- Departamento de Matemáticas, Universidad Nacional de Colombia, AK 30 45-03 postal code 111321, Bogotá Colombia
- PBX +57 1 316 5000 ext. 13163
- I am interested in Mathematical Logic, in particular Model Theory (Stability in Metric Non-Elementary Classes).edit
In my PhD thesis a version of Shelah's Presentation Theorem in the setting of Metric Abstract Elementary Classes was proved, where we claimed that the new function symbols are not necessarily uniformly continuous. In this paper we... more
In my PhD thesis a version of Shelah's Presentation Theorem in the setting of Metric Abstract Elementary Classes was proved, where we claimed that the new function symbols are not necessarily uniformly continuous. In this paper we provide a proof they are in fact uniformly continuous.
Research Interests:
In this paper, we prove that if $\kappa$ is a almost strongly compact cardinal, then any MAEC with L\"owenheim-Skolem number below $\kappa$ is $<\kappa$-d-tame.
Research Interests:
In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible,... more
In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible, co-Girard and a V-domain), we provide, as test questions, a proof of a version of the Tarski-Vaught test (Proposition 3.35) and Łoś Theorem (Theorem 3.62) in our setting.
Research Interests:
We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and... more
We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves.
Research Interests:
We remark that forcing on fiber bundles of structures of first order languages is not a compatible semantics with the pullback (of fiber bundles) and we describe a semantics which behaves well with respect to it. This new semantics uses... more
We remark that forcing on fiber bundles of structures of first order languages is not a compatible semantics with the pullback (of fiber bundles) and we describe a semantics which behaves well with respect to it. This new semantics uses parallel transport and allows to introduce two different types of extensions for the formulae: vertical and horizontal extensions.
Research Interests:
We study versions of limit models adapted to the context of metric abstract elementary classes. Under superstability-like assumptions, we prove some generalizations of theorems from [GrVaVi]. We prove criteria for existence and uniqueness... more
We study versions of limit models adapted to the context of metric abstract elementary classes. Under superstability-like assumptions, we prove some generalizations of theorems from [GrVaVi]. We prove criteria for existence and uniqueness of limit models in the metric context. 1. Preliminaries Why limit models and towers? The Model Theory of metric structures can be approached in a fruitful way from the Abstract Elementary Class perspective — extending in some senses the framework of First Order ContinuousModel Theory [BeBeHeUs] and in other senses beneVtting from the enormous richness of the Stability Theory in Abstract Elementary Classes. Other authors (Hirvonen [Hi] in her thesis with Hyttinen, Usvyatsov and Shelah) have provided essentially two major frameworks for dealing with contexts outside “generalized Vrst order”. Hirvonen and Hyttinen have developed a solid framework for categoricity transfer of metric AEC and for the study ofא0-stable classes of metric structures (a good...
Research Interests:
Research Interests:
We prove that cats (compact abstract theories) satisfy a version of Tarski-Vaught test, a version of DLST (downward Löwenheim-Skolem Tarski) theorem using density character instead of cardinality and the DAP property (disjoint... more
We prove that cats (compact abstract theories) satisfy a version of Tarski-Vaught test, a version of DLST (downward Löwenheim-Skolem Tarski) theorem using density character instead of cardinality and the DAP property (disjoint amalgamation property.
Research Interests:
In this work we present some general categorial ideas on Abstract Elementary Classes (AECs) %\cite{She}, inspired by the totality of AECs of the form $(Mod(T), \preceq)$, for a first-order theory T: (i) we define a natural notion of... more
In this work we present some general categorial ideas on Abstract Elementary Classes (AECs) %\cite{She}, inspired by the totality of AECs of the form $(Mod(T), \preceq)$, for a first-order theory T: (i) we define a natural notion of (funtorial) morphism between AECs; (ii) explore the following constructions of AECs: "generalized" theories, pullbacks of AECs, (Galois) types as AECs; (iii) apply categorial and topological ideas to encode model-theoretic notions on spaces of types %(see Michael Lieberman Phd thesis) ; (iv) present the "local" axiom for AECs here called "local Robinson's property" and an application (Robinson's diagram method); (v) introduce the category $AEC$ of Grothendieck's gluings of all AECs (with change of basis); (vi) introduce the "global" axioms of "tranversal Robinson's property" (TRP) and "global Robinson's property" (GRP) and prove that TRP is equivalent to GRP and GRP entails a...
Research Interests:
Research Interests:
A. We study notions of independence appropriate for a stability theory of metric abstract elementary classes. We build on previous notions used in the discrete case, and adapt denitions to the metric case. In particular, we study notions... more
A. We study notions of independence appropriate for a stability theory of metric abstract elementary classes. We build on previous notions used in the discrete case, and adapt denitions to the metric case. In particular, we study notions that behave well under ...
We remark that forcing on fiber bundles of structures of first order languages is not a compatible semantics with the pullback (of fiber bundles) and we describe a semantics which behaves well with respect to it. This new semantics uses... more
We remark that forcing on fiber bundles of structures of first order languages is not a compatible semantics with the pullback (of fiber bundles) and we describe a semantics which behaves well with respect to it. This new semantics uses parallel transport and allows to introduce two different types of extensions for the formulae: vertical and horizontal extensions.
