Schedule for: 23w5145 - Neostability

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

It is well known that a differential field K of characteristic 0 is contained in a differential field which is differentially closed and has the property that it K-embeds in every differentially closed field containing K. Such a field is called a differential closure of K, and it is unique up to K-isomorphism: this is because the theory if omega-stable, so that prime models exist and are unique. One can ask the same question about difference fields: do they have a difference closure, and is it unique? The immediate answer to both these questions is no, for trivial reasons: in most cases, there are continuum many ways of extending an automorphism of a field to its algebraic closure. Therefore a natural requirement is to impose that the field K be algebraically closed. Similarly, if the subfield of K fixed by the automorphism is not pseudo-finite, then there are continuum many ways of extending it to a pseudo-finite field, so one needs to add the hypothesis that the fixed subfield of K is pseudo-finite. The main result is: Let K be an algebraically closed difference field of characteristic 0, kappa an uncountable cardinal. Assume that the fixed field of K is pseudo-finite and kappa-saturated. Then there is a kappa-prime model over K, and it is unique up to K-isomorphism. In characteristic p, no such result can hold. I will discuss some of the tools used in the proof. And maybe some applications and/or questions

Büchi automata are the natural extension of finite automata to a model of computation that accepts infinite-length inputs. We say a subset X of the reals is r-regular if there is a Büchi automaton that accepts (one of) the base-r representations of every element of X, and rejects the base-r representations of each element in its complement. We can analogously define r-regular subsets of higher arities, and these sets often exhibit fractal-like behavior--e.g., the Cantor set is 3-regular. In this talk, we will examine the interactions between automata theory, fractal geometry, and neostability. We will generalize the notion of sparsity for finite automata to Büchi automata, and demonstrate how this notion gives rise to a dividing line for expansions of the additive group of reals by closed r-regular sets both in terms of fractal dimension and NIP. This is joint work with Jason Bell.

Kim-independence is a notion of independence that corresponds to forking at a generic scale. There is a good theory of Kim-independence over models in NSOP_1 theories, though the situation over arbitrary sets remains mysterious. We will describe an axiomatic framework that gives the desiderata for a theory of independence over arbitrary sets and show how it lets us give precise meaning to questions about Kim-independence and, for example, its relationship to the existence axiom for non-forking. This work is joint with Itay Kaplan.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider dense pairs of o-minimal groups (that is, an o-minimal expansion of an ordered group with a predicate for a dense elementary substructure). Hieronymi and Nell showed that these pairs are not distal, which raises the question of whether they have a distal expansion. Nell answered this question positively in the case of pure ordered vector spaces, and Chernikov observed that Singer’s theory of closed ordered differential fields provides a distal expansion of dense pairs of real closed ordered fields. In joint work with Fornasiero, we give a positive answer for dense pairs of arbitrary o-minimal expansions of ordered fields.

A pervasive question in model theory is when desirable properties of some structure are preserved when expanding that structure by new definable sets. For example, over the last several years there has been an extensive study focused on preserving stability in expansions of the group of integers. An interesting phenomenon in this work is that all previously discovered examples of stable expansions of the group of integers are, in fact, superstable of U-rank omega. Thus it has remained an open problem to exhibit a strictly stable expansion of the group of integers. In this talk, I will present a solution to this problem, which is obtained in a much broader framework. Specifically, we will investigate preservation of model-theoretic tameness (e.g., stability, simplicity, NIP, NTP2, NSOP1) when expanding the induced structure on a definable stably embedded set by some arbitrary new structure. Joint work with C. d’Elbée, Y. Halevi, L. Jimenez, and S. Rideau-Kikuchi.

In recent work, V. Guingona and M. Parnes defined a notion of K-rank for certain classes of finite structures K. This notion encompasses some previously studied ranks (for example dp-rank) in some contexts. I will survey some of this recent work and the specific contributions of indivisibility (and the related property of definable self-similarity) to simplifying the study of K-ranks. The talk will end with a statement of recent results concerning the preservation of these properties under certain products of classes, and the relationship of these results to an open question. This is joint work with V. Guingona and M. Parnes.

I will discuss a definable version of the $(p,q)$ theorem of Jiří Matoušek from combinatorics for NIP theories, conjectured by Chernikov and Simon

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In a stable theory, the canonical base of a forking extension can be found in the algebraic closure of a Morley sequence of the type itself. The degree of nonminimality measures the length of such a sequence which witnesses that a given type is not minimal. In this talk, we will explain several variations and generalizations as well as the applications so far.

I shall discuss a refinement of the definable (p,q)-theorem, in the setting of distal geometric theories, that takes dimension into account, and consider possible generalisations. This is joint work with Charlotte Kestner and Tsinjo Rakotonarivo.

