**Seminario Flotante **

**de Lógica Matemática de Bogotá**

## Sesiones venideras

## Próxima sesión: 12 de mayo (2021) - Samaria Montenegro (San José)

12.05.21 - 4 pm

**Zoom ID: 910 560 7228 Access Code: Los-1955**

**Samaria Montenegro** - Universidad de Costa Rica

Title: **Groups definable in partial differential fields with an automorphism**

Abstract: In this talk we study groups definable in existentially closed partial differential fields of characteristic 0 with an automorphism which commutes with the derivations. In particular, we study Zariski dense definable subgroups of simple algebraic groups, and show an analogue of Phyllis Cassidy's result for partial differential fields.

This is a joint work with Ronald Bustamente Medina and Zoé Chatzidakis.

## 19 de mayo (2021) - Laura Gamboa (Bogotá)

19.05.21 - 4 pm

**Laura Gamboa** - Universidad de los Andes

Title: **TBA**

Abstract:

## 26 de mayo (2021) - Julián Cano (Bogotá)

05.05.21 - 4 pm

**Zoom ID: 910 560 7228 Access Code: Los-1955**** **

**Julián Cano** - Universidad Nacional de Colombia

Title: **Topological games in Ramsey spaces**

Abstract: Abstract Ramsey theory was originally proposed by Carlson and Simpson in 1990, and further developed by Todorcevic in 2010. Its purpose is to study a class of combinatorial topological spaces (called Ramsey topological spaces) that characterize and unify essential features appearing in those combinatorial frames where the Ramsey property is equivalent to Baire property, such as the Ellentuck space.

In this talk, we will present a general overview on the combinatorial structure of Ramsey topological spaces, analyzing their main features and studying some representative examples. Also, we will give a generalization of Kastanas' game in Ellentuck space, constructing topological games that characterize Baire property for a large family of Ramsey topological spaces.

This is joint work with Carlos Di Prisco.

[This talk will be given in spanish. Slides in english will be available]

## 28 de mayo - Juliette Kennedy (Helsinki)

28.05.21 - 9 am

**Juliette Kennedy **- Universidad de Helsinki

Title: **TBA**

Abstract:

## 09 de junio (2021) - Isabel Müller (Londres)

09.06.21 - 4 pm

**Isabel Müller **- Imperial College

Title: **TBA**

Abstract:

## Sesiones anteriores

## 28 de abril (2021) - Juan Felipe Carmona (Bogotá)

28.04.21 - 4 pm

Zoom ID: 910 560 7228 Access Code: Los-1955

**Juan Felipe Carmona** - Universidad Nacional de Colombia (Bogotá)

Title: **Definably amenable groups in continuous logic**

Abstract: In this talk, we show how to generalize the notion of definable amenability to definable groups in continuous structures. In analogy to the classical first-order case, we show that continuous stable groups and pseudo-compact groups are all definably amenable. Secondly, we give a characterization of this class of groups for dependent theories in terms of f-generic types. Finally, we define extreme amenability in this context and show that, under certain conditions, randomizations of classical definably amenable groups are extremely amenable.

This is joint work with Alf Onshuus.

Recording: here

## 23 de abril (2021) - Menachem Magidor (Jerusalén)

23.04.21 - 9 am

Zoom: 995 2698 4412. **Password**: least product of two consecutive odd numbers that is greater than 100.

**Menachem Magidor** - Universidad Hebrea de Jerusalén

Title: **Inner Models Constructed from Generalized Logics**

**Abstract:** The most well known inner model of Set Theory , the constructible universe L is constructed by stages , where at successor stage we take all definable subsets of the previous stage . Here "definable" means "definable by first order logic". The theory of L is well developed and there are relatively few set theoretical questions that can not be decided if one assumes V = L. On the other hand there are very canonical Set Theoretic objects which do not exist in L. (e.g. 0^#.)

Naturally one may ask : What inner model will result if we replace "first order definable" by a stronger notion of definability , like definability in a stronger logic L. For instance if "definable" is interpreted as "definable in second order logic", then by a classical result of Myhill and Scott, the resulting model is **HOD**-The inner models of the sets which are hereditarily ordinal definable.

The problem with **HOD** that it is very alterable: There is very little that one say about it which can not changed by forcing. On the other hand there are generalized logics which , which generalizes first order logic, but are weaker than full second order logic and such that the model constructed from them allows rich theory.

Examples : The logic in which one can express the fact that a linear order has cofinality ω , the logic in which one can express the fact that two sets have the same cardinality, the ”stationary logic” in which one can express the fact that there are stationarily many countable subsets of the universe satisfying some property.

This study touches on some basic issues of definability in Set Theory as well as issues about the relations between different generalized logics. In the talk we shall survey some of known results about such inner models and state some interesting open problems.

Most of the results are joint work with J. Kennedy and J. Väänänen

## 14 de abril - Nicolás Nájar (Bogotá) -[postponed]

14.04.21 - 4 pm (postponed)

**Nicolás Nájar** - Universidad Nacional de Colombia (Bogotá)

Title: **TBA**

Abstract:

## 9 de abril - Henry Towsner (Filadelfia)

09.04.21 - 9 am

Zoom: 910 560 7228. Password: Los-1955

Henry Towsner - University of Pennsylvania

Title: Higher-arity VC dimension and hypergraph regularity

Abstract: One interpretation of finite VC dimension is that a graph with finite VC dimension is one which can be approximated by rectangles - this is essentially the content of the version of Szemeredi regularity for finite VC graphs. Shelah introduced an analog of VC dimension for higher arity relations, called, of course, k-VC dimension. Jointly with Chernikov, we show that, with the right higher arity generalization of a rectangle - the k-ary cylinder intersection set - the hypergraphs with finite k-VC dimension are those which can be approximated by k-ary cylinder intersection set. Since many of these notions are somewhat complicated, this talk will focus on introducing the definitions and motivating the proof.

