Andrés Villaveces

Around independence and domination in metric abstract elementary classes: assuming uniqueness of limit models

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.

LIMIT MODELS IN METRIC ABSTRACT ELEMENTARY CLASSES: THE CATEGORICAL CASE

We study versions of limit models adapted to the context of metric abstract elementary classes. Under categoricity and superstability-like assumptions, we generalize some theorems from [7, 15-17]. We prove criteria for existence and uniqueness of limit models in the metric context.

Sheaves of Metric Structures

We introduce sheaves of metric structures and develop their basic model theory. The metric sheaves defined here provide a way to construct new metric models on sheaves a strong generalization of the ultraproduct construction, with the additional property of having the theory of the resulting model controlled by the topology of a given space. More specifically, a metric sheaf

Simplicity via Complexity: Sandboxes, Reading Novalis

\mathfrak {A}

Hilbert Spaces with Generic Predicates

is defined on a topological space X such that each fiber is a metric model. A new model, the generic metric model, is obtained as the quotient space of the sheaf through an appropriate filter of open sets. Semantics in the generic model is completely controlled and understood by the forcing rules in the sheaf and Theoremi¾?3. This work extends early constructions due to Comer [5] and Macintyre [12] and later developments due to Caicedo [3], to the context of continuous logic. We illustrate these concepts by studying the metric sheaf of the continuous cyclic flow on tori.

Limit Models in Classes with Amalgamation

We prove a version of a small index property theorem for strong amalgamation classes. Our result builds on an earlier theorem by Lascar and Shelah (in their case, for saturated models of uncountable first-order theories). We then study versions of the small index property for various non-elementary classes. In particular, we obtain the small index property for quasiminimal pregeometry structures.

Zalamea, Fernando. Filosofía sintética de las matemáticas contemporáneas. Bogotá: Editorial Universidad Nacional de Colombia, Colección Obra Selecta, 2009. 231 p

When the invitation to a meeting on simplicity arrived, I initially had a sense that the “simplicity question” seemed unidirectional and perhaps all too well posed. It seemed initially to play too well with my own experience. (As a mathematician, one of my most sustained, energy-draining, and time-consuming struggles often seems to be with various forms of simplification.) The philosophical issue seemed almost flat. Yet lurking beneath the surface of this seemingly relentless simplification there is an opposite movement, a dual force. Looking more carefully underneath that unidirectional move toward simplification, I started finding strong elements of a move in the opposite direction, perhaps aptly called “complexification.” This paper centers on the spiraling, back-and-forth movement between simplification and complexification, and on the central role of complexification as part of the simplification process.