Teoria modeli
Dane kontaktowe:
środa
14:15
16:00
HS
O seminarium
Terminy i tematyka spotkań
środa, 05-06-2024 - 16:15, HS
Bohr compactification and type-definable connected component of modules, rings and semidirect product of groups
Mateusz Rzepecki
For a model M and a topological space C we say that a map f: M -> C is definable if for any two disjoint closed sets in C their preimages by f are separable by a definable set.
For a definable structure N in a model M we say that f is a definable compactification of N if f is a compactification of N and f is a definable map.
We say that a definable compactification of N is universal if every definable compactification of N factors by f via a continuous map.
It turns out that if N is a group (Gismatullin, Penazzi & Pillay) or a ring (Gismatullin, Jagiella & Krupiński) then N/N^{00}_M is the universal definable compactification of N, where N^{00}_M is the type-definable connected component of N over M.
A module can be represented as N in two ways. The first one is a definable abelian group and a ring that is a part of the language. The second one is a definable abelian group and a definable ring.
This talk: In this talk we will prove that for a definable module N (in both senses) N/N^00_M is a universal definable compactification of N (the definition of N^00_M for a module will be given). We will also analyze how N^00_M depends on the type-definable connected component of the abelian group. To do this we will prove a theorem that shows how connected components help in creating sets that are closed under commutative addition (generalization of the proof by Krzysztof Krupiński for approximate rings). Using the theorem, we will describe the type-definable connected component of modules, rings and semidirect products of groups. We will also show that in many cases of structures analyzed during this talk adding homomorphism/monomorphism/automorphism/differentiation to the structure N does not change the type-definable connected component of N. During the talk we will come across a few open questions.
This is a joint work with Krzysztof Krupiński and is a part of a bigger project with Grzegorz Jagiella.
środa, 22-05-2024 - 16:15, HS
Maximal WAP and Tame quotients.
Adrián Portillo
Continuation of the previous topic.
środa, 15-05-2024 - 16:15, HS
Maximal WAP and Tame quotients.
Adrián Portillo
We study maximal WAP and tame (in the sense of topological dynamics) quotients of S_X(M), where M is a monster model of a complete theory T and X is a 0-type-definable set. Namely, let F_{WAP} be the finest closed, Aut(M)-invariant equivalence relation on S_X(M) with WAP quotient, and let F_{Tame} be the finest closed, Aut(M)-invariant, equivalence relation on S_X(M) with tame quotient. We show good behaviour of F_{WAP} and F_{Tame} under changing the monster model M. Namely, we prove that if M'≻M is a bigger monster model and F’_{WAP} and F’_{Tame} are the counterparts of F_{WAP} and F_{Tame} computed for M’, then the images of F’_{WAP} and F’_{Tame} under the restriction map r:S_X(M')→ S_X(M) coincide with F_{WAP} and F_{Tame}, respectively. Using these results, we show that the Ellis groups of the quotients of S_X(M) by F_{WAP} and F_{Tame} (as Aut(M)-flows) do not depend on the choice of the monster model M.
This is joint work with Krzysztof Krupiński
środa, 08-05-2024 - 16:15, HS
Inner ultrahomogeneous groups
Tomasz Rzepecki
Continuation of the previous topic
środa, 24-04-2024 - 16:15, HS
Inner ultrahomogeneous groups
Tomasz Rzepecki
We say that a group Γ is inner ultrahomogeneous if every isomorphism between finitely generated subgroups of Γ can be extended to an inner automorphism.
Examples of such Γ are the first three symmetric groups, all existentially closed locally finite groups and the universal locally recursively presentable group. Every group is contained in an inner ultrahomogeneous group.
It turns out that having finite exponent is very restrictive for such Γ (for example, it implies that there is no embedded copy of (Z/2Z)^6, and no abelian subgroups of order greater than 2^{100}), though it is not clear whether there are such finite exponent groups with more than 6 elements.
On the other hand, inner ultrahomogeneous groups of infinite exponent are necessarily very wild model-theoretically (for example, they all have the strict order property, the tree property of the second kind and n-IP for all n). They also have lots of countable abelian subgroups, and they tend to be uniformly simple (group-theoretically) and to have ample generic automorphisms (for countable Γ).
These observations strengthen and generalise previous observations about unstability and ample generic automorphisms of Hall's universal group by many authors.
In my talks, I will state the main results, prove some crucial lemmas about abelian subgroups of inner ultrahomogeneous groups, and apply them to prove some model-theoretic properties. Time permitting, I will also discuss some examples in depth and explain how to obtain ample generic automorphisms.
środa, 17-04-2024 - 16:15, HS
A few words on metric ultraproducts.
Jakub Gismatullin
My talk focuses on ultraproduct constructions, mainly in the context of
groups equipped with (invariant) metrics. I will present some new
constructions of simple and amenable groups based on metric
ultraproducts and discuss connections with open problems. I will also
pay attention to concepts from continuous logic related to the talk
(especially the compactness theorem in the metric setting). This is a
joint work with Krzysztof Majcher and Martin Ziegler.
środa, 10-04-2024 - 16:15, HS
Generalized indiscernibles and n-dependence in continuous logic
Adrián Portillo
Continuation of the previous topic.
środa, 03-04-2024 - 16:15, HS
Globally valued fields - foundations
Michał Szachniewicz
I will talk about foundations of globally valued fields, including continuous logic and a translation theorem between various structures describing a GVF. If time permits, I will talk about definability of intersection theory over a GVF - a joint work with Pablo Destic.
środa, 27-03-2024 - 16:15, HS
Fields without points in a Brody hyperbolic variety.
Michał Szachniewicz
I will talk about a joint project with Vincent Jinhe Ye, about model companion of the theory of fields that do not have rational points in some fixed variety. Previously, Will Johnson and Jinhe Ye proved that if the forbidden variety is a genus greater or equal two curve, then the model companion exists. We were able to extend this result for a higher dimensional Broody hyperbolic variety V. The resulting model companion is called VXF and many results from the previous known case of curves immediately generalise. I will talk about them and about some open questions that arose in the project.
środa, 20-03-2024 - 16:15, HS
Existentially closed valued difference fields and positive NTP2.
Jan Dobrowolski
In a work in progress with F. Gallinaro and R. Mennuni, we aim to understand the class of existentially closed valued difference fields (of equicharacteristic zero). I will give an overview of the project, focussing on the following two questions:
1. Is the above class NTP2 (in the sense of positive logic)?
2. Is the reduct of an e.c. valued difference field to a difference field a model of ACFA?
Motivated by the first question, we generalise several results about NTP2 to the positive setting, strengthening in particular the results of M. Kamsma obtained under the assumption of thickness (which we drop).
środa, 13-03-2024 - 16:15, HS
Affine Keisler Dynamics
Daniel Hoffmanm
It is an ongoing project with Kyle Gannon, but I am happy to present some results, hoping to get feedback and advice from the audience about the future developments. In the project, we are interested in passing from the action of Aut(M) on S(M) to an action of measures on Aut(M) on Keisler measures on S(M). Our method is by introducing to model theory a space of measures for which the convolution is well defined. This new space of measures is closely related to finitely additive probability measures on Aut(M), so to the paradoxical decomposition problem of Aut(M) and amenability of the theory. Anyway, the first goal is to provide a bridge between some Ellis semigroup and spaces of measures on Aut(M), which I plan to explain in more detail.
środa, 06-03-2024 - 16:15, HS
Affine Keisler Dynamics
Daniel Hoffmanm
It is an ongoing project with Kyle Gannon, but I am happy to present some results, hoping to get feedback and advice from the audience about the future developments. In the project, we are interested in passing from the action of Aut(M) on S(M) to an action of measures on Aut(M) on Keisler measures on S(M). Our method is by introducing to model theory a space of measures for which the convolution is well defined. This new space of measures is closely related to finitely additive probability measures on Aut(M), so to the paradoxical decomposition problem of Aut(M) and amenability of the theory. Anyway, the first goal is to provide a bridge between some Ellis semigroup and spaces of measures on Aut(M), which I plan to explain in more detail.
środa, 28-02-2024 - 16:15, HS
Generalized indiscernibles and n-dependence in continuous logic
Adrián Portillo
Generalized indiscernible sequences, first introduced by Shelah, and the modelling property, introduced by Scow, play an important role in model theory, specially in the characterization of dividing lines. We show that these notions and tools naturally extend to continuous logic. Moreover, the continuous modelling property is equivalent to having Embedding Ramsey Property (As in the classical case). We study and characterize n-dependence ($NIP_n$) for continuous logic and in hyperdefinable sets through the collapse of generalized indiscernible sequences. (As done by Chernikov, Palacín and Takeuchi for first order theories).
