BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alfred Geroldinger (University of Graz\, Austria) DTSTART:20260713T153000Z DTEND:20260713T155500Z DTSTAMP:20260710T111704Z UID:CANT2026/14 DESCRIPTION:Title: On factorizations of zero-sum sequences over abelian torsion groups I\nby Alfred Geroldinger (University of Graz\, Austria) as part of Combina torial and additive number theory seminar (CANT 2026)\n\nLecture held in S cience Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nLet $G $ be an additive abelian group and let $G_0\\subseteq G$ be a subset. A ze ro-sum sequence over $G_0$ is an unordered string of terms from $G_0$ (rep etition of terms allowed) such that the sum of terms is $0$. The study of zero-sum sequences dates back over 60 years\, and while they have often be en studied for purely combinatorial interest\, the original motivation was due to connections with factorization in rings of integers in algebraic n umber fields. In the last few decades\, the connection between factorizati ons of zero-sum sequences and factorization of elements in rings of intege rs was made more precise and extended into much more general algebraic set tings. This then allows the algebraic structure of factorization to be stu died via combinatorial properties of zero-sum sequences. We briefly review this connection\, making all notions concrete\, and then turn our focus t o the combinatorial part. In the past\, there has been much focus on finit e groups\, and more recently\, on subsets of finitely generated groups. Ho wever\, very little was known in the case of non-finitely generated abelia n groups. In part\, this is because common invariants used to study factor ization\, such as the Davenport Constant\, are no longer guaranteed to be finite. In order to better understand factorization in the setting of infi nite abelian torsion groups\, we introduce a new technique measuring the s ize of a sequence not by the number of its terms but rather by its cross n umber\, $\\sum_{i=1}^{\\ell} \\frac{1}{\\text{\\rm ord} (g_i)}$\, where th e $g_i\\in G_0 \\subseteq G$ are the terms in the sequence. Cross numbers have previously been used almost solely for finite groups. In order to ada pt their use into the infinite torsion group setting\, we need to introduc e a new invariant\, $\\mathsf K_{\\mathsf{inf}}(G_0)$\, defined as the inf imum of all cross numbers of nontrivial zero-sum sequences with terms from $G_0$. This then sets up dichotomy between when $\\mathsf K_{\\mathsf{inf }}(G_0)=0$ and when $\\mathsf K_{\\mathsf{inf}}(G_0)>0$. In this first par t of two talks\, we focus on when $\\mathsf K_{\\mathsf{inf}}(G_0)=0$\, an d show that factorization of zero-sum sequences can be very ill-behaved un der this assumption. In the follow-up talk\, we then instead consider when $\\mathsf K_{\\mathsf{inf}}(G_0)>0$ and see that this instead guarantees that factorization must be well-behaved\, as measured by the finiteness of several commonly factorization metrics. This is joint work with David J. Grynkiewicz and Guoqing Wang.\n LOCATION:https://researchseminars.org/talk/CANT2026/14/ END:VEVENT END:VCALENDAR