Let $p$, $q$ be two types in $\text{DCF}_0$, over the same set of parameters. Assume that they are non-orthogonal. General stability tells us that there are n,m such that $p^(n$) and $q^(m)$ are non-weakly orthogonal. In my talk, I will explain how to use semiminimal-analysis, Zilber's trichotomy for $\text{DCF}_0$ and Borovik-Cherlin for$\text{ ACF}_0$ to bound $n$ and $m$. This is joint with Freitag and Moosa.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The equivalence of $\text{NSOP}_1$ and $\text{NSOP}_3$, two model-theoretic complexity properties, remains open, and both the classes $\text{NSOP}_1$ and$\text{NSOP}_3$ are more complex than even the simple unstable theories. And yet, it turns out that classical geometric stability theory, in particular the group configuration theorem of Hrushovski (1992), is capable of controlling classification theory on either side of the $\text{NSOP}_1\text{-SOP}_3$ dichotomy, via the expansion of stable theories by generic predicates and equivalence relations. This allows us to construct new examples of strictly $\text{NSOP}_1$ theories. We introduce generic expansions corresponding, though universal axioms, to definable relations in the underlying theory, and discuss the existence of model companions for some of these expansions. In the case where the defining relation in the underlying theory $T$ is a ternary relation $R(x, y, z)$ coming from a surface in $3$-space, we give a surprising application of the group configuration theorem to classifying the corresponding generic expansion $T^R$. Namely, when $T$ is weakly minimal and eliminates the quantifier $\exists^∞$, $T^R$ is strictly $\text{NSOP}_4$ and $\text{TP}_2$ exactly when$R$ comes from the graph of a type-definable group operation; otherwise, depending on whether the expansion is by a generic predicate or a generic equivalence relation, it is simple or $\text{NSOP}_1$.

Given a finite subset A of size N in an ambient affine or projective space. A line with at least three distinct elements in A is called a triple line. There are at most O(N^2)-many triple lines, since a line is determined by two distinct points. And this upper bound can be achieved for certain families of finite sets of unbounded size on cubic curves. In this talk, I will discuss triple lines on smooth cubic surfaces over the complex numbers. We proved that for any finite subset A of a cubic surface X, if A has quadratically many triples lines and the size of A is large enough, then all triples lines are concentrated on a cubic curve given by the intersection of P and A for some plane P. This is a joint work with Martin Bays and Jan Dobrowolski.

We show that Keisler measures with infinite packing numbers do not admit any dfs extensions. This is joint work with Gabriel Conant and James Hanson.

I will discuss some recent developments on neostable positive theories (in the sense of Ben-Yaacov and Poizat), such as simple, $\text{NSOP}_1$ and NIP positive theories. In particular, I will discuss the main established examples (exponential fields, bilinear spaces over a fixed field, fields with a submodule, automorphisms of ordered abelian groups) and some possible future directions.

I will report to summarize the following 2 papers on ATP. -Jinhoo Ahn and Joonhee Kim, "SOP$_1$, SOP$_2$ and antichain tree property," (2020) submitted. -Jinhoo Ahn, Joonhee Kim, and Junguk Lee, "On the antichain tree property," to appear in J. of Math. Logic. ATP (implying SOP$_1$ and TP$_2$) is a dual notion of SOP$_2$. They show that a formula in some structure can have SOP$_1$ while its finite conjunctions do not have SOP$_2$, and any such a case the formula has ATP. (But S. Mutchnik recently shows that SOP$_1$=SOP$_2$ in the structure level.) $(N, \cdot)$ and atomless Boolean algebras are typical examples having ATP. They investigate further properties of ATP, e.g. ATP with tree-indiscernibility and ATP can be witnessed by a 1-variable formula. They also study algebraic (N)ATP structures. In particular ATP is preserved under Mekler's construction (so that there is a pure group having ATP); a PAC field F has ATP iff the Galois group of F (as a complete system) has ATP; and in the standard language, a henselian valued field of char. (0,0) has ATP iff so has its residue field.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

THIS TALK WILL BE PRESENTED ONLINE AND BROADCAST TO IN-PERSON AND VIRTUAL AUDIENCE. We show that there is a tight connection between NIP integral domains and NIP topological fields, allowing one to translate statements about NIP topological fields into statements about NIP integral domains. Let $K$ be an NIP expansion of field, and $\tau$ be a definable ring topology on $K$. Then $\tau$ is locally bounded, and if $K$ is sufficiently saturated, then $\tau$ is induced by an externally definable NIP subring $R \subseteq K$. We can then translate henselianity facts and conjectures about $R$ into statements about $\tau$. For example, if $K$ has positive characteristic or finite dp-rank, then $\tau$ is ``generalized t-henselian'' in the sense of Dittman et al, meaning that the implicit function theorem holds for polynomial equations. Conjecturally, this holds without the assumptions on characteristic or dp-rank. Similarly, other algebraic properties of $R$ imply facts about $\tau$. For example, a result of Simon shows that $R$ is a semilocal ring, which implies that $\tau$ is a field topology, not merely a ring topology.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss the emerging higher arity generalization of Shelah's classification theory, focusing on $n$-dependence, variants of $n$-stability, $n$-distality, and connections to higher amalgamation/stationarity and hypergraph regularity.