Recording: here

## 10 de marzo (2021) - Alex Kruckman (Middletown, Connecticut)

10.03.21 - 4 pm

Zoom: Link for the meeting

ID: 910 560 7228 Access Code: Los-1955

**Alex Kruckman** - Wesleyan University

Title: **A continuous zero-one law for finite metric spaces**

**Abstract:** The classical zero-one law for finite graphs says that for any sentence phi of first-order logic in the language of graphs, the probability that phi holds in a random finite graph of size n approaches 0 or 1 as n approaches infinity. Moreover, the almost-sure theory (the set of sentences with limiting probability 1) is exactly the theory of the random graph (the Fraïssé limit of the class of finite graphs). As a consequence, the theory of the random graph is pseudofinite. In joint work with Bradd Hart and Isaac Goldbring, building on enumerative results by Kozma, Meyerovitch, Peled, and Samotij, we show that the class of finite metric spaces has a zero-one law for sentences of continuous logic in the pure metric language, and we characterize the almost-sure theory. In contrast to the case of graphs, the almost-sure theory is not equal to the theory of the Urysohn space (the Fraïssé limit of the class of finite metric spaces) - in fact, every model of the almost-sure theory of finite metric spaces is topologically discrete. I will also discuss the question of pseudofiniteness for the continuous theory of the Urysohn space, which remains open.

Recording: here

## 03 de marzo (2021) - Joel (Ronnie) Nagloo (Nueva York)

03.03.21 - 4 pm

Zoom: 995 2698 4412. Password: least product of two consecutive odd numbers that is greater than 100

**Joel Nagloo** - City University of New York (CUNY)

Title: **Geometric triviality in differentially closed fields**

**Abstract:** In this talk we revisit the problem of describing the 'finer' structure of geometrically trivial strongly minimal sets in $DCF_0$. In particular, I will explain how recent work joint with Guy Casale and James Freitag on Fuchsian groups (discrete subgroup of $SL_2(\mathbb{R})$) and automorphic functions, has lead to intriguing questions around the $\omega$-categoricity conjecture of Daniel Lascar. This conjecture was disproved in its full generality by James Freitag and Tom Scanlon using the modular group $SL_2(\mathbb{Z})$ and its automorphic uniformizer (the $j$-function). I will explain how their counter-example fits into the larger context of arithmetic Fuchsian groups and has allowed us to 'propose' refinements to the original conjecture.

## 24 de febrero (2021) - H. Jerome Keisler (Madison)

24.02.21 - 4 pm

**H. Jerome Keisler** - University of Wisconsin-Madison

Title: **Using Ultraproducts to Compare Continuous Structures.**

Zoom: 995 2698 4412. Password: least product of two consecutive odd numbers that is greater than 100

**Abstract:** We revisit two research programs that were proposed in the 1960’s, remained largely dormant for five decades, and then become hot areas of research in the last decade.

The monograph “Continuous Model Theory” by Chang and Keisler, Annals of Mathematics Studies (1966), studied structures with truth values in [0,1], with formulas that had continuous functions as connectives, sup and inf as quantifiers, and equality. In 2008, Ben Yaacov, Bernstein, Henson, and Usvyatsev introduced the model theory of metric structures, where equality is replaced by a metric, and all functions and predicates are required to be uniformly continuous. This has led to an explosion of research with results that closely parallel first order model theory, with many applications to analysis. In my forthcoming paper “Model Theory for Real-valued Structures”, the ”Expansion Theorem” allows one to extend many model-theoretic results about metric structures to general [0,1]-valued structures–the structures in the 1966 monograph but without equality.

My paper “Ultrapowers Which are Not Saturated”, J. Symbolic Logic 32 (1967), 23-46, introduced a pre-ordering $M \trianglelefteq N$ on all first-order structures, that holds if every regular ultrafilter that saturates N saturates M, and suggested using it to classify structures. In the last decade, in a remarkable series of papers, Malliaris and Shelah showed that that pre-ordering gives a rich classification of simple first-order structures. Here, we lay the ground-work for using the analogous pre-ordering to classify [0,1]-valued and metric structures.

Recording: here.

## 17 de febrero (2021) - Thomas Scanlon (Berkeley)

17.02.21 -** 5 pm**** (different time than usual)**

Zoom: 995 2698 4412. Password: least product of two consecutive odd numbers that is greater than 100

**Thomas Scanlon - **University of California - Berkeley.

Title: **Questions about definability in, decidability of, and stability of the field of rational functions**

Abstract: The first-order theory of the field $\mathbb{C}(t)$ of rational functions in one variable over the complex numbers has been investigated since at least the 1950s, but many basic questions remain unresolved. For instance, it is not known whether this theory is decidable nor whether the theory is stable. In this lecture, I will describe how some apparently arithmetically complicated sets may be defined in $\mathbb{C}(t)$, but then how standing conjectures in diophantine geometry suggest that the structures obtained by enriching the field of complex numbers with predicates for these sets may still be decidable and stable. I will also describe some concrete instances of geometric problems whose solutions one way or the other bear upon the decidability of this theory.

