Karthik
Ravishankar.

I’m a fifth-year PhD student in mathematics at the University of Wisconsin–Madison, working with Joe Miller in computability theory. Computability is a branch of mathematical logic that studies the limits of algorithms. Classifying problems by their relative degrees of unsolvability gives rise to the rich structure of the Turing degrees, providing a way to calibrate computational strength.

I’m interested in local and global questions about the Turing and enumeration degrees, particularly questions of structure and definability. In past work on Ahmad pairs, I showed that the left half must be ${\text{low}}_3$ and the right half ${\text{high}}_2$. This separated the two halves and led to natural ${\mathrm{\Pi }}_3$ definitions of ${\text{low}}_3$ and ${\text{high}}_2$ in the local structure. A current project focuses on identifying the class of degrees arising as joins of $𝒦$-pairs in the global structure and exploring candidate definitions of cototality.

In computable structure theory, my past work explored degree spectra and their restriction to the ${\mathrm{\Delta }}_2^0$ degrees, as well as negative results on unions of spectra. A current project examines the information that can be encoded in the double jumps of linear orders.

More recently, I’ve become interested in Lean and the formalization of mathematics. I’m currently working on formalizing some theorems in computability theory in Lean.