In this talk I will briefly go over measurable and generalised measurable structures, giving examples and non-examples. I will then go on to consider the two sorted structure $(V,F, \beta)$ where $V$ is an infinite dimensional vector space over $F$ an infinite field, and $\beta$ a bilinear form on this vector space. In particular I will consider the interaction of different notions of independence when this structure is pseudo finite. I will finish with some questions around generalised measurable structures.

I will start from the definition and several characterizations of stability of hyperdefinable sets. Then I will recall some results on maximal stable quotients of groups definable in NIP theories, including the main result of the paper "On maximal stable quotients of definable groups in NIP theories" by M. Haskel and A. Pillay which says that for a group $G$ definable in a NIP theory there exists a smallest type-definable subgroup $G^{st}$ with stable quotient $G/G^{st}$. In the rest of the talk, I will focus on my recent joint paper with A. Portillo, where we prove a counterpart of the last theorem in the context of types. Namely, for a NIP theory $T$, a sufficiently saturated model $C$ of $T$, and an invariant (over some small subset of $C$) type $p \in S(C)$, there exists a finest relatively type-definable over a small set of parameters from $C$ equivalence relation on the set of realizations of $p$ (in a bigger monster model) which has stable quotient. Our proof is via a non-trivial adaptation of the ideas from the proof of the aforementioned result of Haskel and Pillay, using relatively type-definable subsets of the group of automorphisms of the monster model (as defined in the paper ``On first order amenability'' by E. Hrushovski, A. Pillay, and myself) which I will try to explain.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Local topological dynamics is to topological dynamics just like local stability to stability. Given a definable group G in a model M we want to describe its definable topological dynamics, for example the Ellis group of S_{ext,G}(M). Usually this is hard. It may be easier to do this in a local setting, where instead of considering the algebra of externally definable subsets of G we consider some (d-closed) smaller subalgebras. In particular of special importance are strongly generic subsets of G and generic G-algebras of sets. In several local cases, for explicit G-algebras of sets we are able to describe their maximal generic subalgebras and their Ellis groups. These results are motivated by model-theory, however sometimes they have a broader relevance. For example we prove that if G is an infinite compact topological group, then there is a subset A of G that is strongly generic, but not periodic, and moreover this set A is nice topologically: it has Strong Baire Property (i.e. equals an open set modulo a nowhere dense set). We prove that for pre-compact topological group G, the algebra SBP(G) of subsets of G with strong Baire property is d-closed. This is relevamt to o-minimality, since we also prove that in the o-minimal setting, when G is a definable group, then every externally definable subset of G has the strong Baire property. This is a joint work with Adam Malinowski, a part of his Ph.D. thesis

"Skolem arithmetic" is the complete theory T of the natural numbers with multiplication. From a neostability perspective, T is highly untame, for example having both TP2 and SOP. Nevertheless the theory is quite tractable in some ways, and so provides a useful test case to see how stability-theoretic notions behave outside good dividing lines. I will discuss two theorems in this vein: a classification of the 0-definable stably embedded formulas of T, and a proof that T has weak elimination of imaginaries.

Every semigroup admits a universal (initial) homomorphism into a group, called the Grothendieck group. If the semigroup is an Ellis semigroup (such as in the case of the semigroup of finitely satisfiable types in a group), then this homomorphism is onto and factors through Ellis groups, and in particular, for Ellis groups of automorphism or definable group actions, it seems that it might be a new model-theoretic invariant, which is a quotient of the Ellis group, but is in general distinct.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Haskell, Hrushovski and Macpherson developed the theory of stable domination: a notion that captures when a structure is controlled by its stable part. A prime example is ACVF, where the stable part coincides with all the sets that are internal to the residue field (which is strongly minimal). In this talk we present residue field domination statements for henselian valued fields of equicharacteristic zero, which in essence captures the notion that the structure is controlled by the sorts internal to the residue field.