## Próxima sesión: 12 de febrero (2021) - Javier de la Nuez González (Bilbao)

12.02.21 - 9 am

Zoom: 995 2698 4412. **Password**: least product of two consecutive odd numbers that is greater than 100.

**Javier de la Nuez González** - Universidad del País Vasco

Title: **Some model theory of the curve graph**

Abstract: The curve graph of a surface of finite type is a fundamental object in the study of its mapping class group both from the metric and the combinatorial point of view. I will discuss joint work with Valentina Disarlo and Thomas Koberda where we conduct a thorough study of curve graphs from the model theoretic point of view, with particular emphasis in the problem of interpretability between different curve graphs and other geometric complexes.

## 3 de febrero (2021) - Caroline Terry (Ohio)

03.02.21 - 4 pm

Zoom: 995 2698 4412**. **

**Caroline Terry** - Ohio State University

Title: **Speeds of hereditary properties and mutual algebricity**** **

Abstract: A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,...,n}. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete ``jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollobás, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss. This is joint work with Chris Laskowski.

Click here for a recording of the lecture.

## 27 de enero - Marcos Mazari-Armida (Pittsburgh)

27.01.21 - 4 pm

Zoom: 995 2698 4412.

**Marcos Mazari-Armida** - Carnegie Mellon University

Title: **Characterizing noetherian rings via superstability**

Abstract: We will show how superstability of certain classes of modules can be used to characterize noetherian rings. None of the classes of modules that we will consider are axiomatizable by a complete first-order theory and some of them are not even first-order axiomatizable, but they are all Abstract Elementary Classes. This new way of looking at classes of modules as AECs will be emphasized as I think it can have interesting applications. If time permits we will see how the ideas presented can be used to characterize other classical rings.

Click here for a recording.

# PAUSA (diciembre/enero)

## 9 de diciembre - David Valderrama (Bogotá)

09.12 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**David Valderrama** - Universidad Nacional de Colombia (Sede Bogotá)

Title: **Acerca**** de los (L,n)-modelos**

**Resumen:** El método de los (L,n)-modelos fue introducido por Shelah en [Shelah84] para ofrecer una demostración modelo-teórica del teorema de Paris-Harrington. En esta charla describiré las nociones básicas del método; sin embargo, presentaré una definición distinta de la original: además de clausurar por símbolos de función, clausuramos con los términos de una fórmula \phi fija. A la nueva definición la denominamos (\phi,n)-modelo, y a la colección de todos estos como la clase de los (L,n)-modelos. La razón del cambio se debe a cuestiones técnicas que se explicarán en la charla; no obstante, voy a mostrar que la teoría desarrollada en [Switzer19], con sus respectivas adecuaciones, se mantiene con la nueva definición. Por último, presentaré la demostración del teorema de Paris-Harrington usando los (L,n)-modelos con nuevas construcciones que complementan la prueba de Shelah.

**Bibliografía: **

[Switzer19] - C. Switzer, *Independence in Arithmetic: The Method of (L, n)-Models*, Mathematics ArXiv. arXiv: 1906.04273 [math.LO], 2019.

[Shelah84] - S. Shelah, *On logical sentences in PA*, Studies in Logic and the Foundations of Mathematics, 112, 12 1984.

Note: the talk will be given in Spanish, with slides in English. Here is a translation of the title and abstract.

**Title:** Around (L,n)-models

**Abstract:** Shelah developed the machinery of (L,n)-models in [Shelah84] to reprove the Paris-Harrington theorem and to give an example of a true but unprovable $\Pi_1^0$-sentence. We will review the basic notions of the method, but, we will present a definition differing from the original: in addition to closing by function symbols, we are closing with the terms of a fixed formula. We call the new definition (\phi, n)-model, and the collection of all these the class of (L, n)-models. The reason for the change is due to technical issues that will be explained in the talk; however, we are going to show that the theory developed in [Switzer19], with respective adjustments, remains with the new definition. Finally, we will review Shelah's alternative proof of the Paris-Harrington theorem with new constructions complementing Shelah's proof.

## 2 de diciembre - Sylvy Anscombe (París)

02.12 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**Sylvy Anscombe** - Université de Paris, IMJ-PRG.

Title: **Diophantine subsets and subfields of large fields **

Abstract:

The class of large fields includes plenty of familiar examples: algebraically closed, separably closed, real closed, p-adically closed, pseudo algebraically closed, pseudo real closed, etc. Many of these fields are well-understood model theoretically -- for example, by a quantifier-elimination. In his 2010 paper, Arno Fehm showed any infinite diophantine subset of a perfect large field is not contained in any proper subfield.

We will give a proof of the analogous result for the imperfect case, and explore the link with definability in henselian valued fields.

Click here to see a recording.

## 25 de noviembre - Rosario Mennuni (Münster)

25.11 - 4pm

Zoom: 959 7682 3238 (password: SemLogBog)

**Rosario Mennuni** - Westfälische Wilhelms-Universität Münster

Title: **The domination monoid**

Abstract:

This talk is concerned with the interaction between the semigroup of invariant types and the preorder of domination, i.e. small-type semi-isolation. In the superstable case, the induced quotient semigroup, which goes under the name of "domination monoid", parameterises "finitely generated saturated extensions of U" and how they can be amalgamated independently. In general, the situation is much wilder, and the domination monoid need not even be well-defined.

