominated type stable by translation (stably dominated groups) play an
important role in this structure theory: our main result is that Abelian
groups have a maximal value group internal quotient whose kernel is covered by
stably dominated groups. We also relate stably dominated groups to group
schemes over the valuation ring. Finally, we will use this structure theory to
show that, up to definable isomorphism, every field definable in ACVF is
either the residue field or the valued field itself.
07.06.2018
Martin Bays (Muenster)
Projective geometries arising from Elekes-Szabó problems
Abstract:
This is a continuation of a talk given in December 2017.
I will recall how complex varieties which have asymptotically large
intersections with grids of unbounded size can be seen, exploiting ideas of
Hrushovski, to correspond to projective geometries, and how this leads to a
precise characterisation of such varieties. I will explain moreover a version
of this which allows the co-ordinates to be points in higher dimensional
varieties. I will proceed to describe some consequences for generalised
sum-product estimates and connections to diophantine problems. This is joint
work with Emmanuel Breuillard.
17.05.2018
Arthur Forey
Bounded integral in valued fields
Abstract:
Hrushovski and Loeser have shown that Hrushovski and Kazhdan’s motivic
integration provides a direct approach to relate the motivic Milnor fiber and
invariants of the classical Milnor fiber, e.g. to show that their Euler
characteristics coincide. Their work relies on the existence of both a purely
additive integration morphism and one that respect volume forms. I will
present a new integration morphism on some bounded sets that allows to relate
those two morphisms. This is joint work in progress with Yimu Yin.
03.05.2018
Pierre Touchard
Henselian valued fields, burden and RV-sorts
Abstract:
Artem Chernikov and Pierre Simon have proved that any ultraproduct of
p-adics in the language of rings is of burden one, i.e. inp-minimal. They
show the following Ax-Kochen-Ershov-like result : a Henselian
equicharacteristic $0$ valued field $K$ is inp-minimal if and only if the
RV-sort of $K$ (which I will define) is inp-minimal. We will see how to
extend their result to any residue caracteristic and any burden. If we have
time, we will see as an application that the field of $p$-adics
$\mathbb{Q}_p$ is inp-minimal (or equivalently dp-minimal).
12.04.2018
Wiesław Kubiś
The weak amalgamation property
Abstract:
The amalgamation property of a class of structures is a
well-known concept, important for building universal homogeneous models. We
shall discuss its weakening, discovered by Ivanov in 1999, which turns out
to be relevant in the study of so-called generic objects. Namely, a
countable class of finitely generated structures with the joint embedding
property has a generic limit if and only if it has the weak amalgamation
property.
The results come from joint works with A. Krawczyk, A. Kruckman, and A.
Panagiotopoulos.
25.01.2018
Mohammad Bardestani
Kirillov's orbit method and polynomiality of the faithful dimension of $p$-groups
Abstract:
Let $G$ be a finite group. The faithful dimension of $G$ is
defined to be the smallest possible dimension for a faithful complex
representation of $G$. Aside from its intrinsic interest, the problem of
determining the faithful dimension of $p$-groups is motivated by its
connection to the theory of essential dimension. In this talk, we will
address this problem for groups of the form $\mathbf{G}_p:=\exp(\mathfrak{g}
\otimes_{\mathbb{Z}}\mathbb{F}_p)$, where $\mathfrak{g}$ is a nilpotent
$\mathbb{Z}$-Lie algebra of finite rank, and $\mathbf{G}_p$ is the $p$-group
associated to $\mathfrak{g} \otimes_{\mathbb{Z}}\mathbb{F}_p$ in the Lazard
correspondence. We will show that in general the faithful dimension of
$\mathbf{G}_p$ is given by a finite set of polynomials associated to a
partition of the set of prime numbers into Frobenius sets. At the same
time, we will show that for many naturally arising groups, including a vast
class of groups defined by partial orders, the faithful dimension is given
by a single polynomial. The arguments are reliant on various tools from
number theory, model theory, combinatorics and Lie theory.
22.01.2018 11:00 SR1d (Tee 10:30)
André Nies
Logical aspects of profinite groups
Abstract:
The various subdisciplines of mathematical logic connect in interesting
ways to separable profinite groups.
We present three examples.
1. The complexity of the isomorphism problem can be classified using tools
of descriptive set theory: isomorphism is smooth for finitely generated
profinite groups, but as complicated as isomorphism of countable graphs in
general (with Kechris and Tent).
2. Work of Jarden and Lubotzky can be connected to algorithmic randomness.
For instance, every algorithmically random e-tuple in the absolute Galois
group of Q generates a free profinite group (with Fouche).
3. Finally we ask which profinite groups can be uniquely identified by a
first-order sentence. We give a first example, the Heisenberg group over
the p-adic integers.
11.01.2018
Franziska Jahnke
Inp-minimal (left-)ordered groups are abelian
Abstract:
We will discuss a proof by Jan Dobrowolski and John Goodrick showing that
inp-minimal left-ordered groups are abelian. This builds on work by Pierre
Simon in which he shows the same holds for inp-minimal bi-ordered groups.
21.12.2017
Martin Hils
Definable equivariant retractions of stable completions of abelian varieties
Abstract:
(joint work with Ehud Hrushovski and Pierre Simon)
Using the model theory of algebraically closed valued fields, Hrushovski
and Loeser have obtained topological tameness properties for the
Berkovich analytification $V^{an}$ of an algebraic variety $V$ defined
over a valued field, under very general assumptions. They do the main
work in the stable completion of $V$, a model-theoretic avatar of
$V^{an}$.
Given an abelian variety $A$ defined over an algebraically closed valued
field, we will present the construction of a definable equivariant
strong deformation retraction of the stable completion of $A$ onto its
skeleton, which is a piecewise linear group.
14.12.2017
Philip Dittmann
On the p-Stufe and the model theory of absolute Galois groups
Abstract:
Over fields of characteristic 0, Stufe and Pythagoras number are two
important invariants related to field orderings. I will give a
generalisation to p-valuations as opposed to orderings (following the
tradition of extending study of R to study of Q_p), prove some results
on these invariants in fields algebraic over the rational numbers, and
give a connection to the model theory of absolute Galois groups.
