Algorithmic complexity of structures and their equivalence relations

Descripción del proyecto

Una investigación aporta información nueva sobre la complejidad computacional y descriptiva de los objetos matemáticos

La teoría de estructuras computables cuantifica y estudia la complejidad de las estructuras matemáticas. La forma más común de estudiar la complejidad de estas estructuras es mediante espectros de grados. El objetivo del proyecto ACOSE, financiado por las Acciones Marie Skłodowska-Curie, es mejorar la comprensión de la complejidad computacional y descriptiva de los objetos matemáticos (por ejemplo, estructuras y relaciones) y sus problemas de clasificación asociados. Los investigadores proporcionarán información adicional sobre la relación subestimada entre la complejidad descriptiva de un conjunto y el espectro de grados bajo una relación de equivalencia dada. Las nuevas técnicas desarrolladas podrían ayudar a vincular de forma más fidedigna los campos de la teoría descriptiva de conjuntos y la teoría de estructuras computables.

Objetivo

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.