BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Antonio Nakid Cordero (University of Wisconsin) DTSTART:20250410T180000Z DTEND:20250410T190000Z DTSTAMP:20260423T021236Z UID:OLS/172 DESCRIPTION:Title: Ma rtin's conjecture in the enumeration degrees\nby Antonio Nakid Cordero (University of Wisconsin) as part of Online logic seminar\n\n\nAbstract\n Martin's conjecture is a long open problem that seeks to prove the empiric al observation that "naturally occurring" Turing degrees are well-ordered. The conjecture posits that the only natural constructions of incomputable degrees arise from iterations of the Turing jump. Even though the full co njecture remains open\, several significant partial results have been obta ined both in the Turing degrees and by translating the conjecture to other degree structures.\n\n The study of the enumeration degrees has gained r elevance in recent years for their applications to effective mathematics a nd for their structural connections to the Turing degrees. In this settin g\, Martin's conjecture is relevant due to the existence of a definable co py of the Turing degrees inside the enumeration degrees and two natural op erations that extend the Turing jump: the enumeration jump and the skip. H owever\, the unique features of the enumeration degrees pose challenges to even formulating an analogue to Martin's conjecture.\n\n I will present a surprising positive result based on Bard's local approach to the uniform Martin's conjecture. From this\, we can prove part 1 of Martin's conjectu re for uniformly Turing-to-enumeration invariant functions. Additionally\, I discuss several counterexamples\, including an invariant function in th e enumeration degrees that fails to be uniformly invariant.\n LOCATION:https://researchseminars.org/talk/OLS/172/ END:VEVENT END:VCALENDAR