07.12.2017
Martin Bays
The geometry of combinatorially extreme algebraic configurations
Abstract:
Given a system of polynomial equations in m complex variables with
solution set of dimension d, if we take finite subsets X_i of C each of
size at most N, then the number of solutions to the system whose ith
co-ordinate is in X_i is easily seen to be bounded as O(N^d).
We ask: when can we improve on the exponent d in this bound?
Hrushovski developed a formalism in which such questions become amenable
to the tools of model theory, and in particular observed that incidence
bounds of Szemeredi-Trotter type imply modularity of associated
geometries. Exploiting this, we answer a (more general form of) our
question above. This is part of a joint project with Emmanuel
Breuillard.
30.11.2017
Tim Clausen
Valued strongly complete abelian profinite groups
Abstract:
Let A be an elementary substructure of a strongly complete abelian
profinite group and fix a valuation v on (A,+) with good properties. Then
the valued abelian group (A,+,v) has quantifier elimination in a
reasonable language and is dp-minimal.
23.11.2017
Zoé Chatzidakis
Notions of difference closures of difference fields.
Abstract:
It is well known that the theory of differentially closed fields of
characteristic 0 has prime models (over differential subfields) and that
they are unique up to isomorphism.
One can ask the same question for the theory ACFA of existentially
closed difference fields (recall that a difference field is a field with
an automorphism).
In this talk, I will first give the trivial reasons of why this question
cannot have a positive answer. It could however be the case that over
certain difference fields, prime models (of the theory ACFA) exist and
are unique. Such a prime model would be called a difference closure of
the difference field K. I will show by an example that the obvious
conditions on K do not suffice.
I will then consider the class of aleph-epsilon saturated models of
ACFA, or of kappa-saturated models of ACFA. There are natural notions of
aleph-epsilon prime model and kappa-prime model.
It turns out that for these stronger notions, if K is an algebraically
closed difference field of characteristic 0, with fixed subfield F
aleph-epsilon saturated, then there is an aleph-epsilon prime model over
K, and it is unique up to K-isomorphism. A similar result holds for
kappa-prime when kappa is a regular cardinal.
None of this extends to positive characteristic.
16.11.2017
Annette Bachmayr
Large fields in differential Galois theory
Abstract:
Large fields were introduced by Florian Pop in the 1990's and have played
an important role in Galois theory. Examples of large fields are pseudo
algebraically closed fields, fields which are complete with respect to
absolute values and more generally fraction fields of Henselian domains.
In joint work with David Harbater, Julia Hartmann and Florian Pop, we
solve the inverse differential Galois problem over k(x) where k is a large
field of infinite transcendence degree.
The first part of my talk will be about large fields and then I will
explain what differential Galois theory is before I present the result
mentioned above.
09.11.2017
Aleksandra Kwiatkowska
Ważewski dendrites
Abstract:
We will discuss a class of very interesting one-dimensional
metric continua called Ważewski dendrites. The homeomorphism group of a
Ważewski dendrite is isomorphic to an automorphism group of a certain
countable structure. Using the Kechris-Pestov-Todorcevic correspondence
relating topological dynamics and Ramsey theory, we will compute universal
minimal flows of those homeomorphism groups of Ważewski dendrites whose
universal minimal flow is metrizable.
26.10.2017
Elisabeth Bouscaren
Groups of finite rank, the socle theorem and some applications
Abstract:
We will recall first some results about groups of finite rank (Morley Rank or
U-rank) and orthogonality, in particular the existence of the socle and the
"socle theorem". Then we will explain how this theorem can be used in the
context of applications of model theory to algebraic geometry. The first
application appeared in Hrushovski's orginal proof of the Mordell-Lang
Conjecture ( 1994). The theorem was used more recently slightly differently in
joint work with F. Benoist and A. Pillay.
19.10.2017 10:45
Katrin Tent
The complexity of topological group isomorphism
Abstract:
(joint with A. Kechris and A. Nies)
Countable first order structures can be studied via their
automorphism groups. This motivates our study of the complexity of the
isomorphism relation for various classes of closed subgroups of $S_\infty$.
We use reducibility between equivalence relations on Polish spaces.
For profinite, locally compact, and Roelcke precompact groups, we show
that the complexity is the same as the one of countable graph isomorphism.
25.07.2017 11:00-12:00 SR5
Javier de la Nuez Gonzales
Generalizing trees: planar graphs and dp-minimality.
Abstract:
We review some known results on the stability of the theory of
superflat and ultraflat graphs and show how the planar case admits a
geometrically meaningful expansion which, while failing to be stable in
general, will still be dp-minimal NIP. This generalizes the example of a tree
with the inbetweenness relationship.
24.07.2017
Mohammed Bardestani
Some results and questions in extremal group theory.
Abstract:
Extremal combinatorics studies how large or how small a
collection of finite objects can be, if it has to satisfy certain
restrictions. For instance, in an n-element set, what is the largest number
of k-element subsets that can pairwise intersect one another? This
question, which has been answered by Erdos, Ko and Rado, can be considered
as a typical question in this area.
The Erdos-Ko-Rado theorem can be considered in the context of the group
actions. In this talk, I will present some results showing how the
classification of minimal finite simple groups, established by Thompson,
can be applied to answer some problems in the context of extremal group
theory.
26.06.2017
Françoise Delon
C-minimal valued fields
Abstract:
An important achievement in recent model theory
is the o-minimality of the real exponential field.
The theory of C-minimal fields has been developed as a possible analogue
of o-minimality fitting certain tame expansions of
algebraically closed valued fields.
The behaviour at infinity of definable functions demonstrates
however an essential difference:
with Pablo Cubides-Kovacsics, we have shown
that C-minimal fields valued in Q are polynomially bounded.
22.06.2017
Carlos Alfonso Ruiz Guido
Nilpotents in model theory?