Nevertheless, this object has been used to formulate AKE-type results, can be computed in various natural examples, and there is heuristic evidence that well-definedness may hold under NIP. I will give an overview of the subject and present some results on these objects from my thesis.

Click here to see the video recording.

## 18 de noviembre - Alexander Berenstein (Bogotá)

18.11 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**Alexander Berenstein -** Universidad de los Andes

Title: **Expansions of vector spaces with a generic submodule **

Abstract:

(Joint work with C. d'Elbée and E. Vassiliev)

Let (V,+,0,...) be an expansion vector space over a field F such that the definable closure agrees with the vector span and assume the expansion eliminates the quantifier Exists infinity. Let R be a suring of F and consider the expansion of (V,+,0,...) with a predicate for a module over R. We study such expansions when the module is generic and show the generic expansions preserve several model theoretic tame properties like stability, NIP and NTP2.

Click here to see the video recording.

## 11 de noviembre - Sacha Post (Bogotá)

11.11 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**Sacha Post -** Universidad de los Andes

Title: **Lie groups and definability **

Abstract:

It is well known (Pillay, 1980) that a group G definable in an o-minimal expansion of the reals can be equipped with a Lie group structure; that is a topology making G a smooth variety such that the multiplication and inverse maps are smooth. It is then natural to ask whether the contrary is true, that is if any Lie group is actually definable in an o-minimal expansion of **ℝ** . We cannot expect this to be true in full generality since definable groups must have finitely many connected components but we still get some nice results for connected Lie groups. In 2016 A. Conversano, A. Onshuus and S. Starchenko gave a criterion for solvable Lie groups. After setting the general frame of work we will recall this solvable criterion. We will continue with a criterion for the case when the group is linear and we will also deal with non linear Lie groups that have “nice Levi decomposition”. If time allows it we will give a few words about recent work on the general case.

Click here to see the video!

## 6 de noviembre - Samson Abramsky (Oxford)

06.11 - 9 am

Zoom: 912 4399 3381 (password: SemLogBog)

**Samson Abramsky -** University of Oxford

Title: **The logic of contextuality**

Abstract (joint work with Rui Soares Barbosa): Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks.

We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work.

These contrast with traditional quantum logic a la Birkhoff--von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful.

We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches.

We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations.

This construction has a surprisingly broad range of uses.

We apply it in the study of a number of issues, including:

- establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras;

- formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen--Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics;

- investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations;

- developing some work towards a logical characterisation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.

**Click ****here**** to see the video!**

## 28 de octubre - Alf Onshuus (Bogotá)

28.10 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**Alf Onshuus -** Universidad de los Andes

Title: **Omega categorical dependent structures of ordinal th-rank. **

Abstract:

Classification problems in model theory (understanding and characterizing all theories in a fixed language that have certain properties) are a recurrent theme that has only been solved under very particular assumptions. Totally categorical structures can be classified by results of Lachlan and Hrushovski. Certain classes of pseudofinite structures were classified by Cherlin and Hrushovski in the book "Finite Structures with Few Types".

In this talk we will give a sketch of how these classifications were achieved and talk about the classification problem for omega categorical dependent super rosy theories. Some results we will talk about include:

Characterization of omega categorical th-rank one dependent structures.

Coordinatization for omega categorical dependent omega categorical structures of finite th-rank.

Every omega categorical dependent super rosy structure has finite th-rank.

A corollary of these results is that for any fixed countable language, there are only countably many finitely homogeneous relational super rosy dependent theories (modulo isomorphism).

All results are joint work with Pierre Simon. I will give definitions and intuitions for all the terms mentioned above.

Click here to see the recording.

## 23 de octubre - Amador Martín Pizarro (Friburgo)

23.10 - 9 am - OJO: día distinto

Zoom: 912 4399 3381 (password: SemLogBog)

**Amador Martín Pizarro - **Albert-Ludwigs-Universität Freiburg

Title: **Arithmetic progressions and complete amalgamation **

Abstract:

Hindman’s theorem states that, given a finite colouring of the natural numbers, there is an infinite monochromatic set such that all the finite sums of its elements enumerated in increasing order have again the same color. In particular, there is a monochromatic triangle (x,y, x+y). A related question is Roth’s theorem on arithmetic progression, which asks whether a subset $A$ of the natural numbers of positive (upper) density contains an arithmetic progression of length 3, that is, a tuple $(a, a+b, a+2b)$ in $A\times A\times A$. Finitary versions of Roth’s theorem study subsets $A$ of $\{0,\ldots, N\}$ whose density is greater than a fixed lower bound, and ask whether the same holds for sufficiently large $N$.

We will report on recent work with Daniel Palacin on how to prove Roth’s theorem in the context of pseudo-finite groups with the associated counting measure, using techniques from geometric model theory, and particularly, (a version of) complete amalgamation problems, resonating with the independence theorem in simple theories. In this talk, we will not discuss the technical aspects of the proof, but present the main ideas to a general audience with a familiarity in mathematical logic.

Recording: click here.