środa, 21-02-2024 - 14:15,
On compactifications of first-order structures
Grzegorz Jagiella
(work in progress, joint with K. Krupiński) I will talk about the natural generalization of classical notions of compactifications of various topological-algebraic structures, such as groups, rings or modules, to the appropriately defined category of first-order topological structures. I will present some categorical background and the previous work on this subject (mostly via universal algebra). I will show how compactifications of such first-order structures arise uniformly as quotients of their saturated extensions by certain type-definable bounded coungruences. The compactifications include the [definable] "universal" (or "Bohr") compactification of a structure, as well as its "[definable] profinite completion", obtained through the quotients by certain canonical model-theoretic "00" and "0" congruences. For e.g. pure groups, these congruences essentially coincide with the well-known 00- and 0- connected model-theoretic components. I will provide some examples of the construction with existing applications, and (if time permits) a further generalization to "approximate submodels".
piątek, 26-01-2024 - 14:15, HS
Model theoretic events
Kyle Gannon
How do we randomly sample an infinite sequence from a first order structure? What model theoretic properties might hold on almost all random sequences? Can we do probability theory in this setting in a meaningful way? This talk takes these questions seriously. Building on the body of work around Keisler measures, we construct a plausible framework to engage with probabilistic phenomena. This is joint work with James Hanson.
środa, 24-01-2024 - 14:15, HS
The class of Krasner Hyperfields is not elementary
Piotr Błaszkiewicz
In this talk we wish to prove that the class of Krasner (factor) Hyperfields is not elementary. This talk will be a continuation of my previous talk at this seminar, and will be based on the definitions and results from the paper "The hyperring of adèle classes" by Connes and Consani, I presented before. Some of the useful definitions are in the attachment to this email.
We will prove an algebraic claim using results from field theory (such as Chebotarev’s Density Theorem), which will imply that our class is not closed under elementary equivalence, hence it is not elementary.
This is joint work with Piotr Kowalski.
środa, 17-01-2024 - 14:15, HS
On relationships between some Ramsey properties
Grzegosz Jagiella
Continuation of the previous topic
środa, 10-01-2024 - 14:15, HS
On relationships between some Ramsey properties
Tomasz Rzepecki
Continuation of the previous topic
środa, 03-01-2024 - 14:15, HS
On relationships between some Ramsey properties
Tomasz Rzepecki
Continuation of the previous topic
środa, 20-12-2023 - 14:15, HS
On relationships between some Ramsey properties
Tomasz Rzepecki
Continuation of the previous topic
środa, 13-12-2023 - 14:15, HS
On relationships between some Ramsey properties
Krzysztof Krupiński
The classical notion of ERP (the embedding Ramsey property)
of a Fraisse structure (or the corresponding Fraisse class) translated
to the context of a first order theory $T$ yields the notion of EERP
(the elementary embedding Ramsey property) as defined in my joined
paper with J. Lee and S. Moconja. We also introduced more general
(i.e. weaker) properties DEERP [or EDEERP], where the colorings in
question are assumed to be definable [resp. externally definable],
which leads to important characterizations and consequences in terms
of suitable dynamical properties of $T$. Hrushovski does something
else: he does not assume any definability properties of the colorings,
but he replaces complete types by formulas which brings out the unity
in instances of structural Ramsey theory (which hopefully will be
discussed in future seminars).
I will show directly from the definitions that, assuming
$\aleph_0$-categoricity, Hrushovski's notion of $T$ being everywhere
Ramsey is equivalent to EERP. I will also explain it using a
well-known characterization of ERP in terms of generalized
indiscernibles. Moreover, while preparing this talk, I observed that
if the language is countable, then being everywhere Ramsey implies
$\aleph_0$-categoricity, which I will also present. As a corollary, we
get that if a countable theory $T$ is everywhere Ramsey, then it has
EERP, whereas the converse fails, as any countable
non-$\aleph_0$-categorical theory with EERP is not everywhere Ramsey.
If time permits, I will also prove equivalences between several
definitions of being Ramsey from Hrushovski's paper
środa, 29-11-2023 - 14:15, HS
Model completeness of rational points of algebraic groups
Piotr Kowalski
This is joint work with Daniel Max Hoffmann, Chieu-Minh Tran and Jinhe Ye. We show that if G is a simply connected semisimple group scheme over integers and K is a model complete field, then the pure group of rational points G(K) is model complete. I will introduce all these notions and present a proof of the model completeness of G(K).
środa, 08-11-2023 - 14:15, HS
Structural Ramsey theory and definability patterns
Tomasz Rzepecki
In the seminar, I will introduce the Ramsey property as defined by Hrushovski in "Definability patterns and their symmetries", and discuss how it relates to the Ramsey property for omega-categorical structures à la Kechris, Pestov and Todorčević.
Then I will describe the results related to this property from the above paper, including the canonical Ramsey expansion of a complete first order theory.
środa, 18-10-2023 - 14:15, HS
Variants of existential closedness for fields with operators
Jakub Gogolok
There are various interesting classes of "big" fields in algebra, such as (pseudo)algebraically closed, separably closed or large fields. In recent years, differential variants of those classes were introduced and studied, leading to very interesting results. In this talk we want to show how to treat all these notions in a uniform way, as variants of existential closedness in appropriate categories, for operators more general than derivations. Under certain natural assumptions we prove that some classes of "generalized existentially closed fields with operators" are elementary classes and give nice (uniform) geometric axioms for them. This unifies and generalizes many results from the literature, and also answers some open questions.
środa, 11-10-2023 - 14:15, HS
Compactifications of pseudo-finite and pseudo-amenable groups.
Anand Pillay
(Joint with G. Conant and E. Hrushovski).
We give simplified and correct accounts of results of Pillay
on compactifications of pseudo-finite groups.
We develop a suitable continuous logic framework for dealing with
definable homomorphisms from pseudo-amenable groups to compact Lie
groups. We obtain a uniform analogue of Bogolyubov's Lemma for sets
of positive measure in discrete amenable groups.
środa, 04-10-2023 - 14:15, HS
Measures on bounded PAC fields
Nick Ramsey
We will describe a construction for producing Keisler measures on
bounded perfect PAC fields. We will explain how the existence of
these measures entails that all groups definable in bounded perfect
PAC fields, and even in unbounded Frobenius fields, are definably
amenable. This was motivated by the now-solved question of whether
every group definable in a simple theory is definably amenable and
also out of a desire to better understand key examples for (the
currently non-existent) $NSOP_1$ group theory. This work builds on our
earlier constructions of measures for e-free PAC fields and a related
construction due to Will Johnson. This is joint work with Zoé
Chatzidakis.
środa, 14-06-2023 - 14:15, HS
Towards model theory of hyperfields
Piotr Błaszkiewicz
Hyperfields play a special role in the model theory of the henselian valued fields of mixed characteristic. They are used, in the form of so called $RV$-sorts, to provide numerous results, such as relative quantifier elimination.
These particular objects are well known and well understood by the model theorists following the subject, however there is no model theory developed for the general theory of hyperfields.
The goal of this talk is to look at the theory of hyperfields in general. We shall investigate some natural model theoretical questions one needs to ask when approaching this theory. We will focus on the question of the possibility of first order axiomatization of the class of Krasner factor hyperfields, and provide some partial results based on the work The hyperring of adèle classes of Alain Connes and Caterina Consani.
środa, 07-06-2023 - 14:15, HS
Measure doubling of small sets in SO(3,R)
Tran Chieu Minh
In a recent work, we show that if A is an open subset of
SO(3,R) with sufficiently small normalized Haar measure, then
\mu(A^2) >3.99 \mu(A).
Our result was conjectured by Breuillard and Green around 2010 in the
context of finding continuous counterparts of product theorems for
groups of Lie type by Helfgott, Pyber-Szabo, and Breuillard-Green-Tao.
In less precise forms, the question traces back to much earlier works
of Henstock and Macbeath in the 50s.
In this talk, I will discuss this result and its proof highlighting
the fact that ideas from neostable group theory serve both as actual
ingredients of the argument and as conceptual principles behind the
stage. (Based on joint work with Yifan Jing and Ruixiang Zhang)
środa, 31-05-2023 - 14:15, HS
On locally compact models of approximate groups and rings and their applications
Krzysztof Krupiński
By an approximate subring of a ring we mean an additively symmetric subset $X$ such that $X\cdot X \cup (X +X)$ is covered by finitely many additive translates of $X$. I will discuss the proof of the theorem which says that each approximate subring $X$ of a ring has a locally compact model, i.e. a ring homomorphism $f \colon \langle X \rangle \to S$ for some locally compact ring $S$ such that $f[X]$ is relatively compact in $S$ and there is a neighborhood $U$ of $0$ in $S$ with $f^{-1}[U] \subseteq 4X + X \cdot 4X$ (where $4X:=X+X+X+X$). This $S$ is obtained as the quotient of the ring $\langle X \rangle$ interpreted in a sufficiently saturated model by its type-definable ring connected component, and the main point is to prove that this component always exists. The theorem leads (and may lead to more) structural or even classification results on approximate subrings, and I will present several nice applications.
