Algorithmic complexity of structures and their equivalence relations

Projektbeschreibung

Forschung beleuchtet die rechnerische und beschreibende Komplexität mathematischer Objekte

Die berechenbare Strukturtheorie quantifiziert und erkundet die Komplexität mathematischer Strukturen. Die gängigste Methode hierfür sind Gradspektren. Das im Rahmen der Marie-Skłodowska-Curie-Maßnahmen finanzierte Projekt ACOSE zielt darauf ab, unser Verständnis der rechnerischen und beschreibenden Komplexität mathematischer Objekte (wie Strukturen und Beziehungen) und der damit verbundenen Klassifizierungsprobleme zu vertiefen. Die Forschenden werden weitere Einblicke in die zu wenig beachtete Beziehung zwischen der deskriptiven Komplexität einer Menge und dem Gradspektrum unter einer gegebenen Äquivalenzrelation gewinnen. Die neu entwickelten Techniken könnten dazu beitragen, die Bereiche der deskriptiven Mengenlehre und der berechenbaren Strukturtheorie eng miteinander zu verknüpfen.

Ziel

We will investigate the algorithmic complexity of mathematical structures and its connection with notions of complexity studied in descriptive set theory at. The main subject area of the planned research is computable structure theory — an area of logic concerning itself with the computational complexity of countable mathematical structures. Mathematicians usually consider structures up to some equivalence relation. For example, a number theorists works in the standard model of arithmetic, the natural numbers with addition and multiplication, but it is of little interest to him whether he works in the canonic representation or in some isomorphic copy as this does not impact his work.
However, for computational matters the choice of representation is highly important. Therefore one usually measures the algorithmic complexity of a structure by its degree spectrum, the set of Turing degrees of structure equivalent to the structure under a given equivalence relation.
Degree spectra are the main subject of investigation in computable structure theory. A natural way to think of degree spectra is as sets of subsets of the natural numbers, and these sets are studied in descriptive set theory.

So far the relation between the descriptive complexity of a set and whether it can be a degree spectrum under a given equivalence relation has been overlooked. The goal of this project is to relate the descriptive complexity of sets with their realizability as degree spectra under some equivalence relation.
We plan to obtain new results and develop new techniques which will be beneficial to both descriptive set theory and computable structure theory and hope to form a lasting connection between those fields.

The fellowship will be carried out over 36 months, 24 months at the University of California, Berkeley under supervision of Prof. Antonio Montalbán and 12 months at TU Wien under the supervision of Professors Ekaterina Fokina and Matthias Baaz.