{"id":1065,"date":"2022-02-25T19:30:03","date_gmt":"2022-02-25T18:30:03","guid":{"rendered":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/?page_id=1065"},"modified":"2023-12-12T13:00:49","modified_gmt":"2023-12-12T12:00:49","slug":"courses-2022","status":"publish","type":"page","link":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/courses\/courses-2022\/","title":{"rendered":"Courses 2022"},"content":{"rendered":"

Dal 28 novembre 2022 al 02 dicembre 2022 la Dott.ssa Laura Cossu, University of Graz, terra’ un corso di dottorato su: Factorizations in monoids and rings<\/span><\/p>\n


\nMany problems in algebra involve the decomposition of certain elements of a ring (or more generally<\/span>
of a monoid) into a product of certain other elements (hereinafter generically referred to as building<\/span>
blocks) that are in some sense minimal. The classical theory of factorization investigates factorizations<\/span>
in which the building blocks are atoms, i.e., non-unit elements of a monoid that are not products of<\/span>
two non-units. For example, it is well known that every non-zero non-unit of a Dedekind domain (more<\/span>
generally, of a Noetherian domain) can be written as a finite product of atoms and that in general<\/span>
such decompositions are not unique. On the other hand, examples of factorizations that lie beyond the<\/span>
scope of the classical theory include additive decompositions into multiplicative units in rings; cyclic<\/span>
decompositions of permutations in the symmetric group of degree n; idempotent factorizations of the<\/span>
\u201csingular elements\u201d of a monoid; and so on.<\/span>
After an introductory overview of the history and main results in the classical framework, in this<\/span>
course, we combine the language of monoids and preorders (partial orders that are not necessarily an-<\/span>
tisymmetric) to make first steps towards the construction of a \u201cunified theory of factorization\u201d.<\/span> In<\/span>
particular, we prove an abstract existence theorem that recover, among others, a classical theorem of<\/span>
Cohn on atomic factorizations in cancellative monoids, a classical result by Anderson and Valdes-Leon<\/span>
on \u201cirreducible factorizations\u201d in commutative monoids, but also a theorem by Erdos on idempotent fac-<\/span>
torizations of square singular matrices over fields. We also introduce a notion of \u201cminimal factorization\u201d,<\/span>
suitable for the new abstract setting, and we qualify and quantify the related non-uniqueness properties<\/span>
in specific cases but also in some generality. Examples will help to motivate and illustrate the theory.<\/span>
Prerequisites:<\/span> Standard knowledge of (basic) algebraic structures and (binary) relations.<\/span>
Duration of the course:<\/span> 10 hours.<\/span>
Exam:<\/span> Written. Writing of a short essay.<\/span>
References<\/span>
[1] L. Cossu and S. Tringali,<\/span> Abstract Factorization Theorems with Applications to Idempotent Factor-<\/span>
izations<\/span>, e-print (Aug. 2021),<\/span> arXiv:2108.12379<\/span>.<\/span>
[2] L. Cossu and S. Tringali,<\/span> Factorization under Local Finiteness Conditions<\/span>, e-print (Aug. 2022),<\/span>
arXiv:2208.05869<\/span>.<\/span>
[3] A. Geroldinger and F. Halter-Koch,<\/span> Non-Unique Factorizations. Algebraic, Combinatorial and An-<\/span>
alytic Theory<\/span>, Pure Appl. Math.<\/span> 278<\/span>, Chapman & Hall\/CRC, Boca Raton (FL), 2006.<\/span>
[4] S. Tringali,<\/span> An Abstract Factorization Theorem and Some Applications<\/span>, J. Algebra 602 (2022), 352\u2013<\/span>
380.<\/span>
1<\/span><\/p>\n

Il calendario delle lezioni \u00e8 il seguente: Luned\u00ec 28 novembre dalle 16:00 alle 18:00 Aula B Marted\u00ec 29 novembre dalle 16:00 alle 18:00 Aula A Mercoled\u00ec 30 novembre dalle 16:00 alle 18:00 Aula A Gioved\u00ec 1 dicembre dalle 16:00 alle 18:00 Aula A Venerd\u00ec 2 dicembre dalle 16:00 alle 18:00 Aula A<\/p>\n

 <\/p>\n

\n

Dal 20 al 30 giugno i Visiting Professors M. Lucia e P. Sicbaldi terranno\u00a0un\u00a0corso intitolato\u00a0“Geometric\u00a0Analysis”\u00a0per\u00a0un\u00a0totale\u00a0di 20 ore di lezione (dieci ciascuno). Le informazioni si trovano al link: https:\/\/www.unica.it\/unica\/page\/it\/antonio_greco_avs_geometric_analysis<\/a><\/p>\n

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Dal 28 novembre 2022 al 02 dicembre 2022 la Dott.ssa Laura Cossu, University of Graz, terra’ un corso di dottorato su: Factorizations in monoids and rings Many problems in algebra involve the decomposition of certain elements […]<\/p>\n","protected":false},"author":3006,"featured_media":0,"parent":161,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1065","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages\/1065","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/users\/3006"}],"replies":[{"embeddable":true,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/comments?post=1065"}],"version-history":[{"count":13,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages\/1065\/revisions"}],"predecessor-version":[{"id":1273,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages\/1065\/revisions\/1273"}],"up":[{"embeddable":true,"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/pages\/161"}],"wp:attachment":[{"href":"https:\/\/dottorati.unica.it\/matematicaeinformatica\/wp-json\/wp\/v2\/media?parent=1065"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}