środa, 24-05-2023 - 14:15, HS
On locally compact models of approximate groups and rings and their applications
Krzysztof Krupiński
By an approximate subring of a ring we mean an additively symmetric subset $X$ such that $X\cdot X \cup (X +X)$ is covered by finitely many additive translates of $X$. I will discuss the proof of the theorem which says that each approximate subring $X$ of a ring has a locally compact model, i.e. a ring homomorphism $f \colon \langle X \rangle \to S$ for some locally compact ring $S$ such that $f[X]$ is relatively compact in $S$ and there is a neighborhood $U$ of $0$ in $S$ with $f^{-1}[U] \subseteq 4X + X \cdot 4X$ (where $4X:=X+X+X+X$). This $S$ is obtained as the quotient of the ring $\langle X \rangle$ interpreted in a sufficiently saturated model by its type-definable ring connected component, and the main point is to prove that this component always exists. The theorem leads (and may lead to more) structural or even classification results on approximate subrings, and I will present several nice applications.
środa, 10-05-2023 - 14:15, HS
On locally compact models of approximate groups and rings and their applications
Krzysztof Krupiński
In the first talk, I will give an overview of some fundamental issues concerning locally compact models of approximate subgroups, finishing with a discussion of approximate subrings.
The other two talks will be devoted to my results on approximate subrings.
By an approximate subring of a ring we mean an additively symmetric subset $X$ such that $X\cdot X \cup (X +X)$ is covered by finitely many additive translates of $X$. I will discuss the proof of the theorem which says that each approximate subring $X$ of a ring has a locally compact model, i.e. a ring homomorphism $f \colon \langle X \rangle \to S$ for some locally compact ring $S$ such that $f[X]$ is relatively compact in $S$ and there is a neighborhood $U$ of $0$ in $S$ with $f^{-1}[U] \subseteq 4X + X \cdot 4X$ (where $4X:=X+X+X+X$). This $S$ is obtained as the quotient of the ring $\langle X \rangle$ interpreted in a sufficiently saturated model by its type-definable ring connected component, and the main point is to prove that this component always exists. The theorem leads (and may lead to more) structural or even classification results on approximate subrings, and I will present several nice applications.
środa, 26-02-2020 - 16:15, 602
Amenability and definability
Krzysztof Krupiński (University of Wrocław)
The general motivation standing behind this research is to understand
relationships between dynamical and model-theoretic properties of
definable [topological] groups and between dynamical properties of groups
of automorphisms of first order structures and model-theoretic properties
of the underlying theories. More specifically, our goal is to understand
model-theoretic consequences of various notions of amenability.
Among the notions of amenability that we are interested in are: definable amenability of a definable group, classical amenability of a topological group, and, more generally, [weak] definable topological amenability of a definable topological group. We also introduce and study amenable theories.
The consequences of amenability that we obtain are the appropriate versions of G-compactness: for first order theories this is the equality of Lascar strong types and Kim-Pillay strong types; for definable [topological] groups this is the equality of suitably defined connected components $G^{000}$ and $G^{00}$ of the group $G$ in question.
Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some results about measures and measure-like functions.
My series of talks will be based on my preprint “Amenability and definability” joint with Ehud Hrushovski and Anand Pillay. In the first series of talks, I will focus on the context of definable [topological] groups; the second series will be devoted to our new notion of amenable theory.
Among the notions of amenability that we are interested in are: definable amenability of a definable group, classical amenability of a topological group, and, more generally, [weak] definable topological amenability of a definable topological group. We also introduce and study amenable theories.
The consequences of amenability that we obtain are the appropriate versions of G-compactness: for first order theories this is the equality of Lascar strong types and Kim-Pillay strong types; for definable [topological] groups this is the equality of suitably defined connected components $G^{000}$ and $G^{00}$ of the group $G$ in question.
Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some results about measures and measure-like functions.
My series of talks will be based on my preprint “Amenability and definability” joint with Ehud Hrushovski and Anand Pillay. In the first series of talks, I will focus on the context of definable [topological] groups; the second series will be devoted to our new notion of amenable theory.
środa, 22-01-2020 - 16:15, 602
Difference sheaves and torsors
Piotr Kowalski (University of Wrocław)
I will describe some results from the joint paper with Marcin Chałupnik "Difference sheaves and torsors" (available at https://arxiv.org/abs/1912.06886). This work is not about model theory, but it is partially motivated by model theory of difference fields.
We develop sheaf theory in the context of difference algebraic geometry. We introduce categories of difference sheaves and develop the appropriate cohomology theories. As specializations, we get difference Galois cohomology, difference Picard group and a good theory of difference torsors.
We develop sheaf theory in the context of difference algebraic geometry. We introduce categories of difference sheaves and develop the appropriate cohomology theories. As specializations, we get difference Galois cohomology, difference Picard group and a good theory of difference torsors.
środa, 08-01-2020 - 16:15, 602
Elementary equivalence theorem for PAC structures
Junguk Lee (University of Wrocław)
We introduce a notion of PAC structures, which generalizes perfect PAC fields and we provide an elementary equivalence theorem for PAC structures.
It is well known that elementary equivalences of PAC fields are controlled by their Galois groups: For given two PAC fields, if their Galois groups are isomorphic over Galois group over Galois group over a common subfield, then two PAC fields are elementary equivalent. The Key ingredient of this result is the Embedding Lemma for PAC fields, proved by M. Jarden and U. Kiehne.
We generalize Embedding Lemma for PAC fields into PAC structures and using our generalized Embedding Lemma, we deduce the elementary equivalence theorem for PAC structures. This is a joint work with J. Dobrowolski and D. M. Hoffmann.
It is well known that elementary equivalences of PAC fields are controlled by their Galois groups: For given two PAC fields, if their Galois groups are isomorphic over Galois group over Galois group over a common subfield, then two PAC fields are elementary equivalent. The Key ingredient of this result is the Embedding Lemma for PAC fields, proved by M. Jarden and U. Kiehne.
We generalize Embedding Lemma for PAC fields into PAC structures and using our generalized Embedding Lemma, we deduce the elementary equivalence theorem for PAC structures. This is a joint work with J. Dobrowolski and D. M. Hoffmann.
środa, 18-12-2019 - 16:15, 602
Residue field domination in certain Henselian valued fields
Clifton Ealy (Western Illinois University)
The notion of the domination of a type by its stable part was introduced in the work of Haskell, Hrushovski, and Macpherson. In joint work with Haskell and Marikova, we showed that the key ingredient was not so much stability but the residue field, and we obtained similar results for real closed valued fields (where the residue field is unstable). Recently, with Haskell and Simon, we have generalized these results to Henselian valued fields of equicharacteristic zero with bounded Galois group.
In this talk I will explain the idea of residue field domination, sketch some of the proofs, and use the results to at least partially describe forking in such fields relative to the value group and residue field.
In this talk I will explain the idea of residue field domination, sketch some of the proofs, and use the results to at least partially describe forking in such fields relative to the value group and residue field.
środa, 11-12-2019 - 16:15, 602
Generic derivations on o-minimal structures
Antongiulio Fornasiero (University of Florence)
Let $T$ be a complete, model complete o-minimal theory extending the theory $\mathrm{RCF}$ (in some language $L$). We study derivations $\delta$ on models $M \models T$. We introduce the notion of a $T$-derivation: a derivation which is compatible with the $L(\varnothing)$-definable $\mathcal{C}^1$-functions on $M$.
The theory of $T$-models with a $T$-derivation has a model completion $\mathrm{T^{\delta}_G}$. The derivation in models $(M, \delta) \models \mathrm{T^{\delta}_G}$ behaves “generically,” it is wildly discontinuous, and its kernel is a dense elementary $L$-substructure of $M$.
If $T = \mathrm{RCF}$, then $\mathrm{T^{\delta}_G}$ is the theory of closed ordered differential fields ($\mathrm{CODF}$) as introduced by Michael Singer. We are able to recover many of the known facts about $\mathrm{CODF}$ in our setting.
Among other things, $\mathrm{T^{\delta}_G}$ has $T$ as its open core, and $\mathrm{T^{\delta}_G}$ is distal.
We also examine the case of finitely many commuting $T$-derivations.
Joint work with Elliot Kaplan.
The theory of $T$-models with a $T$-derivation has a model completion $\mathrm{T^{\delta}_G}$. The derivation in models $(M, \delta) \models \mathrm{T^{\delta}_G}$ behaves “generically,” it is wildly discontinuous, and its kernel is a dense elementary $L$-substructure of $M$.
If $T = \mathrm{RCF}$, then $\mathrm{T^{\delta}_G}$ is the theory of closed ordered differential fields ($\mathrm{CODF}$) as introduced by Michael Singer. We are able to recover many of the known facts about $\mathrm{CODF}$ in our setting.
