Logan Heath

I am a Ph.D. student at the University of Wisconsin–Madison studying mathematical logic with Uri Andrews. My interests are in the area of computable model theory and degree spectra of theories (the spectrum of a complete theory in a computable language is the collection of Turing degrees of countable models of the theory). Questions I am interested in generally have one of two forms:

1. If S is the spectrum of a theory, what sorts of theories can have S as their spectrum? For example, the TA degrees (degrees of nonstandard models of true arithmetic) are known to be the spectrum of a theory, but not the spectrum of any ω-stable theory (see Andrews and Knight (2013) and Andrews et al. (2017)). Can the TA degrees be the spectrum of a stable or even superstable theory? This question has led me to consider a related collection of degrees: the subuniform upper bounds of the arithmetic sets. This collections is the spectrum of a superstable, atomic theory, but not the spectrum of any small theory. See my talks on this topic here and here.

2. For a given collection C of Turing degrees, is C the spectrum of a theory? An instance of this questions is, "when is the union of the upper cones for a pair of incomparable Turing degrees a theory spectrum?" We know that when the pair of degrees is arithmetically equivalent, then we can obtain the corresponding union of cones as a theory spectrum and that if the pair of degrees are not arithmetically comparable, then we cannot obtain the union of cones as a theory spectrum. See my talk on this topic at the Midwest Computability Seminar here.

I am on the job market as of Fall 2026.

e-mail: laheath (at) wisc (dot) edu
Office: 516 Van Vleck Hall