Simple theories

Simple theories
Name	Simple theories
Field	Model theory
Notable	Kim independence; Lascar strong type

Simple theories Simple theories are a class of complete first-order theories studied in model theory that generalize stable theories while retaining robust independence notions. They were formulated to analyze independence phenomena in settings where stability fails, connecting work by researchers studying forking, dividing, and rank. Simple theories interface with results and constructions from prominent figures and institutions in logic.

Introduction

Simple theories arose from efforts by researchers influenced by results at conferences such as the International Congress of Mathematicians and workshops at institutions like the Institute for Advanced Study and the Mathematical Sciences Research Institute. Early developments built on techniques introduced by contributors associated with the University of California, Berkeley, the Hebrew University of Jerusalem, and the University of Chicago. Key names connected to the subject include Saharon Shelah, Byunghan Kim, Ehud Hrushovski, Alfred Tarski, Jonathan Los, and Michael Morley. Foundational papers were discussed in venues linked to the American Mathematical Society, the London Mathematical Society, and the Association for Symbolic Logic.

Basic Definitions and Examples

A theory T is called simple when dividing satisfies a local character and when forking and dividing coincide in the appropriate sense introduced in seminars at the University of Notre Dame and the University of California, Los Angeles. Canonical examples include the theory of infinite random graphs studied in contexts related to the Erdős–Rényi model, the generic countable graph related to work at the University of Cambridge, and certain theories of trees examined by researchers at the University of Oxford. Other examples derive from expansions considered by groups at the Max Planck Institute for Mathematics and constructions influenced by the Jerusalem Model Theory Group.

Independence and Forking

Independence in simple theories is formalized via non-forking, a relation refined by the notion of forking introduced in seminars at the University of Chicago and elaborated in papers circulated through the American Mathematical Society. Kim independence, developed by Byunghan Kim and collaborators, provides a symmetric independence relation analogous to forking in stable theories and is central to work connected to the Association for Symbolic Logic meetings. Lascar strong types and their analysis, advanced by scholars affiliated with Paris Diderot University and the Hebrew University of Jerusalem, play a role in studying automorphism groups and orbit equivalence in simple contexts.

Model-Theoretic Properties and Theorems

Important theorems for simple theories include analogues of stability-theoretic facts such as the independence theorem, chain conditions, and uniqueness of non-forking extensions; these were developed in collaboration among researchers linked to the Institute for Advanced Study, the Max Planck Institute for Mathematics, and the Kurt Gödel Research Center. Results about ranks, such as U-rank and SU-rank, and their behavior in simple contexts were explored in papers presented at conferences organized by the European Mathematical Society and the American Mathematical Society. Structural theorems relating to definable groups in simple theories stem from work influenced by groups at the University of Illinois at Urbana–Champaign and the University of Michigan.

Classification and Relations to Other Theories

Simple theories sit in the classification hierarchy near stable and NIP theories; comparisons and separations have been investigated by authors associated with the California Institute of Technology, the Princeton University, and the Massachusetts Institute of Technology. The relationship between simplicity and properties like NIP, NTP2, and NSOP1 has been a topic at workshops held by the Association for Symbolic Logic and research clusters at the University of Toronto. Examples showing strict inclusions and independence of properties were constructed in collaboration across groups at the University of Pennsylvania and the University of Cambridge.

Applications and Examples in Mathematics and Logic

Applications of simple theories appear in algebraic examples such as certain differential fields analyzed by teams at the University of Chicago and the University of Illinois at Urbana–Champaign, in combinatorial structures related to the Erdős–Rényi model and work from the Institute for Advanced Study, and in categorical logic discussions at the University of California, Berkeley. Connections to descriptive set theory and automorphism groups have been pursued by researchers affiliated with the University of Bonn and the Hebrew University of Jerusalem. Further examples and applications have been developed in collaboration with the Institute of Mathematics of the Polish Academy of Sciences and presented at meetings of the European Mathematical Society.

Category:Model theory