Among other things, $\mathrm{T^{\delta}_G}$ has $T$ as its open core, and $\mathrm{T^{\delta}_G}$ is distal.
We also examine the case of finitely many commuting $T$-derivations.
Joint work with Elliot Kaplan.
środa, 04-12-2019 - 16:15, 602
A Fraisse theorem for IB-homogeneous relational structures
Andrés Aranda (TU Dresden)
Fix a finite relational language $L$. An $L$-structure $M$ is called ultrahomogeneous if every isomorphism between finite induced substructures is restriction of an automorphism of $M$. Ultrahomogeneous $L$-structures have $\omega$-categorical theories with elimination of quantifiers and large automorphism groups with a finite number of orbits on tuples of any finite length $n$, so they are of interest in group theory and model theory.
The classical Fraisse theorem establishes a correspondence between ultrahomogeneous structures and classes of finite structures satisfying, in addition to the obvious restrictions, the joint embedding property and the amalgamation property.
In the early 2000s, the notion of homomorphism-homogeneity was introduced by Cameron and Nesetril, with further refinements by Lockett and Truss. In total, there are $18$ natural classes of homomorphism-homogeneous structures, but Fraisse theorems were not known for most of them until Coleman's work from last year, in which Fraisse theorems were identified and proved for $12$ of the $18$ classes. In this talk, I will present a Fraisse theorem for structures in which any isomorphism between finite substructures is restriction of a global bijective monomorphism.
The classical Fraisse theorem establishes a correspondence between ultrahomogeneous structures and classes of finite structures satisfying, in addition to the obvious restrictions, the joint embedding property and the amalgamation property.
In the early 2000s, the notion of homomorphism-homogeneity was introduced by Cameron and Nesetril, with further refinements by Lockett and Truss. In total, there are $18$ natural classes of homomorphism-homogeneous structures, but Fraisse theorems were not known for most of them until Coleman's work from last year, in which Fraisse theorems were identified and proved for $12$ of the $18$ classes. In this talk, I will present a Fraisse theorem for structures in which any isomorphism between finite substructures is restriction of a global bijective monomorphism.
środa, 27-11-2019 - 16:15, 602
Valuational weakly o-minimal structures with quantifier elimination
Jana Marikova (Western Illinois University)
Let $R$ be an o-minimal expansion of a group in a language in which $\mathrm{Th}(R)$ eliminates quantifiers, and let $C$ be a predicate realized as a valuational cut in $R$. We describe, in terms of symmetry of non-forking of certain Morley sequences, the class of such structures with quantifier elimination that are universally axiomatizable, after adding constants. This class properly contains the structures $(R,V)$, where $R$ is an o-minimal field and $V$ is a convex subring with o-minimal residue field. This is joint work with C. F. Ealy.
środa, 20-11-2019 - 16:15, 602
Ultraprodukty metryczne group - prostota i średniowalność (kontynuacja)
Jakub Gismatullin (University of Wrocław)
W trakcie referatu omówię grupy metryczne, tzn. grupy z niezmienniczymi metrykami oraz ultraprodukty takich grup. W dalszej części podam warunki prostotę i średniowalność ultraproduktów metrycznych grup oraz przykłady, np. pewne klasy grup liniowych nad nieskończonymi ciałami, grupy Higmana-Thompson oraz grupa IET.
środa, 13-11-2019 - 16:15, 602
Model thoery of Galois actions of torsion Abelian groups
Piotr Kowalski (University of Wrocław)
This is joint work with Ozlem Beyarslan. Let $A$ be a torsion Abelian group. We give an algebraic criterium for $A$ equivalent to companionability of the theory of fields with actions of $A$ by group automorphisms, and we show that the obtained theory is simple.
środa, 23-10-2019 - 16:15, 602
Ultraprodukty metryczne group - prostota i średniowalność
Jakub Gismatullin (University of Wrocław)
W trakcie referatu omówię grupy metryczne, tzn. grupy z niezmienniczymi metrykami oraz ultraprodukty takich grup. W dalszej części podam warunki prostotę i średniowalność ultraproduktów metrycznych grup oraz przykłady, np. pewne klasy grup liniowych nad nieskończonymi ciałami, grupy Higmana-Thompson oraz grupa IET.
środa, 09-10-2019 - 16:15, 602
Bohr compactifications of groups and rings
Grzegorz Jagiella (University of Wrocław)
(joint work with Jakub Gismatullin and Krzysztof Krupiński)
Definable topological dynamics shows that the classical Bohr compactification of a discrete group $G$ can be seen as a special case of "definable" Bohr compactifications. In turn, the definable Bohr compactification of a definable group can be described in terms of its model-theoretic components. The calculation of such components for some classical matrix groups, such as $UT_n(R)$ for a commutative, unital ring $R$, naturally leads to the development of ring analogues to the components of groups. In my talk, I will give the precise definitions of ring components and develop their preliminary theory. I will then describe the components of the groups $UT_n(R)$ and use them to give the precise description of their definable Bohr compactifications, including classical Bohr compactifications of the groups $UT_n(\mathbb{Z})$, e.g. the discrete Heisenberg group.
piątek, 04-10-2019 - 16:15, 603
The formalism of generalised operators and related model theoretic questions
Moshe Kamensky (Ben-Gurion University)
The formalism of generalised operators, due to Moosa and
Scanlon, is designed to deal uniformly with the algebraic and geometric
questions around fields with additive operators, such as derivations and
automorphisms, as well as their combinations.
My talk will consist of three parts: In the second part I will explain
my formulation of the formalism, where the such operators are presented
as generalised monoid actions. I will discuss some general
constructions, along with examples for particular cases. In the third
part, I will survey what is known about the model theory of fields with
such operators (mostly due to Moosa and Scanlon), and discuss some
elements of joint work with Beyerslan, Hoffman and Kowalski. In the
first part, I will survey some general facts about affine schemes that
might be required as background.
środa, 05-06-2019 - 16:15, 602
Strongly minimal compact Kähler threefolds
Jakub Gogolok
We will briefly recall basic results regarding the model theory of compact complex manifold, especially the Zilber trichotomy and some examples of strongly minimal manifolds. Next we will talk about a conjecture on strongly minimal Kähler manifolds. This conjecture implies, that odd-dimensional strongly minimal Kähler manifolds are tori. We will present the main ideas of the proof by Demailly et al. of the result saying that this holds in the case of dimension three. We will also say something about the higher-dimensional case.
środa, 03-04-2019 - 16:15, 602
Omega-categorical groups, rings, and, bilinear forms of finite burden.
Jan Dobrowolski (University of Wrocław)
After briefly introducing the context, I will discuss the main ideas of the proof of the result saying that omega-categorical groups of finite burden are virtually abelian-by-finite, generalizing a result of Kaplan-Levi-Simon. This is a joint work with F. Wagner.
środa, 27-03-2019 - 16:15, 602
Interpretability of Galois groups of first order structures
Junguk Lee (University of Wrocław)
Let $T$ be a complete theory with uniform EI and QE in a language $L$. Let $\mathfrak{C}$ be a monster model of $T$ and let $K$ be a small substructure of $\mathfrak{C}$. We show that the sorted complete system of Galois group of $K$ is uniformly interpretable in the $L_P$-structure $(M,K)$ for any elementary substructure of $\mathfrak{C}$ containing $K$, where the language $L_P$ is an expansion of $L$ by adding a new unary predicate $P$. Using this interpretability result, we give a description of types in PAC structures in terms of sorted complete systems. This is a joint work with Daniel M. Hoffmann.
środa, 20-03-2019 - 16:15, 602
Weak independence theorem for PAC structures
Daniel Hoffmann (University of Warsaw)
I would like to present several highlights from a current project with Junguk Lee. We are studying PAC substructures (e.g. existentially closed substructures) of some monster model of a stable theory and our goal is to describe such substructures with the Galois theory. It turns out that there is a possibility for defining independence relation in a saturated PAC substructure and this perspective independence relation descends from the absolute Galois group. Our results generalize a recent theorem of Zoe Chatzidakis.
środa, 06-03-2019 - 16:15, 602
On Simon's paper „Linear orders in NIP structures”.
Slavko Moconja (University of Wrocław)
This is the first one from several talks during which we plan to read the paper „Linear order in NIP structure” by Pierre Simon. The main result of this paper is that every unstable NIP theory admits a V-definable linear order, which partially answers a famous open question whether every unstable NIP theory admits a definable linear order.
środa, 27-02-2019 - 16:15, 602
Some properties of U-rank and m-rank
Urszula Dobrowolska (University of Wroclaw)
We make some observations about behaviour of U-rank when passing to reducts, and then we prove some analogous results for small profinite structures.
środa, 23-01-2019 - 16:15, 604
Companionability of torsion Abelian group actions on fields.
