BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Paul Baginski (Fairfield University) DTSTART:20250521T193000Z DTEND:20250521T195500Z DTSTAMP:20260423T024834Z UID:CANT2025/36 DESCRIPTION:Title: Arithmetic Progressions\, Nonunique Factorization\, and Additive Combina torics in the Group of Units Mod $n$\nby Paul Baginski (Fairfield Univ ersity) as part of Combinatorial and additive number theory (CANT 2025)\n\ nLecture held in CUNY Graduate Center - Science Center (4th floor).\n\nAbs tract\nFor integers $0\\lt a\\leq b$\, the arithmetic progression $M_{a\,b }=a+b\\mathbb{N}$ is closed under multiplication if and only if $a^2\\equi v a \\mod b$. Any such multiplicatively closed arithmetic progression is c alled an arithmetic congruence monoid (ACM). Though these $M_{a\,b}$ are m ultiplicative submonoids of $\\mathbb{N}$\, their factorization properties differ greatly from the unique factorization one enjoys in $\\mathbb{N}$. \n\nIn this talk we will explore the known factorization properties of the se monoids. When $a=1$\, these monoids are Krull and behave similarly to a lgebraic number rings\, in that they have a class group which controls all the factorization. Combinatorially\, factorization properties correspond to zero-sum sequences in the group. However\, when $a\\gt 1$\, these monoi ds are not Krull and thus do not have a class group which fully captures t he factorization behavior. Nonetheless\, an ACM can be associated to a fin ite abelian group\, whose additive combinatorics relate to the factorizati on properties of the ACM. We will pay particular attention to the factoriz ation property of elasticity and its connection to sequences in the group which attain certain sums while avoiding others.\n LOCATION:https://researchseminars.org/talk/CANT2025/36/ END:VEVENT END:VCALENDAR