BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Arkady Berenstein DTSTART:20240313T003000Z DTEND:20240313T013000Z DTSTAMP:20260423T011111Z UID:NIS2024/14 DESCRIPTION:Title: Noncommutative surfaces\, clusters\, and their symmetries\nby Arkady Berenstein as part of Seminar-Type Workshop on Noncommutative Integrable S ystems\n\n\nAbstract\nThe aim of my talk (based on joint work in progress with Min Huang and Vladimir Retakh) is to introduce and study certain nonc ommutative algebras $A$ for any marked surface. These algebras admit nonco mmutative clusters\, i.e.\, embeddings of a given group $G$ which is eithe r free or one-relator (we call it triangle group) into the multiplicative monoid $A^\\times$. The clusters are parametrized by triangulations of the surface and exhibit a noncommutative Laurent Phenomenon\, which asserts t hat generators of the algebra can be written as sums of the images of elem ents of $G$ for any noncommutative cluster. If the surface is unpunctured\ , then our algebra $A$ can be specialized to the ordinary quantum cluster algebra\, and the noncommutative Laurent Phenomenon becomes the (positive) quantum one. \n\n It turns out that there is a natural action of a certai n braid-like group $Br_A$ by automorphisms of $G$ on each cluster in a com patible way (this is\, indeed\, the braid group $Br_n$ if the surface is a n unpunctured disk with n+2 marked boundary points). If surface is punctur ed\, the algebra $A$ admits a family of commuting automorphisms which will give new clusters and new "tagged" noncommutative Laurent Phenomena. \n\ nThere are important elements in $A$ assigned to each marked point\, which we refer to as noncommutative angles (or h-lengths). They belong to the g roup algebra of each cluster group and are invariant under all noncommutat ive cluster mutations. This eventually gives rise to noncommutative integr able systems on unpunctured cylinders and other surfaces which\, in partic ular\, recover the ones introduced by Kontsevich in 2011 together with the ir Laurentness and positivity.\n LOCATION:https://researchseminars.org/talk/NIS2024/14/ END:VEVENT END:VCALENDAR