# 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).

# Partitions of well-founded trees: Three connections with model theory – Tehran (v) – 2/21

For the IPM (Tehran), the lecture Partitions of well-founded trees: Three connections with model theory.

Abstract: in 2003, Komjáth and Shelah proved a partition theorem on scattered order types; these in turn could be understood as partition relations for classes of well-founded trees. Recently, two different kinds of applications of the same partition relation have been used in infinitary logic and in model theory: one by Väänänen and Velickovic on games related to Shelah’s logic $L^1_\kappa$, another by Shelah and myself on the “canonical tree” of an AEC (a generalization of the Scott sentence for an abstract elementary class). I will describe the Komjáth-Shelah result in the first part and then narrow in the applications (with more details on the second one, from some recent joint work with Shelah). Time permitting, I will also address other interactions between partition relations and model theoretic issues.

# AECs and notions of existential closure – London (v) – 11/20

For the joint Imperial College and Queen Mary University, London, I gave the lecture

in November 2020.

Abstract: I will provide a general overview of AECs with emphasis on their connection with various abstract notions of “existential closure”. I will present a recent joint result of myself together with Shelah (2019) on axiomatizability of AECs in infinitary logic, and examples of their interaction with closure notions. As an example, I will present aspects of a construction of a canonical “existential closure” for locally finite groups due to Shelah, and their connection with more general stability-theoretical questions.

# Partitions of well-founded trees and two connections with model theory. Paris (v) – 6/10

For the Paris Logic Seminar, I gave the lecture

Abstract: in 2003, Komjath and Shelah proved a partition theorem on scattered order types; these in turn could be understood as partition relations for classes of well-founded trees. Recently, two different kinds of applications of the same partition relation have been used in infinitary logic and in model theory: one by Väänänen and Velickovic on games related to Shelah’s logic $L^1_\kappa$, another by Shelah and myself on the “canonical tree” of an AEC (a generalization of the Scott sentence for an abstract elementary class). I will describe the Komjath-Shelah result in the first part and then narrow in the applications (with more details on the second one, from some recent joint work with Shelah).

# On the interplay between Abstract Elementary Classes and Categorical Logic – Bogotá (v), 6/20

For the Seminario Flotante de Lógica Matemática de Bogotá, I gave the lecture 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.

# One Puzzling Logic, Two Approximations… and a Bonus. Helsinki (v), 5/20

For the Helsinki Logic Seminar, I gave (virtually) the lecture

One Puzzling Logic, Two Approximations… and a Bonus.

Abstract: The puzzling logic (called $L^1_\kappa$ for $\kappa$ a singular strong limit cardinal) I will speak about was introduced by Saharon Shelah in 2012. The logic $L^1_\kappa$ has many properties that make it very well adapted to model theory, despite being stronger than$L_{\kappa,\omega}$. However, it also lacks a good syntactic definition.

With Väänänen, we introduced the first approximation (called $L^{1,c}_\kappa$,) as a variant of $L^1\kappa$ with a transparent syntax and many of the strong properties of Shelah’s logic.

The second approximation (called Chain Logic), while not new (it is due to Karp), has been revisited recently by Dzamonja and Väänänen) also in relation to Shelah’s $L^1_\kappa$ and the Interpolation property.

I will provide a description of these three logics, with emphasis on their relevance to model theory.
As a bonus, I will make a connection between these logics and axiomatizing correctly an arbitrary AEC. This last part is joint work with Shelah.