## 16 de octubre - Silvia Barbina (Cambridge)

16.10 - 9 am - OJO: día distinto

Zoom: **912 4399 3381** (password: SemLogBog)

**Silvia Barbina** - The Open University (UK)

Title: **The theory of the universal-homogeneous Steiner triple system**

Abstract: A Steiner triple system is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite Steiner triple systems has a Fraïssé limit, the countable homogeneous universal Steiner triple system M. In joint work with Enrique Casanovas, we have proved that the theory T of M has quantifier elimination, is not small, has TP2, NSOP1, eliminates hyperimaginaries and weakly eliminates imaginaries. In this talk I will review the construction of M, give an axiomatisation of T and prove some of its properties.

Click here to view the recording.

## semana del 5 de octubre - Receso del seminario

## 30 de septiembre - John Goodrick (Bogotá)

30.09 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**John Goodrick** - Universidad de los Andes

Title: **The model theory of ordered Abelian groups **

Abstract: Ordered Abelian groups, considered as structures in the language {+, <}, have long been studied by logicians and are the subject of much recent study in model theory. In terms of the classic Shelahian dichotomies, they are always unstable but also always NIP, by results of Gurevich and Schmitt from the 1960's. In fact, they even proved a quantifier elimination result, which Cluckers and Halupczok reproved and reformulated as a kind of "quantifier elimination relative to colored ordered sets."

Within NIP theories in general, there are finer dichotomies, such as the distinction between strong NIP and non-strong NIP theories, and the dichotomy (within strong NIP theories) of finite dp-rank theories versus those without finite dp-rank. These dichotomies give a useful measure of the complexity of the theory of an ordered Abelian group. For example, Jahnke-Simon-Walsberg in 2015 showed that an ordered Abelian group (G, +, <) has dp-rank 1 just in case it has no "singular primes" (a prime p such that G/pG is infinite). This was recently generalized to give a slightly more complicated characterization of those ordered Abelian groups whose theories are of finite dp-rank (independently found by Halevi-Hasson, Dolich-Goodrick, and Farré): they are those which have finitely many singular primes and also satisfy an additional condition on definable convex subgroups. We will explain this and talk about connections with Johnson's recent proof of the Shelah Conjecture for finite dp-rank fields.

Click here to see the recording.

## 23 de septiembre - André Platzer (Pittsburgh)

23.09 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**André Platzer** - Carnegie Mellon University

Title: **Logical Foundations of Cyber-Physical Systems**

Abstract: Logical foundations of cyber-physical systems (CPS) study systems that combine cyber aspects such as communication and computer control with physical aspects such as movement in space. CPS applications abound. Ensuring their correct functioning, however, is a serious challenge. Scientists and engineers need analytic tools to understand and predict the behavior of their systems. That's the key to designing smart and reliable control.

This talk identifies a mathematical model for CPS called multi-dynamical systems, i.e. systems characterized by combining multiple facets of dynamical systems, including discrete and continuous dynamics, but also uncertainty resolved by nondeterministic, stochastic, and adversarial dynamics. Multi-dynamical systems help us understand CPSs better, as being composed of multiple dynamical aspects, each of which is simpler than the full system. The family of differential dynamic logics surveyed in this talk exploits this compositionality in order to tame the complexity of CPS and enable their analysis.

In addition to providing a strong theoretical foundation for CPS, differential dynamic logics have also been instrumental in verifying many applications, including the Airborne Collision Avoidance System ACAS X, the European Train Control System ETCS, several automotive systems, mobile robot navigation with the dynamic window algorithm, and a surgical robotic system for skull-base surgery. The approach is implemented in the theorem prover KeYmaera X.

Click here for a recording of the lecture.

## 16 de septiembre - Artem Chernikov (Los Ángeles)

16.09 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**Artem Chernikov** - University of California at Los Angeles - UCLA

Title: **Incidence counting and trichotomy in o-minimal structures**

**Abstract: **Zarankiewicz’s problem in graph theory asks to determine the largest possible number of edges |E| in a bipartite graph G = (E, V_1, V_2) with the parts V_1 and V_2 containing m and n vertices, respectively, and such that G contains no complete bipartite subgraphs on k vertices. Graphs definable in o-minimal (or more generally distal structures) enjoy stronger bounds than general graphs, providing an abstract setting for the Szemerédi-Trotter theorem and related incidence bounds. We obtain almost optimal upper and lower bounds for hypergraphs definable in locally modular o-minimal structures, along with some applications to incidence counting (e.g. the number of incidences between points and boxes with axis parallel sides on the plane whose incidence graph is K_{k,k}-free is almost linear). We explain how the exponent appearing in these bounds is tightly connected to the trichotomy principle in o-minimal structures.

Joint work with Abdul Basit, Sergei Starchenko, Terence Tao and Chieu-Minh Tran.

Recording here.

## 11 de septiembre - Matteo Viale (Turín)

11.09 - 9 am --- OJO (note different day and time!)

Zoom: 912 4399 3381 (password: SemLogBog)

**Matteo Viale** - Università di Torino

Title: **Tameness for set theory**

Abstract: We show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship.

Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a -property formalized in an appropriate language for second or third order number theory is forcible from some T extending ZFC + large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T.

Part (but not all) of our results are conditional to the proof of Schindler and Asperò that Woodin’s axiom (*) can be forced by a stationary set preserving forcing.

Recording here.

## 2 de septiembre - John Baldwin (Chicago)

02.09 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**John Baldwin** - University of Illinois at Chicago

Title: **On Strongly Minimal Steiner Systems: Zilber’s Conjecture, Universal Algebra, and Combinatorics**

Abstract: A k-Steiner system is a collection of points and lines such that each line has k points. We discuss the 170 year history of this topic and its connection with quasigroups/Latin squares. Theorem [BP20]. For every k there is a strongly minimal k-Steiner system (M, R) (R is collinearity). Classical result. If k is a prime power, there is a coordinatization of (M, R) by a quasigroup.