Abstract:
Boris Zilber wonders to what extent the duality between algebras and
model theoretic objects (zariski geometry-like objects) can be extended. It is
know that for algebraic varieties the correspondence is precise as long as we
restrict to normal algebraic varieties. I will present a structure that
extends this correspondence when the algebraic varieties are non reduced but
have gorenstein singularities. There are two open questions about this
structures:
1) Nilpotents in algebraic geometry correspond to the infinitesimal study of
zariski closed sets. Is the proposed structure needed? (consider that we
already have a notion of infinitesimal coming from specializations).
2) The use of nilpotents is not exclusive of ACF, they appear naturally in
other contexts like ACFA, CCM. Is it possible to extend the ideas of this
structure to those theories?
07.06.2017 14:30
Alex Lubotzky
First order rigidity of high-rank arithmetic groups
Abstract:
The family of high rank arithmetic groups is class of groups
which is playing an important role in various areas of mathematics. It
includes SL(n,Z), for n>2, SL(n, Z[1/p] ) for n>1, their finite index
subgroups and many more. A number of remarkable results about them have been
proven including; Mostow rigidity, Margulis Super rigidity and the
Quasi-isometric rigidity.
We will talk about a new type of rigidity: "first order rigidity".
Namely if D is such a non-uniform characteristic zero arithmetic group
and E a finitely generated group which is elementary equivalent to it
(i.e., the same first order theory in the sense of model theory) then E is
isomorphic to D.
This stands in contrast with Zlil Sela's remarkable work which implies that
the free groups, surface groups and hyperbolic groups (many of whose are
low-rank arithmetic groups) have many non isomorphic finitely generated
groups which are elementary equivalent to them.
Joint work with Nir Avni and Chen Meiri.
29.05.2017
Maciej Malicki
Automorphism groups of homogeneous metric structures and consequences of the
existence of ample generics
Abstract:
We will define a simple criterion for a homogeneous, complete metric
structure X implying that its automorphism group satisfies all the main
consequences of the existence of ample generics: the automatic continuity
property, the small index property, and uncountable cofinality for non-open
subgroups. It turns out that it holds for the Urysohn space, the Lebesgue
probability measure algebra, and the Hilbert space. We will also formulate
a condition for X implying that every homomorphism of its automorphism
group into a separable group with a left-invariant, complete metric is
trivial, and we will verify it for the Urysohn space, and the Hilbert space.
24.05.2017 (Wednesday) 16:00 in SR1B
Dmitry Sustretov
Asymptotic integration and non-Archimedean geometry
Abstract:
Let X be a family of complex algebraic manifolds over affine line with
the origin removed. If one is interested in the study of the family
near the origin one can consider the base-change of X by Laurent
series C((t)) induced by the natural inclusion C[t,t^{-1}] \to C((t)). The
resulting algebraic variety can also be regarded as a non-Archimedean
analytic space, and its model-theoretic incarnation is a fibre of X over a
point t of valuation 1, in a model of the theory of algebraically closed
valued fields. In this talk I will explain how asymptotics of certain
integrals over X_s as s tends towards 0 can be related to non-Archimedean
geometry of X_t using tools from model theory, such as Robinson fields and
o-minimal cell decomposition.
18.05.2017
Franziska Jahnke
Dreaming about NIP fields
Abstract:
We discuss the main conjectures about NIP fields, the implications
between them and what their consequences are. In particular, we show that the
two most common versions of the conjecture that an infinite NIP field is
separably closed, real closed, or 'p-adic like' are equivalent. This is joint
work in progress with Sylvy Anscombe.
15.05.2017
Peter Sinclair
Stable fields of finite dp-rank
Abstract:
The relationship between logical and algebraic properties of
groups and fields has been an active area of research within model theory
for over 40 years. I will present a recent result I have made in this area,
that stable fields of finite dp-rank are algebraically closed. In addition
to a rough sketch of my work, I will discuss its relationship to recent
results of Johnson on dp-minimal fields, as well as the fundamental result
of Cherlin and Shelah on superstable fields.
08.05.2017
André Nies
The complexity of topological group isomorphism
Abstract:
We study the complexity of the isomorphism relation for various classes of
closed subgroups of the group of permutations on N. We use the setting of
Borel reducibility between equivalence relations on Polish spaces.
For profinite, locally compact, and Roelcke precompact groups, we show
that the complexity is the same as the one of countable graph isomorphism.
For oligomorphic groups, we merely establish this as an upper bound.
Joint work with Alexander Kechris and Katrin Tent
27.04.2017
Martin Ziegler
Zigzags in Free-N-Pseudospaces (with Katrin Tent)
Abstract:
Zigzags were introduced by Tent in her paper on Free-N-Pseudospaces. We use
them systematically for a new axiomatization of Free-N-Pseudospaces and a new
proof of Tent's description of forking.
24.04.2017
Gönenç Onay
The model companion of the valued fields with a non-valuational
automorphism
Abstract:
The model companion of a theory T is a model complete T_M such that every
model of T embeds in a model of T_M. For instance, the model companion of
the theory of fields is the theory of the algebraically closed fields and the
model companion of the theory of ordered fields is the theory of real
closed fields.
By a result of Kikyo, it is known that the theory of non trivilally valued
fields with an automorphism preserving the valuation ring does not admit a
model companion.
We will show however that if we don't require that the automorphism
preserve the valuation ring
then the model companion exists. During the talk basic model theoretical
notions needed will be explained. This is a joint work with O.Beyarslan, D.
Hoffman and D.Pierce.
20.04.2017
Sylvy Anscombe
Viewing free-homogeneous structures as `generalised measurable'
Abstract:
I will speak about a new(ish) generalisation (with Macpherson,
Steinhorn, and Wolf) of (MS-)measurable structures via examples. I
will show how to view 'free homogeneous structures' as generalised
measurable, for example the generic triangle-free graph.
30.01.2017
Alejandra Garrido
Maximal subgroups of groups of intermediate growth
Abstract:
Studying the primitive actions of a group corresponds to studying its maximal
subgroups.
In the case where the group is countably infinite, one of the first questions one
can ask is whether there are any primitive actions on infinite sets; that is,
whether there are any maximal subgroups of infinite index.
