Section: New Results

Degree spectra of Polish spaces

• Participants: Mathieu Hoyrup, Takayuki Kihara, Victor Selivanov

Mathematical objects can encode information. An obvious example is given by subsets of the plane: a text printed on a sheet of paper is a subset of the plane conveying information. However, when the object is submitted to deformations, what information can still be conveyed? What information is invariant under such deformations?

It is the core question in computable structure theory: for instance, what can be encoded in an infinite graph, which can be decoded from the structure itself and not from a particular presentation of the graph? Mathematically, what information is robust under graph isomorphism? It happens that much information can be encoded, for instance by using the lengths of the cycles in the graph.

Albegraic structures have been thoroughly studied from this perspective. However, the study of topological structures is almost inexistant, and more difficult (they are continuous while algebraic structures are often discrete). For instance, what information can be encoded in a subset of the plane, which is stable under continuous deformations (homeomorphisms)?

We have tackled this question during the visit of Takayuki Kihara and Victor Selivanov, and obtained many interesting results. For instance, we have proved that no direct information can be encoded (for instance, no infinite binary sequence can be extracted by an algorithm, unless the sequence is already computable). However, limit information can be encoded (for instance, a binary sequence can be encoded in such a way that a double-sequence converging to it can be extracted from the object by an algorithm). It is still open whether a single limit is possible.

A paper is still in preparation.