**Theorem** (Baldwin)

If k is a prime power there is a strongly minimal quasigroup (M, R, ∗) inducing a k-steiner system.

**Theorem** ([BV20]). In ‘most’ cases no such quasigroup is definable in (M, R).

More generally, with a slight qualification, the Steiner systems and Hrushovski’s original example a) do not admit an ∅-definable binary function b) do not admit elimination of imaginaries. The proof involves a detail analysis of finite subsets (closed under the action of certain groups) of algebraic closure of a finite sets. The ‘most’ depends upon which group is most appropriate. This analysis suggests a refinement of the Zilber trichotomy to distinguish among the diverse examples of non-trivial, non-locally modular, non-field like theories. Connections with combinatorial issues [CW12] will appear depending on time.

**References**

[BP20] John T. Baldwin and G. Paolini. Strongly Minimal Steiner Systems I. Journal of Symbolic Logic, 2020? arXiv:1903.03541.

[BV20] John T. Baldwin and V. Verbovskiy. Dcl in Hrushovski Constructions. in preparation, 2020.

[CW12] P. J. Cameron and B. S. Webb. Perfect countably infinite Steiner triple systems. Australas. J. Combin., 54:273-278, 2012

Click here for a recording of the talk.

## 26 de agosto - Daniel Calderón (Toronto)

26.08 - 4 pm

Zoom: 959 7682 3238 (password: SemLogBog)

**Daniel Calderón **- University of Toronto

Title: **Nullity notions on the real line**

Abstract: Borel conjecture asserts that all strong measure zero subsets of the real line are countable. The interest of this problem is two-fold: in one hand, it gives a connection between abstract set theory and problems in analysis and on the other hand the proof of its consistency, due to Laver, contains the first use of countable support iterated forcing (this will produce such deep developments as the Proper Forcing Axiom).

Strong measure zero subsets of the reals can be characterized in various ways: algebraically (Galvin--Mycielski--Solovay), through selection principles, topological games, and Ramsey-theoretic methods (Scheepers), and by the mean of tools coming from geometric measure theory (Besicovitch and Zindulka). This motivates a systematic study of \emph{nullity notions on the real line}; a hierarchy of subsets of the reals whose measure-theoretic nature lies in between being countable and being a Lebesgue-null set.

# PAUSA (julio/agosto)

## 1° de julio - Juan Ignacio Agudelo (Bogotá)

01.07 - 4 pm

Zoom: https://zoom.us/j/148975473

**Juan Ignacio Agudelo** - Universidad Nacional de Colombia (Bogotá)

Title: **On the space of stably dominated types of ACVF**

Abstract: I will describe some model-theoretic ideas around the work of Hrushovski and Loeser on ACVF, with emphasis on the pro-definable structure and its connections to non-archimedean geometry.

## 24 de junio - Andrés Felipe Uribe (Medellín)

24.06 - 4 pm

Zoom: https://zoom.us/j/148975473

**Andrés Felipe Uribe **- Universidad Nacional de Colombia (Medellín)

Título: **Un problema de independencia en topología general: la conjetura del espacio normal de Moore, caso separable **(slides in English, talk in Spanish)

Resumen: La conjetura del espacio normal de Moore es un problema que se refiere a la metrización de espacios topológicos, planteado por F. B. Jones en 1937 y que se convirtió en uno de los problemas más importantes de la historia de la topología general. El propósito de la charla es explorar algunas relaciones entre teoría de conjuntos y la topología general, presentando las implicaciones que tienen tanto el Axioma de constructibilidad como el Axioma de Martin en los espacios normales de Moore y establecer qué papel juegan en la independencia del caso separable de la conjetura en cuestión.

Bibliografía:

-Jones, F. B. (1937). Concerning normal and completely normal spaces. *Bulletin American Mathematical Society *47, p. 671-677.

-Tall, F. D. (1969). Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problem, University of Wisconsin, Madison, (PhD).

-Parra-Londoño, Carlos M. & Uribe-Zapata, Andrés F. (2020). La independencia de una versión débil de la conjetura del espacio normal de Moore. *Rev. Integr. Temas Mat. *38, Nr. 1, p. 43-54.

## 17 de junio - Paulo Soto (Bogotá)

17.06 - 4 pm

Zoom: https://zoom.us/j/148975473

**Paulo Soto **- Universidad de los Andes (Bogotá)

Título: **Pseudofinitud y pseudocompacidad en lógica continua** (slides in English, talk in Spanish)

Resumen: La noción de pseudofinitud en lógica de primer orden ha probado ser una herramienta útil e interesante en las últimas décadas, con aplicaciones importantes en combinatoria y teoría de grafos. El propósito de la charla es definir el paralelo adecuado de la noción de pseudofinitud en lógica continua, explorar algunas nociones equivalentes y exponer un resultado sobre leyes 0-1 para los espacios métricos finitos, en respuesta parcial a la pregunta sobre la pseudofinitud de la esfera de Urysohn.

Bibliografía:

Ben Yaacov, I. (2015). Fraïssé limits of metric structures.