The study of maximal subgroups of countably infinite groups has so far mainly
concerned groups of either polynomial or exponential word growth and in the case
where there are maximal subgroups of infinite index, there are uncountably many.
It is natural to investigate this question for groups of intermediate growth, for
instance, some groups of automorphisms of rooted trees.
I will report on some recent joint work with Dominik Francoeur where we show that
some such groups of intermediate growth have exactly countably many maximal
subgroups of infinite index.
23.01.2017
Aleksandra Kwiatkowska
Groups of measurable functions
Abstract:
We study groups L_0(G) of measurable functions defined on ([0,1], \lambda),
where \lambda is the Lebesgue measure, with values in a topological group
G. We will focus on groups G which are automorphism groups of countable
structures and we will investigate properties of L_0(G) groups related to
conjugacy classes, such as meagerness of topological similarity classes and
the existence of a cyclically dense conjugacy class. In the talk I will
state and explain a number of results on L_0(G) groups obtained in a recent
joint work with Maciej Malicki.
09.01.2017
Silvain Rideau
Imaginaries in pseudo-p-adically closed fields.
Abstract:
In her PhD thesis, Samaria Montenegro proved many new results about
pseudo-p-adically closed fields (a class that contains both
pseudo-algebraically closed fields, and p-adically closed fields). But one
result is still missing, especially if one compares with the other, very
similar class she was also interested in, pseudo-real-closed fields:
elimination of imaginaries.
In this talk, I will explain how it can be proved using generalizations to the
non-simple setting of the classical elimination of imaginaries argument using
higher amalgamation (eg. ACFA and pseudo-algebraically closed fields).
This is joint work with Samaria Montenegro.
19.12.2016
Pierre Touchard
Henselian valued fields of equicharacteristic 0 and inp-minimality
Abstract:
I will present a paper of Chernikov and Simon, where they prove an Ax-Kochen
type result on preservation of inp-minimality for Henselian valued fields
of equicharacteristic 0. We will work in the RV-language, i.e. a language
of valued fields with a sort for RV:=K*/(1+m), where m is the maximal ideal
of the valuation ring of K. As a corollary, we will see that any
ultra-product of p-adics is inp-minimal.
12.12.2016
Zaniar Ghadernezhad
Constructing countable generic structures
Abstract:
In this talk we present a new method of constructing countable generic
structures. We introduce some conditions that guarantees the constructed generic
structure has the algebraic closure property. Moreover, we investigate dividing
and forking in the theory of the constructed generics. This method generalizes
the well-known construction method of building generic structures using
pre-dimension functions and it can be also seen as an attempt to
comprehend/generalize the original Hrushovski's generic construction (although
it was not the initial motivation). Using this new method it is very easy to
build generic structures that their theories are not simple. Time permitting we
investigate TP_2 of the non-simple generics that are obtained using this method.
5.12.2016
Martin Bays
Kummer theory for split semiabelian varieties, and the model theory of
their exponential maps.
Abstract:
Let A be an abelian variety over a number field k_0 and consider its
product G = A \times \G_m^n with a power of the multiplicative group. Then
G admits Kummer theory over Tor(G): if b in G is in no proper algebraic
subgroup, then there is a bound, uniform in n, on the number of orbits under
the Galois group over k_0(Tor(G),b) of an nth division point of b. I will
explain the relevance of this and related results to the model theory of
the exponential map of G(\C), and sketch a proof.
28.11.2016
Martin Hils
Model theory of separably closed valued fields
Abstract:
We will present some recent model-theoretic results for separably closed
valued fields of finite degree of imperfection. In particular we will treat
the classification of imaginaries by the 'geometric sorts' and the
'metastability' of the theory, in the sense of Haskell, Hrushovski and
Macpherson who had initiated the geometric model theory of valued fields in
their work on algebraically closed valued fields.
This is joint work with Moshe Kamensky and Silvain Rideau.
14.11.2016
Rémi Jaoui
Orthogonality to the constants for geodesic differential equations
Abstract:
Let $S \subset \mathbb{R}^n$ be a smooth algebraic subset. We consider
the system $(E)$ of differential equations which describes the motion of
a free particle with $n$ degrees of freedom and fixed energy $E_0$
constrained to move on the submanifold $S$. As $S \subset \mathbb{R}^n$
is a algebraic subset, this system is given by polynomial differential
equations and gives rise to a definable set of the theory
$\mathbf{DCF}_0$ of differentially closed fields.
In my talk, I will be interested in the model-theoretic behaviour of
such definable sets. More precisely, I will describe a general criterion
of orthogonality to the constants for types over the field of real
numbers (with trivial derivation). Then, using this criteria, I will
explain how to deduce from structural results on the dynamics of the
system $(E)$, that if $S$ is compact and the restriction of the
euclidian metric in $\mathbb{R}^n$ to $S$ has strictly negative
curvature, then the generic type of the system $(E)$ is orthogonal to
the constants.
7.11.2016
Mohammad Bardestani
Chromatic numbers of structured cayley graphs
Abstract:
We will study the chromatic number of Cayley graphs of algebraic
groups that arise from algebraic constructions. Using Lang-Weil bound and
representation theory of finite simple groups of Lie type, we will
establish lower bounds on the chromatic number of these graphs. Using
Weil’s bound for Kloosterman sums we will also prove an analogous result
for SL2 over finite rings.
31.10.2016
Aleksandra Kwiatkowska
The Lelek fan and the Poulsen simplex as Fraisse limits
Abstract:
I will describe the Lelek fan and the Poulsen simplex in the
Fraisse-theoretic framework in the context of categories enriched over
metric spaces, developed by Kubis, and derive consequences on their
universality and homogeneity. I will show as an application, strengthening
a result of Kawamura, Oversteegen, and Tymchatyn, that for every two
countable dense subsets of end-points of the Lelek fan there exists a
homeomorphism of the Lelek fan mapping one set onto the other.
This is joint work with Wieslaw Kubis.
20.10.2016 (Donnerstag) 14:00 SR1D
Itay Kaplan
On omega-stable graphs which are not countably colorable
Abstract:
This is joint work in progress with Elad Levi.
