# 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.

# Sobre el Main Gap de Shelah: una visión panorámica. Bogotá, 5/20

Participación en Tertulias Matemáticas: 2. Teoría del Main Gap (Shelah)

con presentación inicial Sobre el Main Gap de Shelah: una visión panorámica.

Haga clic en la imagen para ver la intervención en su navegador. Mediante comandos (n para “next”, p para “previous”, etc. puede navegar). Un comando útil es t: puede agregar o quitar texto informativo.

# Enlaces muy tenues: Lógica y Teoría del Índice. Bogotá, 4/20

Participación en Tertulias Matemáticas: 1. Teoría del Índice (Atiyah-Singer)

con brevísima intervención llamada Enlaces muy tenues: Lógica y Teoría del Índice.

Haga clic en la imagen para ver la intervención en su navegador. Mediante comandos (n para “next”, p para “previous”, etc. puede navegar). Un comando útil es t: puede agregar o quitar texto informativo.

# Lógica(s) y Topología(s) – desde Stone hasta Lurie / Bogotá, 2/20

Para el III Simposio Carlos Ruiz en la Universidad Nacional (Bogotá) di la charla Lógica(s) y Topología(s) – desde Stone hasta Lurie.

Resumen: En esta charla hago un recuento de tres teoremas de dualidad: Stone (1936), Makkai (1988 – generalizando resultados anteriores con Reyes) y Lurie (preprint, 2019) también llamados “Teoremas de Completitud Conceptual” en algunos contextos. Señalo el rol central de la topología como herramienta de reconstrucción (de la sintaxis a partir de la semántica) y planteo otros dos escenarios de reconstrucción análogos (trabajos conjuntos con Ghadernezhad – 2017 y con Shelah – en proceso).