Piotr Kowalski (University of Wrocław)
This is joint work (in progress) with Ozlem Beyarslan. In our previous work (building on my joint work with D. Hoffmann), we showed that Galois actions of finitely generated virtually free groups are companionable. In this work, we deal with the case of infinitely generated torsion Abelian groups. So far, we have obtained partial results suggesting that companionability is closely related with the size of Prufer p-subgroups. I will present first a general model-theoretic background (chains of theories), and then proceed towards the examples of (non)companionable actions in this context.
środa, 09-01-2019 - 16:15, 604
On simple groups definable in valued fields
Jakub Gismatullin (University of Wrocław)
I will discuss my joint work with D. Macpherson, I. Halupczok on the following problem: given a henselian valued field of characteristic $0$, possibly equipped with analytic structure, describe the possibilities for a definable group $G$ in the valued field sort which is definably almost simple, that is, has no proper infinite definable normal subgroups. I will also explain our results for an algebraically closed valued field $K$ in characteristic $p$, but assuming also that the group is a definable subgroup of $\mathrm{GL}(n,K)$.
środa, 05-12-2018 - 16:15, 604
Definable sets in linearly ordered structures
Slavko Moconja
We will discuss the description of definable sets in some linearly ordered structures.
środa, 28-11-2018 - 16:15, 604
Some applications of model theory in computer science
Szymon Toruńczyk (University of Warsaw)
I will present a few basic applications of model theory in theoretical computer science, e.g. in verification, databases, and algorithms. I will also discuss some further perspectives on employing (ideas from) stability theory to solve algorithmic problems concerning graphs.
środa, 28-11-2018 - 16:15, 604
Some applications of model theory in computer science
Szymon Toruńczyk (University of Warsaw)
I will present a few basic applications of model theory in theoretical computer science, e.g. in verification, databases, and algorithms. I will also discuss some further perspectives on employing (ideas from) stability theory to solve algorithmic problems concerning graphs.
środa, 21-11-2018 - 16:15, 604
Definable actions, weakly almost periodic actions, and continuous logic stability (joint work with E. Hrushovski and K. Krupiński)
Anand Pillay (University of Notre Dame)
We show the connection between definability of an action of a definable group in a structure M on a compact space X, the notion of weakly almost periodic actions, and stability in continuous logic. As a consequence show that if M is sufficiently saturated and we are given a definable action of a group G definable in M on a compact space X, then X supports a G-invariant (Borel probability) measure. This yields to a negative answer to a question of Krupiński.
środa, 14-11-2018 - 16:15, 604
Finding polynomial orbits.
Tadeusz Pezda
For a ring $R$, and a polynomial $F\in R[X]$ we consider an
$n$-tuple $x_0,x_1,...,x_{n-1}$ of distinct elements of $R$ satisfying
$F(x_0)=x_1,F(x_1)=x_2,...,F(x_{n-1})=x_0$, and call it a " cycle " of
length $n$. We shall outline a way how to examine which $n$ may appear as
the length of a cycle, when $R=Z_K$, i.e. $R$ is the ring of integral
elements in a finite extension $K$ of $Q$ (and is the main object of study
in algebraic number theory).
środa, 07-11-2018 - 16:15, 604
Quantifier elimination of pure extensions of abelian groups
Martin Ziegler
In 3-sorted structures (A,B,C), where B is a pure extension of A and C the quotient B/A, we give a quantifier elimination which translates a formula phi(x,..) into a formula psi(r(x,..),..), where the r,.. are simple definable functions from B to A and C^eq and psi speaks only about A and C.
środa, 10-10-2018 - 16:15, 604
Model theory of fields with free operators in positive characteristic
Piotr Kowalski (Uniwersytet Wrocławski)
This is joint work with Özlem Beyarslan, Daniel Hoffmann and Moshe Kamensky, which is available here: https://arxiv.org/abs/1806.00464 . I will describe the set-up (introduced by Moosa and Scanlon) of "B-operators" on rings. This set-up includes derivations, Hasse-Schmidt derivations and endomorphisms. In the paper, we give algebraic conditions about a finite algebra B over a field of positive characteristic, which are equivalent to the companionability of the theory of fields with B-operators (i.e. the operators coming from homomorphisms into tensor products with B). We show that, in the most interesting case of a local B, these model companions admit quantifier elimination in the "smallest possible" language and they are strictly stable. We also describe the forking relation there.
środa, 03-10-2018 - 16:15, 604
Jordan permutation groups and reducts of trees
David Bradley-Williams (Universität Düsseldorf)
When a structure, M, is presented to us (in a language L), it is a
natural problem to try to describe the "reducts" of M, the (relational)
structures on the same domain as M which are definable in L, and to do
this up to first-order interdefinability. When M is omega-categorical
(and countable) this is equivalent to describing the permutation groups
H such that Aut(M) < H < Sym(M) that are closed in the natural Polish
topology on Sym(M). This characterisation allows us to use the theory of
infinite permutation groups to study reducts (and vice-versa).
There is a particularly interesting class of permutation groups called
Jordan groups with the property that if G is a Jordan group then any
closed permutation group containing G is also a Jordan group. This fact
has already been used to study reducts of combinatorial trees (by M.
Bodirsky and H.D. Macpherson) and affine/projective spaces (by I. Kaplan
and P. Simon).
We apply results from the structure theory of Jordan groups and
semilinear orderings to describe the reducts of ordered trees that are
sufficiently homogeneous and, in doing so, identify an infinite class of
maximally closed subgroups of Sym(omega).
środa, 06-06-2018 - 16:15, 604
On Galois groups and PAC substructures
Daniel Hoffmann
In the talk, I will present several results from my latest preprint. It concerns the question whether a profinite group can occur as a Galois group of some Galois extension inside a monster model of a previously chosen stable theory. Does assuming projectivity of our profinite group reflect somehow in the properties of the Galois group? Yes - it turns out that any projective profinite group is the absolute Galois group of some definably closed substructure and in some circumstances it is even the absolute Galois group of a pseudo-algebraically closed substructure.
środa, 30-05-2018 - 16:15, 604
Localization
Ludomir Newelski
Localization is an important idea of stability theory and model theory in general. I will speak of localization of definable topological dynamics.
środa, 23-05-2018 - 16:15, 604
Boundedness and absoluteness of some dynamical invariants (continuation)
Krzysztof Krupiński
This will be a continuation of Ludomir's talks about our joint paper with Pierre Simon. Ludomir presented the proofs of boundedness and absoluteness of Ellis groups of the flows of the form (Aut(C), SX(C)), where X is type-definable over the empty set and C is a monster model of a given theory. During my talks, I will focus on boundedness and absoluteness of minimal ideals of Ellis semigroups of such flows. I will also discuss some more specific results on minimal ideals and on Ellis groups in the NIP environment.
środa, 16-05-2018 - 16:15, 604
Applications of model theory to the study of Roelcke precompact groups and their actions
Todor Tsankov from Paris Diderot University
Roelcke precompact groups are exactly the topological groups that can
be realized as automorphism groups of omega-categorical structures (in
continuous logic). In this talk, I will discuss a model-theoretic
framework for the study of those groups and their dynamical systems as
well as two concrete applications. The talk is based on joint work
with Itaï Ben Yaacov and Tomás Ibarlucía.
środa, 09-05-2018 - 16:15, 604
Boundedness and absoluteness of some dynamical invariants
Krzysztof Krupinski
This will be a continuation of Ludomir's talks about our joint paper with Pierre Simon. Ludomir presented the proofs of boundedness and absoluteness of Ellis groups of the flows of the form (Aut(C), SX(C)), where X is type-definable over the empty set and C is a monster model of a given theory. During my talks, I will focus on boundedness and absoluteness of minimal ideals of Ellis semigroups of such flows. I will also discuss some more specific results on minimal ideals and on Ellis groups in the NIP environment.
środa, 11-04-2018 - 16:15, 604
Homology theory of types in model theory
Junguk Lee
J. Goodrick, B. Kim, and A. Kolesnikov introduced a notion of amenable
collection of functors to a proper category C to develop some simplicial
homology theory in a category. An amenable collection of functors satisfies
good extension and localization properties. A typical example comes from
model theory by considering the category of small subsets of a fixed
monster model with partial elementary embeddings.
Given an amenable collection, we define homology groups and under
(n+1)-complete amalgamation, the n-th homology group is just a set of
homology classes of n-chains of a certain simple form, called n-shell.
Specially, the first homology group is always given by homology classes of
1-shell.
By classifying all possible minimal 2-chains having 1-shell boundaries, we
can compute the first homology groups. In model theory, the first homology
group of a strong type of a model is the abelianization of Lascar group.
For a given abstract group G, we get an amenable collection of functors by
considering G-action on itself by left multiplication. In this case, the
first homology group is the abelianization of G.