Research Interests:
We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the me tric to take values in a well-behaved quantale. As a proof of concept we show that the result of [BZ15] on... more
We broaden the framework of metric abstract elementary classes
(mAECs) in several essential ways, chiefly by allowing the me
tric to take values in a well-behaved quantale. As a proof of concept we show that the result of [BZ15] on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves.
(mAECs) in several essential ways, chiefly by allowing the me
tric to take values in a well-behaved quantale. As a proof of concept we show that the result of [BZ15] on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves.
Research Interests:
We study notions of independence appropriate for a stability theory of metric abstract elementary classes (for short, MAECs). We build on previous notions used in the discrete case, and adapt definitions to the metric case. In particular,... more
We study notions of independence appropriate for a stability theory of metric abstract elementary classes (for short, MAECs). We build on previous notions used in the discrete case, and adapt definitions to the metric case. In particular, we study notions that behave well under superstability-like assumptions. Also, under uniqueness of limit models, we study domination, orthogonality and parallelism of Galois types in MAECs.
Research Interests:
Research Interests:
In this paper, we study a stability transfer theorem in d-tame metric abstract elementary classes, in a similar way as in 2, but using superstability-like assumptions which involves a new independence notion (ζ*-independence) instead of... more
In this paper, we study a stability transfer theorem in d-tame metric abstract elementary classes, in a similar way as in 2, but using superstability-like assumptions which involves a new independence notion (ζ*-independence) instead of ℵ0-locality.
Research Interests:
"We study versions of limit models adapted to the context of metric abstract elementary classes. Under superstability-like assumptions, we prove some generalizations of theorems from [GrVaVi]. We prove criteria for existence and... more
"We study versions of limit models adapted to the context
of metric abstract elementary classes. Under superstability-like assumptions, we prove some generalizations of theorems from [GrVaVi]. We prove criteria for existence and uniqueness of limit models in the metric context."
of metric abstract elementary classes. Under superstability-like assumptions, we prove some generalizations of theorems from [GrVaVi]. We prove criteria for existence and uniqueness of limit models in the metric context."
"We study notions of independence appropriate for a sta- bility theory of metric abstract elementary classes. We build on pre- vious notions used in the discrete case, and adapt deVnitions to the metric case. In particular, we study... more
"We study notions of independence appropriate for a sta-
bility theory of metric abstract elementary classes. We build on pre-
vious notions used in the discrete case, and adapt deVnitions to the
metric case. In particular, we study notions that behave well under
superstability-like assumptions."
bility theory of metric abstract elementary classes. We build on pre-
vious notions used in the discrete case, and adapt deVnitions to the
metric case. In particular, we study notions that behave well under
superstability-like assumptions."
In this talk, we prove of some model-theoretical basic facts which hold in Covers of Multiplicative Groups of an Algebraically Closed Field of characteristic 0, in order to study its model-theoretical be- haviour using tools from the... more
In this talk, we prove of some model-theoretical basic facts
which hold in Covers of Multiplicative Groups of an Algebraically Closed
Field of characteristic 0, in order to study its model-theoretical be-
haviour using tools from the Model Theory of Non-Elementary Classes.
which hold in Covers of Multiplicative Groups of an Algebraically Closed
Field of characteristic 0, in order to study its model-theoretical be-
haviour using tools from the Model Theory of Non-Elementary Classes.
En clases elementales abstractas, una noción débil de superestabilidad corresponde a la existencia y unicidad de modelos límite (trabajo de Grossberg, VanDieren y Villaveces). En esta charla presentaremos algunos resultados parciales... more
En clases elementales abstractas, una noción débil de superestabilidad corresponde a la existencia y unicidad de modelos límite (trabajo de Grossberg, VanDieren y Villaveces). En esta charla presentaremos algunos resultados parciales apuntando a probar un resultado similar en el contexto de las clases elementales abstractas métricas.
In this talk, We will give some basic facts in Metric Abstract Elementary Classes. Also, we will give some basic facts relative to a notion of independence which we will use for understanding a version of superstability -uniqueness of... more
In this talk, We will give some basic facts in Metric Abstract Elementary Classes. Also, we will give some basic facts relative to a notion of independence which we will use for understanding a version of superstability -uniqueness of limit models- in this setting.
In this talk, we will prove that the completion of unions of an increasing and continuous chain of reduced towers is reduced and also the density of reduced towers, in the setting of Metric Abstract Elementary Classes. These are important... more
In this talk, we will prove that the completion of unions of an increasing and continuous chain of reduced towers is reduced and also the density of reduced towers, in the setting of Metric Abstract Elementary Classes. These are important steps towards proving the uniqueness of limit models in the setting of Abstract Elementary Classes.
In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible,... more
In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible, co-Girard and a V-domain), we provide, as test questions, a proof of a version of the Tarski-Vaught test (Proposition 3.35) and Łoś Theorem (Theorem 3.62) in our setting.