A graph coloring is a map from the set of vertices to some set X with the
property that two related edges are mapped to different colors. A graph is
countably colorable (cc) if it has a graph coloring with countable image.
A conjecture of Shelah, Erdös and Hajnal states that if a graph is not cc
then it contains all finite subgraphs of the n-shift graph on omega (I will
define this in the talk).
This conjecture was refuted in full generality by Hajnal and Komjath. However
we proved it for omega-stable graphs of U-rank 2. I will discuss the proof and
attempts at generalizations.
13.07.2016
Martin Bays
Pseudofinite-dimensional Schrödinger representations
Abstract:
Following Zilber, we obtain the Schrödinger representation of
the (3-dimensional) Heisenberg algebra on the tempered distributions as an
ultralimit of finite-dimensional representations of certain subgroups of
the Heisenberg group. Moreover, we see that the corresponding Weil
representation can be obtained this way.
This is joint work with Bradd Hart.
11.07.2016
Tim Clausen
Polynomial subgroup growth and groups with NIP
Abstract:
A group has polynomial subgroup growth if the number of subgroups of
index at most n is bounded by some polynomial f(n). I will show that in a
group with NIP every family of uniformly definable subgroups has polynomial
subgroup growth.
27.06.2016
Zaniar Ghadernezhad
An NTP_2+NSOP generic structure that is not simple
Abstract:
In this talk we first introduce a new method to build a new class of
generic structures. This method generalizes the usual construction method of
generic structures that is based on a pre-dimension function. Using this new
method we build a generic structure that its theory is not simple but NTP_2 and
NSOP. This gives a partial answer to a question by Chernikov about the
existence and hierarchy of such theories.
This is a joint-work with Massoud Pourmahdian.
13.06.2016
Jonathan Kirby
Exponential-algebraic closedness and quasiminimality
Abstract:
It is well-known that the complex field $\C$, considered as a structure in the
ring language, is strongly minimal: every definable subset of $\C$ itself is
finite or co-finite. Zilber conjectured that the complex exponential field
$\C_\exp$ is quasiminimal, that is, every subset of $\C$ definable in this
structure is countable or co-countable.
He later showed that if Schanuel's conjecture of transcendental number theory
is true and $\C_\exp$ is strongly exponentially-algebraically closed
then his conjecture holds. Schanuel's conjecture is considered out of reach,
and proving strong-exponential algebraic closedness involves finding solutions
of certain systems of equations and then showing they are generic, the latter
step usually done using Schanuel's conjecture.
We show that if $\C_\exp$ is exponentially-algebraically closed then it
is quasiminimal. Thus Schanuel's conjecture can be dropped as an assumption,
and strong exponential-algebraic closedness can be weakened to
exponential-algebraic closedness which requires certain systems of equations
to have solutions, but says nothing about their genericity.
This is joint work with Martin Bays.
10.06.2016
Stefan Müller-Stach
Periods
Abstract:
We present a more or less elementary introduction to
Kontsevich-Zagier periods and their relation to Nori motives. The latter form
a Tannakian category of mixed motives without assuming any standard
conjectures. The relation gives a lot of insight into transcendance questions.
This is joint work with Annette Huber.
08.06.2016
Franziska Jahnke
NIP henselian valued fields
Abstract:
We show that tame henselian valued fields are NIP if and only if
their residue field is NIP. Moreover, we show that if (K,v) is a henselian
valued field of characteristic (char(K),char(Kv))=(q,p)
for which K^\times/(K^\times)^p is finite, then (K,v) is NIP iff Kv is NIP and
v is roughly tame. This is joint work with Pierre Simon.
06.06.2016 Zakhar Kabluchko
Ultralimits appearing in probability theory
Abstract: We will present an ultralimit construction which appears naturally
in many problems of probability theory.
This construction yields a remarkable metric space whose properties we will
study.
30.05.2016 Charlotte Kestner
Non-forking formulas in distal NIP theories
We give a survey of non-forking formulas in non-stable theories. In
particular we look at recent progress on a problem of Chernikov and Simon:
given a non-forking formula over a model M of an NIP theory, is this
formula a member of a consistent definable family, definable over
M. We give details of a positive answer to this question in distal NIP
theories.
25.05.2016 - 11:00 in N2 (Achtung! Raum und Zeit geändert) Katrin Tent
Infinite sharply multiply transitive groups
Abstract:
The finite sharply 2-transitive groups were classified by Zassenhaus
in the 1930's. They essentially all look like the group of affine linear
transformations x |-> ax + b for some field (or at least near-field) K. However,
the question remained open whether the same is true for infinite sharply
2-transitive groups. There has been extensive work on the structures associated
to such groups indicating that Zassenhaus' results might extend to the infinite
setting. For many specific classes of groups, like Lie groups, linear groups,
or groups definable in o-minimal structures it was indeed proved that all
examples inside the given class arise in this way as affine groups. However, it
recently turned out that the reason for the lack of a general proof was the fact
that there are plenty of sharply 2-transitive groups which do not arise from
fields or near-fields! In fact, it is not too hard to construct concrete
examples. In this talk, we survey general sharply n-transitive groups and
describe how to construct examples not arising from fields.
23.05.2016 Benjamin Brück
Weight in non-standard models of the theory of free groups
As the common first order theory Tfg of non-abelian free
groups is stable, we can use the notion of forking independence in
order to ask whether a set of elements in models of Tfg is
independent or not.
In this theory, there is a unique generic type p0 over the
empty set. The interest in this type comes from the fact that its
realisations in free groups are exactly the primitive elements and
in a free group of finite rank, a set is a maximal independent set
of realisations of p0 if and only if it forms a basis. In
particular, all those sets have the same cardinality.
The aim of this talk is to look at the analogues of this in groups
that share the same theory as free groups but are not free
themselves. Using hyperbolic towers, I will firstly present a
criterion for the maximality of independent sets of realisations
of p0 in those non-standard models and afterwards give a
construction of models of Tfg that contain such maximal
independent sets with arbitrarily large differences in their
sizes. The existence of such sets of different cardinalities can
be expressed by the fact that the type p0 has infinite weight.