*J. Symb. Log., 80*(1), 100–115.Goldbring, I., & Hart, B. (2019). The almost sure theory of finite metric spaces

*arXiv: Logic*.Goldbring, I., & Lopes, V. (2015). Pseudofinite and pseudocompact metric structures

*Notre Dame J. Form. Log., 56*(3), 493–510.Usvyatsov, A. (2008). Generic separable metric structures

*Topology Appl., 155*(14), 1607–1617.

## 10 de junio - Andrés Villaveces (Bogotá)

10.06 at 4 pm

Zoom: https://zoom.us/j/148975473

**Andrés Villaveces **- Universidad Nacional de Colombia (Bogotá)

Title: **On the interplay between Abstract Elementary Classes and Categorical Logic**

Abstract: I will describe two recent lines of interplay between Abstract Elementary Classes and Categorical Logic: the problem of building the "Galois group" of an AEC (building on Lascar and Poizat's work on the "Galois theory of model theory", and on the role of the Small Index Property - joint work of mine with Ghadernezhad) and interpreting $\lambda$-categoricity in terms of properties of classifying topoi (recent work of Espíndola, connected to his ground-breaking work on Shelah's eventual categoricity conjecture). My talk will stress the way these connections appear and the opening of new lines of possibility.

**There was no recording of this session, due to a zoom connection problem.**

## 3 de junio - Rami Grossberg (Pittsburgh)

03.06 - 4 pm

Zoom: https://zoom.us/j/148975473

**Rami Grossberg** - Carnegie Mellon University (Pittsburgh)

Title: **On local & global questions in the theory of Abstract Elementary Classes**

Abstract: In the last 20 years (and more so in the last 10 years), Classification Theory for AECs (Abstract Elementary Classes) witnessed exponential growth, with spectacular results and also leading to a good theory generalizing first-order forking and various independence relations. The driving force was a combination of global questions (like Shelah's categoricity conjecture) and a local question about what properties of models in a fixed cardinality in a given AEC would imply existence of a model in its successor. I will describe some of the questions, results and the interplay between them.

Recording: click here.

## 27 de mayo - Carlos Di Prisco (Caracas / Bogotá)

27.05 - 4 pm

**Carlos Di Prisco **- IVIC (Caracas) y Universidad de los Andes (Bogotá)

Zoom: https://zoom.us/j/148975473

Title: **Ideals and maximal almost disjoint families**

Abstract: We will present several results about families of infinite sets of natural numbers that are almost disjoint. In particular, several recent results concerning the existence of definable maximal almost disjoint families. Almost disjoint families generate ideals of sets that have interesting properties: the complement of such an ideal is a selective coideal. We also present some results about selective and semiselective coideals and forcing notions related to them. In the generic extension of the universe obtained by collapsing a Mahlo cardinal to the first uncountable cardinal every definable set of real numbers is H-Ramsey for every coideal H in a wide class of coideals.

Recording of the session (starts some five minutes into the session): click here.

## 22 de mayo - Zaniar Ghadernezhad (Londres)

22.05 - 9 am

Zoom: https://zoom.us/j/148975473

**Zaniar Ghadernezhad ** - Imperial College London

Title: **Group topologies on automorphism groups of homogeneous structures**

Abstract: Automorphism groups of structures endowed with the topology generated by stabilisers of small subsets are topological groups and indeed when countable they are Polish. The interaction between the topological/dynamical properties of automorphism group of a structure and the logical and combinatorial properties of the structure has been widely studied in recent years. In this talk I will discuss different group topologies on automorphism groups of homogeneous structures and especially focus on minimal group topologies. A Hausdorff topological group is called minimal if it does not admit a strictly coarser Hausdorff group topology. I will provide some background, and discuss several classification of group topologies coarser than so called point-wise convergence topology in the case of automorphism groups of countable homogeneous structures and Urysohn space. This is a joint work with Javier de la Nuez González.

Recording: click here.

## 13 de mayo - Boris Zilber (Oxford)

La conferencia originalmente planeada para mañana (Villaveces) ha sido aplazada al 10 de junio. En su lugar, mañana 13 de mayo tendremos una conferencia organizada por Alexander Cruz en el contexto de la Serie *Perspectivas de la Matemática Contemporánea *y el Grupo de Investigación **Conexión de GALoiS. **Por considerarla de alto interés para quienes siguen este seminario decidimos incorporarla a esta programación también.

**Boris Zilber (University of Oxford)**

Title:** ‘Syntax, definability and geometry’.**

Abstract: I will start by talking about syntax/semantics duality in geometry and then explain some deep insights of Grothendieck in terms of model theory.

**Mayo 13, 10:00 am - ¡OJO hora distinta de la usual!**

Lugar: meet.google.com/iue-vpar-igj

Recording: click here.

## 8 de mayo - Miguel Moreno (Viena)

8.05 - 9 am

Zoom: https://zoom.us/j/148975473

**Miguel Moreno** - Kurt Gödel Research Center (KGRC) for Mathematical Logic (Vienna)

Title: **Consistency of Filter Reflection**

Abstract: Filter reflection is an abstract version of stationary reflection. In this talk we will give the definition of filter reflection and different avatars of it. We will show that filter reflection is compatible with large cardinals, forcing axioms, also V=L. We will also discuss how to force filter reflection and its applications to Generalized Descriptive Set Theory. This is joint work with Gabriel Fernandes and Assaf Rinot.

See https://arxiv.org/abs/2003.08340 for more details.

Click here to watch the recording.

