BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Patrick Lutz (UC Berkeley) DTSTART:20200901T200000Z DTEND:20200901T210000Z DTSTAMP:20260423T022714Z UID:CTA/20 DESCRIPTION:Title: Par t 1 of Martin's Conjecture for Order Preserving Functions\nby Patrick Lutz (UC Berkeley) as part of Computability theory and applications\n\n\nA bstract\nMartin's conjecture is an attempt to make precise the idea that t he only natural functions on the Turing degrees are the constant functions \, the identity\, and transfinite iterates of the Turing jump. The conject ure is typically divided into two parts. Very roughly\, the first part sta tes that every natural function on the Turing degrees is either eventually constant or eventually increasing and the second part states that the nat ural functions which are increasing form a well-order under eventual domin ation\, where the successor operation in this well-order is the Turing jum p. \n\nIn the 1980's\, Slaman and Steel proved that the second part of Mar tin's conjecture holds for order-preserving Borel functions. In joint work with Benny Siskind\, we prove the complementary result that (assuming ana lytic determinacy) the first part of the conjecture also holds for order-p reserving Borel functions (and under AD\, for all order-preserving functio ns). Our methods also yield several other new results\, including an equiv alence between the first part of Martin's conjecture and a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees. \n\nIn my talk\, I will give an overview of Martin's conjecture and then describe ou r new results.\n LOCATION:https://researchseminars.org/talk/CTA/20/ END:VEVENT END:VCALENDAR