Seminar | May 5 | 2:15-3:30 p.m. | 3 LeConte Hall
Denis Hirschfeldt, University of Chicago
I will discuss some recent work and possible future developments in computability theory, with an inevitable bias towards my own interests. The study of Turing reducibility and the Turing degrees has long been the backbone of the area, but many other notions of computability-theoretic comparison, old and new, have been intensely studied in the last few years. These notions, and the ways they allow us to view computability theory as the study of the fine structure arising from interactions between definability and computation, can serve as a guiding principle to exploring the many ways in which the field has expanded over the last couple of decades or so. Some of this work is “pure” computability theory, but much of it has focused on connections with other areas of mathematics. Within mathematical logic, three examples are computable model theory; reverse mathematics, which connects computability theory with proof theory; and the study of the effective mathematics of the uncountable, which often requires an engagement with set theory. I hope to touch on these and other areas to give a glimpse, however incomplete, of the field, eighty years or so after its beginnings.