środa, 21-03-2018 - 16:15, 604
Wymiar Hausdorffa przestrzeni metrycznych definiowalnych w o-minimalnych wzbogaceniach ciała liczb rzeczywistych (według J. Ma{\v r}{\'i}kovej i E. Walsberga)
Roman Wencel
Niech R będzie o-minimalnym wzbogaceniem ciała liczb rzeczywistych. Podczas seminarium zostanie omówione twierdzenie mówiące, że wymiar Hausdorffa dla R-definiowalnej rodziny przestrzeni metrycznych jest funkcją definiowalną przyjmującą wartości w ciele potęg.
W przypadku gdy R jest wielomianowo ograniczona, funkcja ta przyjmuje tylko skończenie wiele wartości.
środa, 14-03-2018 - 16:15, 604
Tame regularity theorems for groups with a distinguished subset
Anand Pillay
(This is joint with Gabriel Conant and Caroline Terry.)
Graph regularity theorems (i.e. Szemeredi) concern decomposing finite graphs
(V,W,R) into a small number of subgraphs (Vi,Wj,R|(V_i×W_j)) most of
which are "almost regular", i.e. subgraphs have approximately the same
density.
When more assumptions are made on the relation R such as uniform stability
or NIP one obtains stronger statements with almost homogeneity in place of
almost regularity.
In the group version we consider finite groups G equipped with a
distinguished subset A and assumptions are made on the relation xy A. One
seeks nice decompositions compatible with the group structure and this is
what I will talk about.
środa, 24-01-2018 - 16:15, 604
Difference modules and difference cohomology
Piotr Kowalski
This is joint work with Marcin Chałupnik. I will talk about our paper (available on https://arxiv.org/abs/1612.06960) which gives some basics about homological algebra of difference representations. We consider both the difference-discrete and the difference-rational case. We define the corresponding cohomology theories and show the existence of spectral sequences relating these cohomology theories with the standard ones. The main motivation of our work is to find a general difference counterpart of the generic cohomology of Cline, Parshall, Scott, and Van Der Kallen.
środa, 22-11-2017 - 16:15, 604
Stationary types in linear orders
Slavko Moconja
In the talk(s), the joint work with Predrag Tanović will be presented. We introduce the notion and do some investigation of stationary types in theories of linear orders. Most importantly, the relation of forking-dependence between realizations of stationary types is investigated, which turned out to be symmetric and transitive. We will show that the existence of some stationary types implies many countable models. Also, if all types over small sets (or just small models) are stationary, we will show that such theories are dp-minimal.
środa, 04-10-2017 - 16:15, 604
Virtually free groups and Galois actions
Piotr Kowalski (Uniwersytet Wrocławski)
I will talk about joint work with Özlem Beyarslan (the last version of our paper is available here: http://www.math.uni.wroc.pl/~pkowa/mojeprace/vfree4.pdf). We showed that for a finitely generated virtually free group G, the theory of actions of G on fields has a model companion, which we call G-TCF. We also gave an algebraic condition on G, which is equivalent to simplicity of the theory G-TCF. Recently, we learnt from Ehud Hrushovski an argument showing that if the group Z × Z embeds into G, then the theory of G-actions on fields does NOT have model companion. I will present this argument as well.
środa, 10-05-2017 - 16:15, 604
Teoria modeli a dynamika topologiczna i grupy polskie
Tomasz Rzepecki
Opowiem o związkach między pojęciami teoriomodelowymi (formuły i teorie stabilne, z NIP) a pojęciami z dynamiki topologicznej (funkcje słabo prawie okresowe, funkcje oswojone).
Opowiem też o twierdzeniu (z powstającej pracy z K. Krupińskim) o tym, że grupa Galois (Lascara) dowolnej teorii jest ilorazem zwartej grupy polskiej, i podobnie dla ilorazów grup typowo definiowalnych przez spójne składowe.
środa, 26-04-2017 - 16:15, 604
Some model theory of Galois actions
Piotr Kowalski
This is joint work with Özlem Beyarslan. For a fixed finitely generated group G, we consider actions of G by field automorphisms. If the theory of such generic actions is first-order axiomatizable, then we say that G-TCF exists. It is well-known that G-TCF exists if G is a free group (the theory ACFA_n), and it is also known that G-TCF exists for a finite G. On the other hand, it is also known that (Z^2)-TCF does not exist.
Using Bass-Serre theory, we give plausible axioms for G-TCF if G is virtually free. We show that in such a case, the theory G-TCF is simple if and only if G is free or finite.
piątek, 21-04-2017 - 14:15, 602
Dynamika topologiczna i grupy definiowalnie średniowalne
Grzegorz Jagiella
W odczycie pokażę pewne wyniki dotyczące dynamiki topologicznej w grupach NIP, których struktury dają się opisać w terminach podgrup oraz ilorazów będących przykładami grup definiowalnie średniowalnych. Wyniki te stanowią teoriomodelowe uogólnienie opisu dynamiki topologicznej w definiowalnych grupach Liego.
Następnie pokażę ich zastosowanie w opisie dynamiki arbitralnych grup definiowalnych w o-minimalnych rozszerzeniach ciał rzeczywiście domkniętych, dowodząc wariantu hipotezy dotyczącej izomorfizmu grup Ellisa. Wskażę też zastosowania dla pewnych klas grup definiowalnych w ciałach z waluacją, w szczególności w "rozszerzeniach Macintyre'a" ciała liczb p-adycznych.
środa, 12-04-2017 - 16:15, 604
Wprowadzenie do silnych typów
Tomasz Rzepecki
W pierwszej części seminarium prof. Ludomir Newelski dokończy swój odczyt "Model i jego podzbiór".
W drugiej części seminarium przypomnę podstawowe wiadomości na temat silnych typów i ich przestrzeni. Między innymi zdefiniuję silne typy Lascara oraz Kima-Pillaya, topologię logiczną na przestrzeniach silnych typów. Powiem też co rozumiemy przez moc borelowską silnego typu, a także o teoriomodelowych grupach Galois i składowych spójnych. Jeżeli wystarczy czasu, przypomnę też pokrótce istotne z tego punktu widzenia podstawowe pojęcia dynamiki topologicznej.
Planowany odczyt stanowi przypomnienie podstawowych pojęć przed właściwym odczytem, który odbędzie się w maju.
środa, 29-03-2017 - 16:15, 604
Model i jego podzbiór
Ludomir Newelski (Uniwersytet Wrocławski)
Załóżmy, że M jest strukturą przeliczalną, zaś Q jest jej
typowo-definiowalnym podzbiorem. Kiedy struktura M jest wyznaczona
jednoznacznie przez strukturę Q? W odczycie udzielimy odpowiedzi na to
pytanie. Udowodnimy także, że tego rodzaju relatywna kategoryczność jest
dość powszechnym zjawiskiem.
środa, 22-03-2017 - 16:15, 604
Własności grup topologicznych postaci L_0(G)
Aleksandra Kwiatkowska (Uniwersytet Wrocławski)
Dla grupy topologicznej G rozważamy grupę topologiczną L_0(G) składającą się z funkcji mierzalnych określonych na ([0,1],λ), gdzie λ jest miarą Lebesgue'a, o wartościach w G. Mnożenie jest punktowe, bierzemy metrykę zbieżności w mierze.
W czasie wykładu skoncentrujemy się na grupach G które są automorfizmami struktur przeliczalnych oraz skupimy się na pojęciach i własnościach związanych z klasami sprzężoności, takich jak topologiczne klasy podobieństwa i istnienie gęstej cyklicznej klasy sprzężoności.
Przedstawię wyniki otrzymane we wspólnej pracy z Maciejem Malickim.
środa, 01-03-2017 - 16:15, 604
Wymiar Hausdorffa przestrzeni metrycznych definiowalnych w o-minimalnych wzbogaceniach ciała liczb rzeczywistych (według J. Maříkovej i E. Weinberga)
Roman Wencel (Uniwersytet Wrocławski)
Niech R będzie o-minimalnym wzbogaceniem ciała liczb rzeczywistych. Podczas seminarium zostanie omówione twierdzenie mówiące, że wymiar Hausdorffa dla R-definiowalnej rodziny przestrzeni metrycznych jest definiowalną funkcją parametrów definiujących daną przestrzeń metryczną.
środa, 01-02-2017 - 16:15, 604
O średnicach teorio-modelowych składowych grup
Jakub Gismatullin
Opowiem o średnicach Lascara składowej G∞ w kilku konkretnych przypadkach, np. dla grup nilpotentnych, rozwiązalnych i innych (np. grup torsyjnych, skończenie generowanych). Podam odpowiedzi na pewne wcześniej zadane pytania. Część zaprezentowanych metod będzie operała się na tw. Petera-Weyla o reprezentacjach unitarnych.
środa, 18-01-2017 - 16:15, 604
Średniowalność, spójne składowe i G-zwartość
Krzysztof Krupiński (Uniwersytet Wrocławski)
W ciągu kilku najbliższych seminariów omówię moją najnowszą pracę wspólną z A. Pillayem. Rozwijamy w niej dynamikę topologiczną dla grup topologicznych definiowalnych w strukturach pierwszego rzędu. W szczególności wprowadzamy pewne nowe topologiczno-teoriomodelowe spójne składowe.
