# El corazón (galoisiano) de una clase elemental abstracta (Bogotá (v), 9/21)

For the Encuentro de topología Carlos Ruiz, Universidad Nacional de Colombia, Bogotá, I gave the lecture El corazón (galoisiano) de una clase elemental abstracta.

# Tra teoria dei modelli e teoria degli insiemi – Torino (v) – 5/21

As part of my Torino Lectures in Set Theory, Special Lecture 3 was given in Italian. The title was Tra teoria dei modelli e teoria degli insiemi.

La terza lezione speciale (parte del corso Teoria degli insieme) sarà questo prossimo giovedì (16:30 ore a Torino). Darò io stesso la lezione stavolta, su connessioni tra teoria dei modelli e teoria degli insiemi. La lezione sarà diretta principalmente a studenti che iniziano il loro apprendimento delle tecniche di forcing e grandi cardinali (la lezione 2, che Vika Gitman ha dato due settimane fa, anche aveva un mix di questi due temi).
Sarà anche la prima volta che parlerò in italiano in pubblico online (forse soltanto per una parte della lezione; dipende di come andrà tutto). Ho invitato gente che in principio può capire la lezione così.

# Two logics, and their connections with large cardinals / Questions for BDGM – New York (v) – 4/21

For the CUNY Graduate Center Set Theory Seminar I gave a series of two lectures with title Two logics, and their connections with large cardinals / Questions for BDGM.

Abstract: In the past couple of years I have been involved (joint work with Väänänen and independently with Shelah) with some logics in the vicinity of Shelah’s $L^1_\kappa$ (a logic from 2012 that has Interpolation and a very weak notion of compactness, namely Strong Undefinability of Well-Orderings, and in some cases has a Lindström-type theorem for those two properties). Our work with Väänänen weakens the logic but keeps several properties. Our work with Shelah explores the connection with definability of AECs.
These logics seem to have additional interesting properties under the further assumption of strong compactness of a cardinal, and this brings them close to recent work of Boney, Dimopoulos, Gitman and Magidor [BDGM].
During the first lecture, I plan to describe two games and a syntax of two logics: Shelah’s $L^1_\kappa$ and my own logic (joint work with Väänänen) $L^{1,c}_\kappa$. I will stress some of the properties of these logics, with any use of large cardinal assumptions.
During the second lecture, I plan to enter rather uncharted territory. I will describe some constructions done by Shelah (mostly) under the assumption of strong compactness, but I also plan to bring these logics to a territory closer to the work of [BDGM]. This second lecture will have more conjectures, ideas, and (hopefully interesting) discussions with some of the authors of that paper.

# Around Shelah’s logic L^1_\kappa – Pittsburgh (v) – 3/21

For the Carnegie Mellon Model Theory Seminar, I gave in March 2021 a series of two lectures with the title Around Shelah’s logic $L^1_\kappa$.

Abstract: In 2012, Shelah introduced a new logic called $L^1_\kappa$. The declared intention was to solve the problem of finding a logic with interpolation and maximality properties with respect to weak forms of compactness. This logic satisfies the declared intention (when the parameter $\kappa$ is a singular strong limit of countable cofinality): the logic $L^1_\kappa$ has interpolation (in itself) and undefinability of well-order. However, the logic’s syntax is given only through a variant of the classical Ehrenfeucht-Fraïssé game and is not built recursively. In recent joint work with Väänänen, we have provided an approximation (from below) to Shelah’s logic that indeed has a recursively built syntax and clarifies some questions on Shelah’s $L^1_\kappa$.

In the first lecture, I will give a presentation of Shelah’s logic, with one proof (of a long string of theorems leading to a Lindström-like characterization). In the second lecture, I will describe our logic (from our joint work with Väänänen) that approximates $L^1_\kappa$ and allows us to understand in a different way some of the properties of Shelah’s logic. If time allows, I will also mention other two approaches (joint work of Väänanen with Dzamonja for one of these approaches and with Velickovic for the other one).

# Axiomatizations of abstract elementary classes and natural logics for model theory: the role of partition relations – Toronto (v) 2/21

For the Toronto Set Theory Seminar, I gave the lecture Axiomatizations of abstract elementary classes and natural logics for model theory: the role of partition relations.

Abstract: Two seemingly unrelated questions (the quest for natural logics of abstract elementary classes on the one hand, and the quest for logics adequate to model theory on the other hand) converge around the same combinatorial core: partition relations for scattered order types (due to Kómjath and Shelah). I will present recent results concerning the first question (and axiomatizing a.e.c.’s – joint work with Shelah) and the second question (joint work with Väänänen).

# Some New Infinitary Logics and a Canonical Tree – New York, 11/19

Abstract: The main recent logic I will describe is Shelah’s infinitary logic $L^1_\kappa$ (from 2012). I will describe some of the reasons for studying this logic (roughly, it is an infinitary logic that has interpolation and a weak form of compactness – therefore particularly well-adapted to model theory, as well as closure under chains) and some of the features lacking (mostly, a workable syntax). I will describe two other logics that have been created in order to capture better the syntax (one of these logics is my joint work with Väänänen, the other one is due originally to Karp and Cunningham and has recently been connected to $L^1_\kappa$ by Dzamonja and Väänänen. Finally I will connect these logics with the problem of axiomatizing abstract elementary classes. In particular, I will describe canonical trees of models that enables one to build a sentence to test models for membership into aecs. This last part is joint work with Shelah.

# Interpolation and model theoretic forcing – some new perspectives (Campinas, Brazil, 12/18)

Interpolation and model theoretic forcing – some new perspectives.

(A lecture in Cantor Meets Robinson – Set theory, model theory and their philosophy. University of Campinas, Brazil, December 2018.)

Model Theoretic Forcing has been interweaved with interpolation theorems in infinitary logic since the early work of Mostowski, Vaught, Harnik and others. I will present some of these historical connections and their effect on Shelah’s much more recent logic $L^1_\kappa$. In particular I will focus on some connections between model theoretic forcing and the model theory of abstract elementary classes.

# Infinitary logic, large cardinals and AECs: some reflections (Montseny, Catalunya, 11/18)

Reflections on Set Theoretic Reflection – Montseny, Catalunya, nov. 2018.

Abstract:

The interaction between infinitary logic and the model theory of abstract elementary classes has had a serious imprint of large cardinals since the inception of AECs. Although later developments in AECs have emphasized a more purely model theoretic treatment, capturing independence-like relations, there are various fundamental questions on the relation between various logics and AECs — and, in some of these, large cardinals are central.
I will discuss some work by Boney on these connections, as well as some recent joint work by Väänänen and myself.

# Propagators and Sheaves (Bogotá, 9/18)

Seminario Lógica y Geometría (Bogotá, 9/18).

Mi charla: Propagators and Sheaves.

Uno de los ejemplos provenientes de la física cuántica que ha sido objeto de varios análisis modelo-teóricos en años recientes ha sido el propagador cuántico (trabajos de Zilber y de Hirvonen-Hyttinen). Aquí proponemos otro enfoque, más cercano a la lógica de haces métricos, en trabajo conjunto con Maicol Ochoa. En particular, damos una construcción mediante espacios de Schwartz que permite enfocar el comportamiento del operador asociado al propagador como límite de operadores que actúan sobre espacios finito-dimensionales.