BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Kirsten Eisenträger (The Pennsylvania State University) DTSTART:20210301T213000Z DTEND:20210301T223000Z DTSTAMP:20260423T005725Z UID:CTA/44 DESCRIPTION:Title: A t opological approach to undefinability in algebraic extensions of the ratio nals\nby Kirsten Eisenträger (The Pennsylvania State University) as p art of Computability theory and applications\n\n\nAbstract\nIn 1970 Matiya sevich proved that Hilbert’s Tenth Problem over the\nintegers is undecid able\, building on work by Davis-Putnam-Robinson. Hilbert’s\nTenth Probl em over the rationals is still open\, but it could be resolved by\ngiving an existential definition of the integers inside the rationals. Proving\nw hether such a definition exists is still out of reach. However\, we will s how\nthat only “very few” algebraic extensions of the rationals have t he property\nthat their ring of integers are existentially or universally definable.\nEquipping the set of all algebraic extensions of the rationals with a natural\ntopology\, we show that only a meager subset has this pro perty. An important tool\nis a new normal form theorem for existential def initions in such extensions. As\na corollary\, we construct countably many distinct computable algebraic\nextensions whose rings of integers are nei ther existentially nor universally\ndefinable. Joint work with Russell Mil ler\, Caleb Springer\, and Linda Westrick.\n LOCATION:https://researchseminars.org/talk/CTA/44/ END:VEVENT END:VCALENDAR