09.05.2016 Elad Levi
Rational functions with algebraic constraints
Abstract: A polynomial P(x,y) over an algebraically closed field k has an
algebraic constraint if the set {(P(a,b),(P(a',b'),P(a',b),P(a,b')|a,a',b,b' \in k}
does not have the maximal Zariski-dimension. Tao proved that if P has an
algebraic constraint then it can be decomposed: there exists Q,F,G \in k[x]
such that P(x1,x2)=Q(F(x1)+G(x2)), or P(x1,x2)=Q(F(x1)*G(x2)). We will
discuss the generalisation of this result to rational functions with
3-variables and show the connection to a problem raised by Hrushovski and
Zilber regarding 3-dimensional indiscernible arrays.
02.05.2016 Ralf Schindler
The continuum hypothesis, MM, and (*).
We will consider two of the guiding questions of contemporary
set theory: What is the strength of MM (Martin's Maximum)?
And how does MM relate to Woodin's axiom (*)? We will
discuss why these questions are interesting (yes, they are!), and
which methods are currently used in order to attack
those problems. This will be a soft talk, not assuming any serious
knowledge of set theory.
25.04.2016 Anna Blaszczok
On Maximal Immediate Extensions of Valued Fields
A valued field extension is called immediate if the corresponding value group
and residue field extensions are trivial. A better understanding of the
structure of such extensions turned out to be important for questions in
algebraic geometry, real algebra and the model theory of valued fields. In
this talk we focus mainly on the problem of the uniqueness of maximal
immediate extensions. Kaplansky proved that under a certain condition, which
he called "hypothesis A", all maximal immediate extensions of the valued field
are isomorphic. We study a more general case, omitting one of the conditions
of hypothesis A. We describe the structure of maximal immediate extensions of
valued fields under such weaker assumptions. This leads to another condition
under which fields in this class admit unique maximal immediate extensions. We
further prove that there is a class of fields which admit an algebraic maximal
immediate extension as well as one of infinite transcendence degree. We
present also the consequences of the above results and of the model theory
of tame fields for the problem of uniqueness of maximal immediate extensions
up to elementary equivalence.
25.01.2016 Martin Bays
On variations on Zilber's exponential-algebraic closedness conjecture
I will present some very recent work on further weakening the remaining
conjecture required to prove quasiminimality of C_exp via Zilber
pseudo-exponentiation techniques.
20.01.2016 Lam Pham
Embedding semigroups in groups: Ore's Theorem
11.01.2016 Daniel Palacín
On nilpotent-by-finite groups
Neumann showed that bounded FC-groups (i.e. groups in which there
is a natural number bounding the size of every conjugacy class) are
finite-by-abelian.
In this talk I shall generalize this notion to bounded FC-nilpotent groups
and prove model-theoretically that bounded FC-nilpotent groups are
precisely the nilpotent-by-finite ones. This is joint work with Nadja
Hempel.
14.12.2015 Arno Fehm
Elementary equivalence of profinite groups
A profinite group is a totally disconnected compact topological
group. Jarden and Lubotzky had shown in 2008 that if two finitely generated
profinite groups are elementarily equivalent in the language of groups, then
they are in fact already isomorphic. Around the same time, Frohn had studied
the theory of abelian profinite groups in the Cherlin-van den Dries-Macintyre
language of inverse systems and reached a similar conclusion for so-called
small abelian profinite groups. A common generalization of these two results
was given recently by Helbig. I will explain these results and discuss some
related questions concerning elementary equivalence (in the language of groups
and in the language of inverse systems) and isomorphism (as abstract groups
and as profinite groups).
07.12.2015 Immanuel Halupczok
A new notion of minimality in valued fields
In the past, various attempts have been made to come up with an
analogue of o-minimality which works for valued fields. This has had a certain
success in special case like in ℚ_p (p-minimality, t-minimality) and in
algebraically closed valued fields (C-minimality, v-minimality). I will
present a new attempt which works well in valued fields of the form k((t)),
for k of characteristic 0. (This is work in progress with Raf Cluckers and
Silvain Rideau.)
30.11.2015 Katrin Tent
Describing finite groups by short first order sentences
23.11.2015 Misha Gavrilovich
On analytic Zariski geometries associated with a group action
We define a structure related to the universal covering space of a complex
algebraic variety, considering in particular the case of a line bundle over an
abelian variety (or rather, its associated C*-bundle). We ask whether these
structures have nice categoricity properties and observe that these properties
seem related to group-theoretic properties of the fundamental group, namely
subgroup separability.
16.11.2015 Franziska Jahnke
Near-henselian fields - valuation theory in the language of rings
(Joint work with Sylvy Anscombe) We consider four properties of fields
(all implying the existence of a non-trivial henselian valuation) and
study the implications between them. Surprisingly, the full pictures look
very different in equicharacteristic and mixed characteristic.
09.11.2015 Martin Bays
Exponential maps and categoricity
Associated to a complex algebraic group G is its
exponential map exp_G : LG --> G. I will discuss various results on
the model theory of expansions of the complex field by such maps, and
related structures, focusing on abelian G, and on the categoricity
problem of giving descriptions of the structures which determine them up
to isomorphism.
02.11.2015 Pierre Simon
Decomposition of types in NIP theories
The class of NIP theories contains both stable theories and o-minimal ones, and one often thinks about NIP structures
as being in some sense combinations of those two extremes. In an attempt to make this intuition precise, I will explain how any type in an NIP
theory can be analysed in terms of a definable component and an "order-like" one.
26.10.2015 Martin Bays
The Group Configuration
This will be the third part on a presentation of the proof of the Group Configuration Theorem.
21.09.2015 Yilong Yang
Quasirandom groups and covering properties
A group is D-quasirandom if all its non-trivial unitary representations have dimensions more than D. This property is obviously not definable in first order logic, and in particular, an
ultraproduct of quasirandom groups will in general fail to be quasirandom.
In this talk, I shall present the covering properties, which is definable in first order logic, and shall characterize the quasirandomness to a certain degree.
