Small Covers over Prisms

In this paper we calculate the number of equivariant diffeomorphism classes of small covers over a prism

Random strings and tt-degrees of Turing complete C.E. sets

We investigate the truth-table degrees of (co-)c.e.\ sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefix-free machine is Turing complete, but that truth-table completeness depends on the choice of universal machine. We show that for such sets of random strings, any finite set of their truth-table degrees do not meet to the degree~0, even within the c.e. truth-table degrees, but when taking the meet over all such truth-table degrees, the infinite meet is indeed~0. The latter result proves a conjecture of Allender, Friedman and Gasarch. We also show that there are two Turing complete c.e. sets whose truth-table degrees form a minimal pair.Comment: 25 page

On the Nonexistence of a Strong Minimal Pair

Two nonzero recursively enumerable (r.e.) degrees $\mathbf{a}$ and $\mathbf{b}$ form a strong minimal pair if $\mathbf{a} \wedge \mathbf{b}=\mathbf{0}$ and $\mathbf{b}\vee \mathbf{x}\geq \mathbf{a}$ for any nonzero r.e. degree $\mathbf{x}\leq \mathbf{a}$. We prove that there is no strong minimal pair in the r.e. degrees. Our construction goes beyond the usual $\mathbf{0}'''$-priority arguments and we give some evidence to show that it needs $\mathbf{0}^{(4)}$-priority arguments

The Complements of Lower Cones of Degrees and the Degree Spectra of Structures

We study Turing degrees a for which there is a countable structure whose degree spectrum is the collection {x : x ≰ a}. In particular, for degrees a from the interval [0′, 0″], such a structure exists if a′ = 0″, and there are no such structures if a″ \u3e 0‴

Elements of Classical Recursion Theory: Degree-Theoretic PROPERTIES AND COMBINATORIAL PROPERTIES

In Recursion Theory (Computability Theory), we study Turing degrees in terms of their degree-theoretic properties and combinatorial properties. In this dissertation we present several results in terms of connections either between these two categories of properties or within each category. Our first main result is to build a strong connection between array nonrecursive degrees and relatively recursively enumerable degrees. The former is a combinatorial property and the latter is a degree-theoretic one. We prove that a degree is array nonrecursive if and only if every degree above it is relatively recursively enumerable. This result has a corollary which generalizes Ishmukhametov’s classification of r.e. degrees with strong minimal covers to the class of n-REA degrees. Then we produce new connections between minimality and jump classes, both are degree-theoretic. By using more and more complicated structures, we can finally build a minimal cover over a minimal degree (which we call a 2-minimal degree) which is GH1, and this is the highest jump class we can reach by finite iterations of minimality. This result answers a question by Lewis and Montalbán