A lecture on metric lattices (and model theory)
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Andrés Villaveces
Jun 6, 2024, 4:22:47 PMJun 6
to C Ward Henson, John Baldwin, Jose Miguel Contreras Mantilla, Gianluca Paolini, Zaniar Ghadernezhad, logbog
Dear all,
Tomorrow Friday at noon in Colombia and Chicago time (10 AM in California, 17:00 UTC) there will be an "ad-hoc" lecture (not officially part of any seminar) by José Miguel Contreras (now finishing his undergraduate studies here at Universidad Nacional) on work he did together with Thomas Sinclair during an exchange program at Purdue last semester. Together, Contreras and Sinclair generalized the axiomatization of Atomless Probability Spaces due to Berenstein and Henson to metric lattices (a class containing, among others, finite partition lattices and Björner’s continuous partition lattice).
You are invited!
Google Meet link: http://meet.google.com/iat-bdgh-qkm
Metric Lattices from a model-theoretical perspective
José Miguel Contreras - Universidad Nacional de Colombia
Abstract: Model theory for metric structures is a branch of model theory developed to solve some of the limitations of first order logic to capture notions related to metric spaces, C*-algebras, Banach spaces, etc. During this talk we will use this framework to present our notion of metric lattices, a kind of lattices with properties that make them suitable to study using this continuous [0,1]-valued logic.
We will begin with a quick introduction to this alternative model theory and an example of its application in the study of Probability Spaces. Then motivated by this case we will define metric lattices, show how they include probability algebras and other interesting examples, as finite partition lattices and Björner’s continuous partition lattice, and prove that they are axiomatizable in this continuous logic despite the possible non-continuity of the ”meet” operation. Finally we will present some results and notions related to metric lattices, such as modularity, definability of the meet operation, distributivity and complements, getting a better understanding of their theory and how to
study it using the advantages of the model theory for metric structures. This talk is based on the research done by Prof. Thomas Sinclair and me, during my research stay at Purdue University with the program UREP-C 2023-II.
We will begin with a quick introduction to this alternative model theory and an example of its application in the study of Probability Spaces. Then motivated by this case we will define metric lattices, show how they include probability algebras and other interesting examples, as finite partition lattices and Björner’s continuous partition lattice, and prove that they are axiomatizable in this continuous logic despite the possible non-continuity of the ”meet” operation. Finally we will present some results and notions related to metric lattices, such as modularity, definability of the meet operation, distributivity and complements, getting a better understanding of their theory and how to
study it using the advantages of the model theory for metric structures. This talk is based on the research done by Prof. Thomas Sinclair and me, during my research stay at Purdue University with the program UREP-C 2023-II.
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avillavecesn.net
Departamento de Matemáticas - Edificio 404
Universidad Nacional de Colombia
AK 30 # 45-03
111321 - Bogotá - Colombia
Tel: +57 601 3165000 ext 13209
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