Das Oberseminar findet in diesem Semester immer **donnerstags** von 11:00 Uhr bis 12:00 Uhr im Raum SR1D statt. Wir treffen uns vorher um 10:30 Uhr zum Tee im Büro
von
Prof. Katrin Tent (8. Stock, Raum 810). Abweichende Zeiten oder Räume werden zeitnah auf der Homepage bekannt gegeben. Bei Fragen wenden Sie
sich bitte an Katrin Tent.

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.

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.

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.