A group is said to have a good covering properties iff it has an element g, and the conjugacy classes of all small powers of g are fast expanding.
These properties will be almost equivalent to quasirandomness if we ignore the cosocle of a group (the intersection of all maximal normal subgroups).
Furthermore, it is preserved under arbitrary products and quotients. We shall also discuss its connections to ultraproduct of quasirandom groups, Bohr
compactifications and ergodic theory results.
11.06.2015 Will Anscombe
The existential theory of equicharacteristic henselian valued fields
We present some recent work - joint with Arno Fehm - in which we give
an `existential Ax-Kochen-Ershov principle' for equicharacteristic
henselian valued fields. More precisely, we show that the existential
theory of such a valued field depends only on the existential theory
of the residue field. In residue characteristic zero, this result is
well-known and follows from the classical Ax-Kochen-Ershov Theorems.
In arbitrary (but equal) characteristic, our proof uses F-V Kuhlmann's
theory of tame fields.
One corollary is an unconditional proof that the existential theory of
F_q((t)) is decidable. We will explain how this relates to the earlier
conditional proof of this result, due to Denef and Schoutens. If there
is time, we will indicate other consequences for the study of
existentially and universally definable henselian valuations.
28.05.2015 Zoé Chatzidakis
Difference fields and algebraic dynamics Joint work with E. Hrushovski.
An algebraic dynamics is given by a pair (V,f), where V is an (irreducible quasi-projective) variety, and f:V --> V a
dominant rational map.
Assume that (V,f) is defined over a field K=k(t), where t is transcendental over k, and that for some integer N, for all n,
V(K) contains "many" points P such that P, f(P), ..., f^n(P) have height less than N (i.e., given an embedding of V into
projective space, these points can be represented by polynomials of degree less than N).
When the degree of f is >1, we showed that this implies that (V,f) has a quotient (W,g) defined over k, and such that
deg(g)=deg(f).
In this talk, I will concentrate on the case deg(f)=1, and show that the same conclusion holds (with dim(W)>0), provided k
is algebraically closed. The proof involves definable Galois theory, and the description of difference varieties which are
"internal to the fixed field".
19.05.2015 Javier de la Nuez Gonzáles
Some fine structure of the complex of curves
The second of Masur and Minsky's foundational paper on the geometry of the complex of curves explores certain affinities between it and locally finite complexes.
This is achieved through a careful analysis of a certain net of geodesics in links, so called "hierarchies". We will try to give an overview of their methods, which have
consequences for the geometry of conjugation in the mapping class group.
07.05.2015 - Samaria Montenegro
Model theory of pseudo real closed fields
The notion of PAC fields has been generalized by Basarab and by Prestel to ordered fields. Prestel calls a field M pseudo real closed (PRC) if M is existentially closed (in the language of rings) in every
regular extension L to which all orderings of M extend. Equivalently, if every absolutely irreducible variety defined over M that has a rational point in every real closure of M, has an M -rational point.
In this talk we will study the class of bounded PRC fields from a model theoretical point of view. We work with the complete theory of a fixed bounded PRC field M in the language of rings expanded with
enough constant symbols. The boundedness condition implies that M has only finitely many orders. Our main theorem is a positive answer to a conjecture of Chernikov, Kaplan and Simon that says: A PRC
field is NTP2 if and only if it is bounded. This also allows us to explicitly compute the burden of types, and to describe forking. Some of these results generalize to bounded PpC fields, using the same
kind of techniques.
30.04.2015 - Daniel Palacín
Around the Canonical Base Property
The canonical base property (CBP) is a property of finite rank theories, which was introduced by Pillay and whose formulation was motivated by results of Campana in complex geometry.
The main feature of such a property is that it provides a dichotomy for types of rank one, and in consequence one can reproduce Hrushovski's proof of Mordell-Lang for function fields in
characteristic zero with considerable simplifications.
In this talk, I will motivate the statement of the CBP and describe some results around the CBP, in particular on definable groups.
16.04.2015 - Dugald Macpherson
Groups definable in valued fields
I will discuss joint work in progress with Jakub Gismatullin and Patrick Simonetta on groups definable (in the home sort) in certain
valued fields, such as ACVF and Q_p. Our main result is a description of simple definable groups which are linear, using Bruhat-Tits
buildings and a result of Prasad. It appears to be open whether there is a nonlinear simple definable group.
23.03.2015 - Monday, 16:00 Uhr! - André Nies
The complexity of similarity relations for Polish metric spaces
We consider the similarity relations of isometry and homeomorphism for
Polish metric spaces. We survey known results on the descriptive
complexity of such relations. For instance, Gao and Kechris showed that
isometry is "orbit complete", i.e. as complex as possible, while Gromov
proved that for compact metric spaces isometry is smooth, which means
simple. Orbit completeness of homeomorphism for compact metric spaces has
only recently been shown by Zilinsky, applying a result of Sabok on
simplices. Using a result of Camerlo and Gao, I will indicate a simple proof
(with Solecki) that in the computable setting, this problem is Sigma-1-1
complete for equivalence relations.
18.02.2015 - Wednesday, 10:45 Uhr! - Javier de la Nuez Gonzales
Bounding the Shelah rank of certain definable sets of the free group
The notion of Shelah rank provides a fine measure of the size of definable sets of tame enough (superstable) fragments of first order theories.
We use the properties of the mapping class group of surfaces to establish lower bounds for the Shelah rank of a certain family of varieties in nonabelian free groups.
Work in progress joint with Rizos Sklinos and Chloe Perin.
08.01.2015 - Nadja Hempel
On FC-nilpotent groups
In this talk we define an approximate notion of centralizers and commutators. We establish basic properties, such as the three-subgroup lemma, for the almost commutators
and centralizers of ind-definable subgroups which are well-known facts in the ordinary case. With these results we are able to generalize Hall's nilpotency criteria and a
theorem due to Fitting to FC-nilpotent subgroups of so called MC~ -group. These are groups with a chain condition on centralizers up to finite index whose crucial property
in this context is that the almost centralizer of any subgroup is definable. They appear naturally in the field of model theory, i.e. any group with a simple theory has this property.