Dowodzimy, że średniowalność grupy w różnych kontekstach implikuje równość pewnych spójnych składowych (odpowiednich dla rozważanego kontekstu), w szczególności odpowiadając na pytanie z mojej wcześniejszej pracy z A. Pillayem i mojej pracy z J. Gismatullinem. Używając tego, uzyskujemy główny wynik pracy: jeśli grupa automorfizmów struktury ω-kategorycznej jest średniowalna (jako grupa topologiczna), to teoria tej struktury jest G-zwarta. Istotnym elementem dowodu, który jest ciekawy sam w sobie, jest przedstawienie grup Galois rozważanej teorii jako ilorazów grupy automorfizmów, zinterpretowanej w modelu monstrum pewnej bogatej struktury, przez odpowiednie spójne składowe przez nas wprowadzone.
środa, 30-11-2016 - 16:15, 604
Średniowalność, spójne składowe i G-zwartość
Krzysztof Krupiński
W ciągu kilku najbliższych seminariów omówię moją najnowszą pracę wspólną z A. Pillayem. Rozwijamy w niej dynamikę topologiczną dla grup topologicznych definiowalnych w strukturach pierwszego rzędu. W szczególności wprowadzamy pewne nowe topologiczno-teoriomodelowe spójne składowe.
Dowodzimy, że średniowalność grupy w różnych kontekstach implikuje równość pewnych spójnych składowych (odpowiednich dla rozważanego kontekstu), w szczególności odpowiadając na pytanie z mojej wcześniejszej pracy z A. Pillayem i mojej pracy z J. Gismatullinem. Używając tego, uzyskujemy główny wynik pracy: jeśli grupa automorfizmów struktury ω-kategorycznej jest średniowalna (jako grupa topologiczna), to teoria tej struktury jest G-zwarta. Istotnym elementem dowodu, który jest ciekawy sam w sobie, jest przedstawienie grup Galois rozważanej teorii jako ilorazów grupy automorfizmów, zinterpretowanej w modelu monstrum pewnej bogatej struktury, przez odpowiednie spójne składowe przez nas wprowadzone.
środa, 16-11-2016 - 16:15, 604
Lematy o regularności
Tomasz Rzepecki
Lemat Szemerédiego o regularności, w pewnym uproszczeniu, mówi że każdy dostatecznie duży graf można podzielić na pewną skończną liczbę części (niezależną od wielkości grafu) o prawie jednakowym rozmiarze, tak że prawie każda para części zachowuje się jak "graf losowy". Znajduje on zastosowanie w teorii grafów, teorii liczb oraz kombinatoryce ekstremalnej.
Wadą lematu jest to, że wielkość grafu konieczna do uzyskania pożądanej "losowości" rośnie bardzo szybko (szybciej niż każda funkcja wykładnicza). Okazuje się jednak, że gdy ograniczymy stopień skomplikowania grafu (np. gdy zażądamy, by był on definiowalny w ciele skończonym, lub rzeczywistym), można uzyskać dużo lepsze szacowania.
W czasie seminarium opowiem o niektórych wariantach i wnioskach z lematu Szemerédiego dla teorii stabilnych i NIP, wraz z powiązaną terminologią, oraz udowodnię wersję lematu dla teorii stabilnych w wersji udowodnionej przez Malliaris oraz Pillay'a.
środa, 26-10-2016 - 16:15, 604
Modele atomowe teorii nieprzeliczalnych
Ludomir Newelski
Pokażę niesprzecznosć z ZFC+\neg CH następujących twierdzeń:
- Istnieje teoria mocy \aleph_1 z jedynym modelem atomowym, który jednak nie jest konstruowalny
- Każda zupełna teoria mocy \aleph_1, która ma nieprzeliczalne modele atomowe, lecz nie ma modeli konstruowalnych, ma 2^{\aleph_1} modeli mocy \aleph_1.
Dowody używają kombinatoryki na \aleph_1 (słaby diament) oraz na prostej
rzeczywistej, odwołują się do metod Shelaha. Są to wyniki Douglasa Ulricha.
wtorek, 18-10-2016 - 16:15, 604?
Differential Galois theory and differential Galois cohomology.
Anand Pillay (University of Notre Dame)
We relate a differential field K having fundamental systems of
solutions for all linear differential equations, to K having trivial
differential Galois cohomology with respect to linear differential algebraic groups.
The model theory of groups of finite Morley rank and superstable groups,
plays a role.
środa, 12-10-2016 - 16:15, 604
Modelowy towarzysz działań Galois grup wirtualnie wolnych, cz. 2
Piotr Kowalski
Teoria ciał z automorfizmem ma modelowego towarzysza (aksjomatyzującego egzystencjalnie domknięte ciała z automorfizmem): teorię ACFA (Chatzidakis-Hrushovski, Macintyre). Innymi słowami, modelowy towarzysz istnieje dla działań Galois grupy Z. Wiadomo też, że modelowy towarzysz istnieje dla działań Galois skończenie generowanej grupy wolnej oraz że nie istnieje dla działań Galois grupy Z\times Z. Poza tym, modelowy towarzysz istnieje dla działań Galois grup skończonych (Sjorgen, Hoffmann-Kowalski) oraz działań Galois Q (Medvedev).
Podczas seminarium opiszę aksjomatyzację modelowego towarzysza działań Galois nieskończonej grupy dihedralnej oraz próby uogólnienia tej aksjomatyzacji do działań Galois grup wirtualnie cyklicznych i (ogólniej) grup wirtualnie wolnych. Jest to wspólna praca z Özlem Beyarslan.
środa, 05-10-2016 - 16:15, 604
Modelowy towarzysz działań Galois grup wirtualnie wolnych
Piotr Kowalski
Teoria ciał z automorfizmem ma modelowego towarzysza (aksjomatyzującego egzystencjalnie domknięte ciała z automorfizmem): teorię ACFA (Chatzidakis-Hrushovski, Macintyre). Innymi słowami, modelowy towarzysz istnieje dla działań Galois grupy Z. Wiadomo też, że modelowy towarzysz istnieje dla działań Galois skończenie generowanej grupy wolnej oraz że nie istnieje dla działań Galois grupy Z\times Z. Poza tym, modelowy towarzysz istnieje dla działań Galois grup skończonych (Sjorgen, Hoffmann-Kowalski) oraz działań Galois Q (Medvedev).
Podczas seminarium opiszę aksjomatyzację modelowego towarzysza działań Galois nieskończonej grupy dihedralnej oraz próby uogólnienia tej aksjomatyzacji do działań Galois grup wirtualnie cyklicznych i (ogólniej) grup wirtualnie wolnych. Jest to wspólna praca z Özlem Beyarslan.
środa, 21-09-2016 - 16:15, 602
An application of descriptive set theory to Kripke models
Shashi Srivastava (Indian Statistical Institute, Kolkata)
We show that logical equivalence, behavioral equivalence and
bisimilarity are equivalent for Kripke models. This is a joint work with E. E. Doberkat.
środa, 08-06-2016 - 16:15, 604
Rzeczywisty ślad grup definiowalnych w strukturach o-minimalnych
Grzegorz Jagiella (University of Haifa)
Klasyczne wyniki pokazują, że gdy G jest definiowalnie zwartą grupą
definiowalną w strukturze o-minimalnej, to G/G00 z topologią logiczną jest
zwartą grupą Liego o wymiarze równym o-minimalnemu wymiarowi G. Na
spotkaniu omówię uogólnienie tych wyników do dowolnych grup działających
wiernie i tranzytywnie na zbiorze definiowalnie zwartym.
W szczególności pokażę, że w tej sytuacji G ma ind-definiowalną podgrupę zawierającą
typowo definiowalną podgrupę, taką że ich iloraz \bar{G} jest grupą Liego
odpowiedniego wymiaru o następującej własności: dla każdego wiernego i
tranzytywnego działania G na definiowalnie zwartym zbiorze X, istnieje
rzeczywista, zwarta rozmaitość \bar{X} (uzyskana jako iloraz X) taka, że
działanie G na X redukuje się do indukowanego działania \bar{G} na
\bar{X}.
(Praca wspólna z K. Peterzilem.)
środa, 11-05-2016 - 16:15, 603
Definable sets and regular types in linear orderings
Slavko Moconja (University of Belgrade)
In the talk we present joint work with Dejan Ilić and Predrag Tanović. We
discuss complete theories of linear orderings with unary predicates and
convex equivalence relations (equivalence relations with convex classes). We
study two problems: the description of definable sets and the number of
non-isomorphic countable models.