Analysis over a valued field works best when it is spherically complete, meaning the intersection of any (non-empty) chain of (valuative) balls is itself non-empty. While desirable, this is patently not a first-order, elementary property. Yet when working in a theory T expanding that of valued fields, while a typical model may not be spherically complete, it can be convenient to move to a elementary extension that is, provided one exists. Of course this is not always possible: if T is not definably spherically complete, there is certainly no spherically complete model. In this talk, I will present joint results with Immi Halupczok proving the existance of sperically complete elementary extensions for certain Hensel minimal expansions of valued fields and ask some related open questions. This will involve a quick exposition of the notions of Hensel minimality recently introduced by R. Cluckers, I. Halupczok, S. Rideau-Kikuchi in the equi-characteristic 0 setting, and those authors together with F. Vermeulen in mixed-characteristic.

We initiate a study of groups definable in 1-h-minimal fields (introduced by Cluckers, Halupczok and Rideau). We prove that (1) Definable groups can be endowed uniquely with a group topology coinciding generically with the affine topology. (2) The construction also equips the groups with a (weak) strictly differentiable manifold structure with respect to which the group is a strictly differentiable (weak) Lie group. We also classify definable fields and prove that 1-dimensional groups are finite-by-abelian-by-finite.

Zilber's Restricted Trichotomy Conjecture predicts that if $K$ is an algebraically closed field, and $\mathcal M=(M,...)$ is a non-locally modular strongly minimal structure interpreted in $K$, then $\mathcal M$ interprets $K$. A preprint of Hasson and Sustretov solves the conjecture when $M$ has dimension 1 over $K$ -- the key idea being the use of Bezout-like theorems to recover (approximately) the tangency relation on $\mathcal M$-definable plane curves through a distinguished point. This talk will outline a new strategy which circumvents the `Bezout' approach, instead recovering an approximation of the Zariski closure operation and then recovering tangency in terms of closure. A key feature of this approach is that it is independent of the dimension of $M$, so can also apply to higher dimensional universes. We will focus mainly on the easiest case, when $K=\mathbb C$; in this case the method yields a complete proof of the conjecture in characteristic zero, independent of the Hasson/Sustretov paper. We will also mention variants of the argument in other settings. Most notably, joint work with J. Ye has produced an analog in the `higher-dimensional' case over algebraically closed valued fields, which in particular is enough to prove Zilber's original conjecture in full generality.

A theory is noetherian if there is a family of definable sets with the descending chain condition such that every definable set is a boolean combination of those in the family. Noetherianity captures some of the desired properties of algebraically closed fields in any characteristic or differentially closed fields in characteristic 0. Noetherian theories are in particular omega-stable and equational. In recent work with M. Ziegler, we show that the theory of proper pairs of algebraically closed fields in any characteristic is noetherian.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We extend the notion of beautiful pairs by Poizat to unstable theories, with a specific interest in such pairs valued fields. In particular, we establish an analogue of Ax-Kochen-Ershov principles in for certain pairs of valued fields. In the specific case of ACVF, we classify all such pairs and deduce the strict pro-definability of various spaces of definable types such as the stable completion introduced by Hrushovski-Loeser and a model theoretic analogue of the Huber analytification of an algebraic variety. This is joint with Pablo Cubides Kovacsics and Martin Hils.

We will discuss some recent progress on the problem of classifying the reducts of the complex field (with named parameters and up to interdefinability). The tools we use include the recent solutions of the Restricted Trichotomy Conjecture in characteristic 0 and a generalized sumproduct result from additive combinatorics. (Joint with Benjamin Castle)

We show that for any $n\geq 3$ the theory of free generalized $n$-gons is complete, strictly stable and strictly $1$-ample yielding a new class of examples in the zoo of stable theories.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

[Joint work with Karim Adiprasito and Ehud Hrushovski] A globally valued field (GVF) is a field K equipped with a measure on the set of valuations (or absolute values) on K , which is *global*, namely, such that for every non-zero x : \int v(x) d \mu(v) = 0 Up to some obvious identifications, this is the same as a field equipped with a family of height functions on the projective spaces P^n(K) satisfying some obvious axioms. From a model-theoretic point of view, GVFs are structures in *unbounded* continuous logic, and as such form an elementary, inductive class. The class of GVFs includes the standard global fields, namely number fields and function fields of curves, as well as non-standard ones (e.g., any ultraproduct of the former). Even the most basic model-theoretic questions regarding GVFs seem quite complicated to answer. To date we still do not have a model companion (one is conjectured, though), which means that our understanding of quantification is limited. Also, for a very long time, we could not say even if the quantifier-free formulas were stable. My goal in this talk is to present the general context of GVFs, as well as some recent progress on the last question, whereby the main obstruction to quantifier-free stability has been removed.