18.12.2014 - Frank Wagner
Approximative Gruppen
(gemeinsam mit JC Massicot)
Eine approximative Gruppe ist eine symmetrische
Teilmenge A einer Gruppe, so dass A.A von endlich vielen
Links-Translaten von A überdeckt wird. Endliche approximative Gruppen
wurden vor Kurzem von Breuillard, Green und Tao klassifiziert; dabei
benutzten sie Ideen von Hrushovski, der mit modelltheoretische Methoden
zeigte, dass eine pseudo-endliche approximative Gruppe einen
Homomorphismus in eine Liegruppe besitzt. Wir zeigen, dass dieser
Homomorphismus auch dann existiert, wenn A lediglich definierbar
mittelbar ist.
04.12.2014 - Daniel Palacin Cruz
On Dependence
A first-order theory is n-dependent if it cannot encode a bipartite random n-hypergraph with a definable edge relation. This notion has been introduced recently by Shelah as a generalization
of dependence (NIP). In this talk I present some examples of n-dependent theories, and characterize n-dependence by counting local types over finite sets.
This is a joint work with Artem Chernikov and Kota Takeuchi.
27.11.2014 - Kai-Uwe Bux
Separability of Free Groups and Surface Groups
Joint work with Oleg Bogopolski
The word problem for a given finitely generated group
is the problem of telling whether a word in the
generators represents the identity element. For any
finitely presented group, this problem has an easy
part: if the word is trivial, then it follows from
the given finitely many relations; hence it is possible
to algorithmically list all trivial words. Thus, in
a group with unsolvable word problem, it is impossible
to algorithmically list the non-trivial words.
For a finitely presented group, it is easy to list all
actions on finite sets: for a proposed action of the
generators, just check whether the relations hold. Hence,
one can algorithmically list all finite quotients of
a finitely presented group. This provides an obvious
way of listing some non-trivial words: put down
those, that represent a non-trivial element in some
finite quotient. A group where this algorithm eventually
finds each non-trivial word is called residually
finite. Residually finite groups have an obvious
solution to the word problem.
The conjugacy problem of telling which words represent
conjugate elements allows for a similar treatment.
In any finitely presented group, it is algorithmically
easy to list all pairs of words representing conjugate
elements. The hard part is to list the pairs of words
representing non-conjugate elements. A group is called
conjugacy separable, if any two non-conjugate elements
stay non-conjugate in some finite quotient. Thus,
finitely presented conjugacy separable groups admit
an obvious solution to the conjugacy problem.
Other classical algorithmic problems can be treated
analogously. Each leads to a corresponding notion of
separability. The problem of telling whether two
finitely generated subgroups are conjugate gives rise
to the notion of subgroup conjugacy separability.
A group is subgroup conjugacy separable if any two
non-conjugate finitely generated subgroups have
non-conjugate images in some finite quotient.
We show that finitely generated free groups and
fundamental groups of closed oriented surfaces are
subgroup conjugacy separable.
21.11.2014 - Immanuel Halupczok
QE in angeordneten abelschen Gruppen
Ich werde eine Sprache vorstellen, in der die Theorie
angeordneter
abelscher Gruppen teilweise Quantoren-Elimination hat. Dies ist ein altes
Resultat von Gurevich und Schmitt, in Zusammenarbeit mit Cluckers neu
bewiesen
(und mit einer etwas anderen Sprache).
13.11.2014 - Zaniar Ghadernezhad
The small index property for automorphism groups of generic structures
The automorphism group of a countable first-order structure with the pointwise convergence topology is a Polish group. Then every open subgroup of the automorphism group has a small index
(less than continuum). The automorphism group of a countable structure has the small index property (denote it by SIP) if every subgroup of small index is open. In this talk, we show that the automorphism
group of Hrushovski's ab-initio generic structures does not have SIP with the standard pointwise convergence topology. However, we show that the automorphism group of Hrushovski's ab-initio generic structure
admits "almost SIP": meaning that every finite index subgroup contains a subgroup that fixes an infinite set of finite dimension.
06.11.2014 - Franziska Jahnke
Uniformly defining the canonical p-henselian valuation
Joint work with Jochen Koenigsmann
Admitting a p-henselian valuation is a weaker assumption on a field than admitting a henselian valuation.
Unlike henselianity, p-henselianity is an elementary property in the language of rings. We are interested in the question when a field admits a non-trivial 0-definable p-henselian
valuation (in the language of rings). We give a classification of elementary classes of fields in which the canonical p-henselian valuation is uniformly 0-definable. Time permitting,
we apply this to show that there is a definable valuation inducing the (t-)henselian topology on any (t-)henselian field which is neither separably nor real closed.
30.10.2014 - David Bradley-Williams
Reducts of a universal binary branching tree
In this talk I will describe joint work with Bodirsky, Pinsker and Pongrácz in which we study the reducts of a binary branching semilinear order.
The main result is a classification of model complete cores of such reducts, from which we also classify the maximally closed supergroups of the automorphism group of this structure.
23.10.2014 - Françoise Delon
Group construction in non-trivial geometric $C$-minimal structures
Joint work by Françoise Delon and Fares Maalouf
Zilber conjectured that a strongly minimal structure interprets an infinite group, or even an infinite field, as soon as it fulfils some conditions, that are clearly necessary.
This conjecture turned out to be false in general. However, together with Ehud Hrushovski they were able to establish that the conjecture holds for what they called ``Zariski structures'',
first order structures with a topology which mimics the Zariski topology. Ya'acov Peterzil and Sergei Starchenko proved a variant of the conjecture for o-minimal structures and we tackle the
conjecture in the class of C-minimal structures. The $C$-minimality condition is an equivalent of strong minimality in the setting of ultrametric structures just as o-minimality is an equivalent
of strong minimality in the setting of ordered structures. Fares Maalouf had constructed an infinite definable group in any geometric $C$-minimal structure, which is non-trivial and locally modular.
We remove the assumption of local modularity but assume that the structure is definably maximal and has no definable bijection in its canonical tree between a bounded interval and an unbounded one.