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.

TBA

Abstract: TBA

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

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

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

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

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

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!

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

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

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

(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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 (!)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

«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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

10.06.2016

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

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

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

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 -

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

Weight in non-standard models of the theory of free groups

As the common first order theory T

09.05.2016

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

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

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

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

Embedding semigroups in groups: Ore's Theorem

11.01.2016

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

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

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

Describing finite groups by short first order sentences

23.11.2015

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

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

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

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

The Group Configuration

This will be the third part on a presentation of the proof of the Group Configuration Theorem.

21.09.2015

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

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

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

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 -

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 -

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 -

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 -

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 -

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 -

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 -

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 -

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 -

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

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

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

21.11.2014 -

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 -

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 -

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 -

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 -

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.