## 29 de abril - Darío García (Bogotá)

29.04 - 4 pm (Bogotá time), 9 pm UTC

Zoom: https://zoom.us/j/148975473

**Darío García** - Universidad de los Andes

Title: **Pseudofinite structures: asymptotic classes, dimensions and ranks** [see slides below]

Abstract: The fundamental theorem of ultraproducts ( Łoś’ Theorem) provides a transference principle between the finite structures and their limits and provides an interesting duality between finite structures and their infinite ultraproducts. This kind of finite/infinite connection can sometimes be used to prove qualitative properties of large finite structures using the powerful known methods and results coming from infinite model theory, and in the other direction, quantitative properties in the finite structures often induce desirable model-theoretic properties in their ultraproducts.

In this talk I will review some concepts on pseudofinite structures, and present joint work with D. Macpherson and C. Steinhorn (cf. [1]) where we explored conditions on the (fine) pseudofinite dimension that guarantee good model-theoretic properties (simplicity or supersimplicity, and finite SU-rank) of the underlying theory of an ultraproduct of finite structures, as well as a characterization of forking in terms of decrease of the pseudofinite dimension. The main examples of structures with these properties are ultraproducts of *asymptotic classes* of finite structures (cf. [2]), which are supersimple of finite SU-rank. If time permits, I will also present recent joint work with A. Berenstein and T. Zou that where we study some constructions that naturally provide examples with infinite SU-rank.

[1] Darío García, Dugald Macpherson, Charles Steinhorn, Pseudofinite structures and simplicity, Journal of Mathematical Logic, vol.15 (2015), no. 01, 1550002

[2] Dugald Macpherson and Charles Steinhorn, One-dimensional asymptotic classes of finite structures, Transactions of the American Mathematical Society, vol. 360 (2008), no. 1, pp. 411–448.

Click here to watch the recording of the seminar. (Password: 1m@D$+pe)

## 24 de abril - Mirna Dzamonja (Norwich / París)

24.04 - 9 am

**Mirna Dzamonja** - University of East Anglia (Norwich) / Membre Associée, IHPST, CNRS-Université Panthéon-Sorbonne, Paris

Title: **On wide Aronszajn trees **(see slides below)

Abstract: A tree of height and size aleph_1, but with no uncountable branches, is called a wide Aronszajn tree. Such trees can be quasi-ordered by the relation of weak embedding. Weak embeddings are given by functions that preserve the strict order, that is

x <_T y implies f(x)<_{T’} f(y)

for a weak embedding between a tree T and a tree T’.

It is known by an application of the sigma-functor of Kurepa that the class of wide Aronszajn trees does not have a maximal element in this order when CH is assumed. In 1994 Mekler and Väänänen asked what happens under MA(omega_1).

In our recent joint work with Shelah, we prove that there is no universal wide Aronszajn tree under MA(omega_1). We find a ccc forcing P and a family of aleph_1 dense sets in the forcing, such that for a given wide Aronszajn tree T, the forcing P forces an Aronszajn tree T’ in which T embeds. On the other hand, by a result of Todorcevic from 2007, to which we give an alternative proof, there is maximal Aronszajn tree under MA(omega_1).

Click here to watch the recording.

## 15 de abril - Xavier Caicedo (Bogotá)

15.04 at 4 pm

Zoom: https://zoom.us/j/148975473

**Xavier Caicedo** - Universidad de los Andes

Title: **Undefinability of rigid classes **

Abstract: The undefinability of well order in L_{∞ω} extends to the undefinability of any proper class of rigid structures, and it is a particular case of a general result for model-theoretic logics. Namely, if L enjoys Beth's definability property relative to an extension L′, and L′ enjoys a weak Feferman-Vaught property for pairs of structures relative to a further extension L′′, then no proper class of rigid structures is definable in L, not even in relativized projective form. It is not difficult to see that L_{∞ω} satisfies the hypothesis taking L′=L′′=L_{∞∞}. The non-definability theorem applies to logics not necessarily closed under negations and yields strong results and applications if we assume (relative) Craig interpolation instead of Beth, or stronger forms of the (relative) Feferman-Vaught property.

## 3 de abril - Jouko Väänänen (Helsinki / Amsterdam)

3.04 - 9 am --- note different time and day! ¡Note que será un viernes a las 9 am!

Zoom: https://zoom.us/j/323926528

**Jouko Väänänen** - University of Helsinki and University of Amsterdam

Title: **Lindström's theorem revisited**

Abstract: Lindstrom’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

Recording of the lecture

## 25 de marzo - Roman Kossak (Nueva York)

25.03 - 4 pm

**Roman Kossak** - City University of New York - Graduate Center

Title: **Kernels of digraphs having local finite height**

Abstract: Under certain assumptions, a nonstandard model of arithmetic admits an assignment of truth values for all of its sentences, standard and nonstandard. This important result in the model theory of arithmetic was proved in 1981 by Kotlarski, Krajewski and Lachlan, with a proof employing a "rather exotic proof-theoretic technology." In 2009, Enayat and Viser gave a much more accessible model-theoretic proof. In 2018, Schmerl isolated the graph-theoretic component of the Enayat-Visser proof, by showing that certain infinite graphs have kernels, from which the theorem can be obtained as a straightforward corollary. This story is an excellent example of how mathematics gets simplified. I will explain all basic concepts and I will outline the proof of Shmerl's result. (Schmerl's paper is at: arXiv:1807.11832.)

(Was held via Google Meet: here is a recording of the seminar session)