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Logic Quantum Mechanics Model Theory
LoQMoT is a series of lectures on research in Quantum Mechanics and related fields of Mathematics, approached from the perspective of Model Theory.
LoQMoT 25 is focused on Model Theory of Continuous Logic and some of its uses in Functional Analysis, Probability Theory, Geometry and the mathematical foundations of Quantum Mechanics.
Schedule
Abstracts of the Lectures
Tutorial (Lógica continua) (Miércoles 15 de octubre, 11:00 a 13:00)
En una sesión inicial previa al evento, el estudiante de doctorado Gustavo Cipagauta ofrece un tutorial introductorio de Lógica Continua, dirigido a cualquier persona que quiera conocer, afinar o recordar definiciones y conceptos básicos del campo, como preparación para el evento. Este tutorial será dado en español; el resto del evento será en inglés.
Stability of theories of probability spaces (Gustavo Cipagauta, Universidad Nacional de Colombia, sede Bogotá).
Abstract: Based on a dialectical classification of the misnamed interpretations of quantum theory, we justify approaches from mathematical logic and, in particular, from model theory, to some of the fundamental problems in this field of physics. In this talk, we are particularly interested in model theory on the continuous logic of probability spaces and random variables. We emphasize the important model-theoretical concept of stability.
Metric AECs and Physics (Andrés Villaveces, Universidad Nacional de Colombia, sede Bogotá).
Abstract: The metric version of AECs has been studied, for different reasons and in slightly different ways, in three places: Jerusalem, Bogotá and Helsinki. I will describe the approach we took around 15 years ago here with my then-student Pedro Zambrano, and later works by Hirvonen and Hyttinen. I will also describe how in our work with Argoty, we used our approach to capture early descriptions in physics. My account of these works will be both historic and forward-looking: I will focus on which aspects of metric AECs generalize which aspects of traditional Continuous Logic, and why I believe they are worthwhile studying in connection with new challenges in Quantum Mechanics.
Mirror maps through logically perfect structures (Alexander Cruz, Universidad Nacional de Colombia, sede Bogotá).
Abstract: Mirror symmetry is an interesting phenomenon in both mathematics and physics. Emerging in physics more than 30 years ago, as a kind of duality between symplectic and complex geometry, has called the attention of mathematicians who have proposed several programs to understand it. One of the central objects in mirror symmetry is the so-called mirror map which is still a bit mysterious. On the other hand, from the model theory side, Boris Zilber has proposed a program for both mathematics and physics based on the idea of logically perfect structures, a notion close to the idea of categoricity and a kind of extension of his program in abstract algebraic geometry based on the idea of Zariski geometries.
In this talk, I will propose a set of ideas looking for understanding mirror maps from the point of view of logically perfect structures. The main idea behind is to explore how model theoretic tools could help to understand the way different geometries are related.
Structural approximation: local versus global (Boris Zilber, Oxford University).
Abstract:
L1 convolution algebras (Alexander Berenstein, Universidad de Los Andes).
Abstract: The theory of L1-Banach lattices is well understood, both when the underlying measure space has atoms and when it is atomless. It is known, for example, that it is w-stable and forking independence is well understood. In this talk we will explore expansions of the form (L_1(G),*), where G is a locally compact and * stands for the convolution. A classical theorem of Wendel and Kanada says that two such structures (L_1(G),) and (L_1(H),) are isomorphic iff the underlying groups G and H are topologically and algebraically isomorphic. We recover a model theoretic version of this result when the underlying groups are discrete. We also show that the expansion is unstable when the underlying group G is non-discrete.
Lindström Theorems for Continuous Logic (Xavier Caicedo, Universidad de Los Andes).
Abstract:
From Model Theory to Foundations of Physics (Boris Zilber, Oxford University).
Abstract: Model Theory is having an increasing impact in research in classical areas of mathematics, ranging from Complex Geometry through Number Theory to Combinatorics. In this talk I begin by discussing its potential applications in Foundations of Physics and subsequently present some results in this direction. Among these, I will explain how Hilbert-space axiomatisation of quantum physics is in effect an axiomatisation in the language of Continuous Logic. In particular, I prove that in this interpretation Dirac - von Neumann axioms of quantum mechanics admit models with (very large) finite universes.
Perspectives (Round-table).
We'll close with a round-table discussion on perspectives for interactions between Quantum Theory and other parts of Mathematics, with emphasis in the role of Model Theory.
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