Event Detail
Fri Oct 9, 2015–Sat Oct 10, 2015 11:10 PM–1 AM |
Logic Colloquium Takayuki Kihara (JSPS Postdoctoral Fellow, Department of Mathematics, UC Berkeley) Recursion Theoretic Methods in Descriptive Set Theory and Infinite Dimensional Topology |
The notion of degree spectrum of a structure in computable model theory is defined as the collection of all Turing degrees of presentations of the structure. We introduce the degree spectrum of a represented space as the class of collections of all Turing degrees of presentations of points in the space. The notion of point degree spectrum creates a connection among various areas of mathematics including computability theory, descriptive set theory, infinite dimensional topology and Banach space theory.
Through this new connection, for instance, we construct a family of continuum many infinite dimensional Cantor manifolds with property C in the sense of Haver/Addis-Gresham whose Borel structures at an arbitrary finite rank are mutually non-isomorphic. This provides new examples of Banach algebras of real valued Baire class two functions on metrizable compacta, and strengthen various theorems in infinite dimensional topology such as Polâ€™s solution to Alexandrovâ€™s old problem.
To prove our main theorem, an invariant which we call degree co-spectrum, a collection of Turing ideals realized as lower Turing cones of points of a Polish space, plays a key role. The key idea is measuring the quantity of all possible Scott ideals (omega-models of WKL) realized within the degree co-spectrum (on a cone) of a given space. This is joint work with Arno Pauly (University of Cambridge, UK).