We define the notion of strong linear binarity of linearly ordered
structures and their complete theories. We prove that any complete theory of
a linear ordering with unary predicates and convex equivalence relations is
strongly linearly binary, and also that every strongly linearly binary
structure is definitonally equivalent to a linear ordering with unary
predicates and convex equivalence relations. In the proof we give the
description of definable sets in linear orderings with unary predicates and
convex equivalence relations.
Further, we study regular types (in the sense of Pillay and Tanović) in
theories of linear orderings with unary predicates and convex equivalence
relations. We prove that every non-algebraic, global, invariant 1-type is
regular and that every non-algebraic complete 1-type over a small set A has
precisely two A-invariant global extensions. As an application we obtain
that these theories have either finitely many or continuum many
non-isomorphic countable models.
środa, 27-04-2016 - 16:15,
Existentially closed fields with finite group actions
Piotr Kowalski
(Joint with Daniel Hoffmann)
We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a (finite) group scheme action.
środa, 20-04-2016 - 13:30,
Grupy definiowalnie średniowalne w teoriach z NIP-em
Krzysztof Krupiński
Będzie to cykl wykładów poświęconych pracy „Definably amenable NIP groups” A. Chernikova i P. Simona. Omówione zostaną różne własności grup definiowalnie średniowalnych w teoriach z NIP-em. W szczególności:
- charakteryzacja definiowalnej średniowalności w terminach typów f-generik oraz ograniczonych orbit;
- charakteryzacja bycia f-generikiem w terminach formuł słabo generycznych oraz miar;
- opis miar ergodycznych przy użyciu typów f-generik;
- dowód hipotezy Newelskiego dla definiowalnie średniowalnych grup z NIP-em.
środa, 06-04-2016 - 16:15,
Relacje równoważności niezmiennicze na działania grup, cz. 2
Tomasz Rzepecki
Będę kontynuował opowiadanie o wynikach ze swojej ostatniej pracy, Equivalence relations invariant under group actions.
W pracy pokazałem, że dla pewnej szerokiej klasy relacji niemienniczych na ciągłe działanie grupy zwartej Hausdorffa, na typowo definiowalne działanie, lub na działanie grupy automorfizmów modelu monstrum, jeżeli wszystkie klasy są domknięte lub typowo definiowalne odpowiednio, to cała relacja jest domknięta lub typowo definiowalna odpowiednio.
Dzięki temu można uzyskać rozszerzenie wyników z mojej ostatniej pracy z K. Krupińskim i A. Pillayem o zastosowaniach dynamiki topologicznej do silnych typów w teorii modeli dotyczące związku między gładkością przestrzeni silnych typów a typową definiowalnością. Ponadto możemy wywnioskować analogiczne twierdzenia dla działań ciągłych grup zwartych (lub, szerzej, działań właściwych grup topologicznych Hausdorffa) na przestrzeniach polskich, oraz dla działań typowo definiowalnych (w miejsce działania grupy automorfizmów).
środa, 30-03-2016 - 18:45,
Around Hrushovski's stabilizer theorem and NTP_2
Pierre Simon (Université Claude Bernard - Lyon 1)
I will discuss some work in progress around Hrushovski's stabilizer theorem from the approximate subgroups paper and applications to NTP2. I will in particular discuss Hrushovski's theorem, present a simpler proof of it and some variations. I will then apply it to definably amenable groups in NTP2 theories.
Joint work with Alf Onshuus and Samaria Montenegro.
środa, 23-03-2016 - 16:15, 603
Relacje równoważności niezmiennicze na działania grup
Tomasz Rzepecki
Opowiem o wynikach ze swojej ostatniej pracy, Equivalence relations invariant under group actions.
W pracy pokazałem, że dla pewnej szerokiej klasy relacji niemienniczych na ciągłe działanie grupy zwartej Hausdorffa, na typowo definiowalne działanie, lub na działanie grupy automorfizmów modelu monstrum, jeżeli wszystkie klasy są domknięte lub typowo definiowalne odpowiednio, to cała relacja jest domknięta lub typowo definiowalna odpowiednio.
Dzięki temu można uzyskać rozszerzenie wyników z mojej ostatniej pracy z K. Krupińskim i A. Pillayem o zastosowaniach dynamiki topologicznej do silnych typów w teorii modeli dotyczące związku między gładkością przestrzeni silnych typów a typową definiowalnością. Ponadto możemy wywnioskować analogiczne twierdzenia dla działań ciągłych grup zwartych (lub, szerzej, działań właściwych grup topologicznych Hausdorffa) na przestrzeniach polskich, oraz dla działań typowo definiowalnych (w miejsce działania grupy automorfizmów).
środa, 09-03-2016 - 16:15, 603
Półgrupy stabilne, wg Yatira Halevi, cz. 3
Ludomir Newelski
Będą przedstawione wyniki dotyczące struktury półgrup (typowo) definiowalnych w strukturach stabilnych. Wyniki te polepszają wcześniejsze wyniki L. Newelskiego. W szczególności, zostanie udowodnione, że półgrupa typów grupy stabilnej jest granicą odwrotną półgrup definiowalnych.
środa, 02-03-2016 - 16:15, 603
Półgrupy stabilne, wg Yatira Halevi, cz. 2
Ludomir Newelski
Będą przedstawione wyniki dotyczące struktury półgrup (typowo) definiowalnych w strukturach stabilnych. Wyniki te polepszają wcześniejsze wyniki L. Newelskiego. W szczególności, zostanie udowodnione, że półgrupa typów grupy stabilnej jest granicą odwrotną półgrup definiowalnych.
środa, 24-02-2016 - 16:15, 603
Półgrupy stabilne, wg Yatira Halevi
Ludomir Newelski
Będą przedstawione wyniki dotyczące struktury półgrup (typowo)
definiowalnych w strukturach stabilnych. Wyniki te polepszają wcześniejsze
wyniki L. Newelskiego. W szczególności, zostanie udowodnione, że półgrupa
typów grupy stabilnej jest granicą odwrotną półgrup definiowalnych.
środa, 16-12-2015 - 16:15, 603
Dynamika topologiczna i złożoność silnych typów, cz. 4
Tomasz Rzepecki
W ciągu kilku najbliższych seminariów omówimy główne wyniki z pracy "Topological dynamics and the complexity of strong types" autorstwa K. Krupińskiego, A. Pillaya i T. Rzepeckiego. W teorii modeli istotną rolę odgrywają pewne grupy Galois danej teorii (głównie grupy Galois Shelaha, Kima-Pillaya oraz Lascara) oraz silne typy (głównie Shelaha, Kima-Pillaya i Lascara), które definiujemy jako klasy ograniczonych, niezmienniczych relacji równoważności rozdrabniających relację posiadania tego samego typu. O ile grupa Galois Kima-Pillaya może być naturalnie wyposażona w strukturę zwartej grupy topologicznej Hausdorffa, o tyle na grupie Lascara analogiczna topologia jest zwarta, ale niekoniecznie Hausdorffa. Podobnie dla przestrzeni silnych typów - iloraz przez ograniczoną relację typowo-definiowalną wyposażony w tzw. topologię logiczną jest zwartą przestrzenią Hausdorffa, natomiast na ilorazie przez ograniczoną, niezmienniczą relację równoważności topologia logiczna nie musi być Hausdorffa i może ona nawet być trywialna (a więc zupełnie bezużyteczna). Prowadzi to do bardzo ogólnego pytania: W jaki sposób traktować grupę Galois Lascara oraz przestrzenie silnych typów jako obiekty matematyczne i jak badać ich złożoność? Rozwijając dynamikę topologiczną dla grupy automorfizmów modelu monstrum, otrzymaliśmy głębokie związki między pewnymi pojęciami z dynamiki topologicznej oraz z teorii modeli. W szczególności przedstawiliśmy grupę Galois Lascara oraz przestrzenie silnych typów (na zbiorze realizacji jednego typu zupełnego nad zbiorem pustym) jako ilorazy pewnej zwartej grupy Hausdorffa. Używając tych wyników oraz wiedzy na temat zwartych grup topologicznych i deskryptywnej teorii mnogości uzyskaliśmy bardzo ogólne rezultaty na temat mocy [borelowskich] przestrzeni silnych typów, w szczególności odpowiedzieliśmy na pewne otwarte pytania. Końcowym wnioskiem jest trychotomia w pełni wyjaśniająca związki między fundamentalnymi własnościami (relatywną definiowalnością, typową definiowalnością, gładkością w sensie mocy borelowskich oraz liczbą klas) ograniczonych, niezmienniczych, analitycznych relacji równoważności.
środa, 08-04-2015 - 16:15, 603
Zbiory silnie generyczne, część 4
Ludomir Newelski
Zbiory silnie generyczne to nowe pojęcie wprowadzone przeze mnie w definiowalnej dynamice topologicznej. Może ono być przydatne również w zwykłej dynamice topologicznej. Przedstawię to pojęcie wraz z kilkoma zastosowaniami.
