Alexei Kolesnikov

Canonical forking in AECs

Boney and Grossberg (BG) proved that every nice AEC has an in- dependence relation. We prove that this relation is unique: In any given AEC, there can exist at most one independence relation that satisfies existence, exten- sion, uniqueness and local character. While doing this, we study more generally properties of independence relations for AECs and also prove a canonicity re- sult for Shelahs good frames. The usual tools of first-order logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theo- ries) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework.

Amalgamation functors and boundary properties in simple theories

This paper continues the study of generalized amalgamation properties begun in [1], [2], [3], [5] and [6]. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and we link the binding group of the groupoids to a certain automorphism group of the monster model, showing that the group must be abelian as well.We also study connections between n-existence and n-uniqueness properties for various “dimensions” n in the wider context of simple theories. We introduce a family of weaker existence and uniqueness properties. Many of these properties did appear in the literature before; we give a category-theoretic formulation and study them systematically. Finally, we give examples of first-order simple unstable theories showing, in particular, that there is no straightforward generalization of the groupoid construction in an unstable context.

Generalized amalgamation and n-simplicity

Abstract We study generalized amalgamation properties in simple theories. We formulate a notion of generalized amalgamation in such a way so that the properties are preserved when we pass from T to T e q or T h e q ; we provide several equivalent ways of formulating the notion of generalized amalgamation. We define two distinct hierarchies of simple theories characterized by their amalgamation properties; examples are given to show the difference between the hierarchies.

n-Simple theories

Abstract The main topic of this paper is the investigation of generalized amalgamation properties for simple theories. That is, we are trying to answer the question of when a simple theory has the property of n-dimensional amalgamation, where two-dimensional amalgamation is the Independence Theorem for simple theories. We develop the notions of strong n-simplicity and n-simplicity for 1≤n≤ω, where both “1-simple” and “strongly 1-simple” are the same as “simple”. For strong n-simplicity, we present examples of simple unstable theories in each subclass and prove a characteristic property of strong n-simplicity in terms of strong n-dividing, a strengthening of the dependence relation called dividing in simple theories. We prove a strong three-dimensional amalgamation property for strongly 2-simple theories, and, under an additional assumption, a strong (n+1)-dimensional amalgamation property for strongly n-simple theories. In the last section of the paper we comment on why strong n-simplicity is called strong.

Type-amalgamation properties and polygroupoids in stable theories

We show that in a stable first-order theory, the failure of higher dimensional type amalgamation can always be witnessed by algebraic structures that we call n-ary polygroupoids. This generalizes a result of Hrushovski in [16] that failures of 4-amalgamation are witnessed by definable groupoids (which correspond to 2-ary polygroupoids in our terminology). The n-ary polygroupoids are definable in a mild expansion of the language (adding a predicate for a Morley sequence).

Multivariate \(C^1\)-Continuous Splines on the Alfeld Split of a Simplex

Using algebraic geometry methods and Bernstein-Bezier techniques, we find the dimension of \(C^1\)-continuous splines on the Alfeld split of a simplex in \(\mathbb {R}^n\) and describe a minimal determining set for this space.

THE HANF NUMBER FOR AMALGAMATION OF COLORING CLASSES

We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in previous work of Baldwin, Kolesnikov, and Shelah. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of Baldwin, Kolesnikov, and Shelah by showing, in ZFC, that the (disjoint) amalgamation property for classes

The equality S1 = D = R

K_\alpha

Categoricity, amalgamation, and tameness

studied in that paper must hold up to

The amalgamation spectrum

\beth_\alpha