Electronic Research Announcements

February 2019 , Volume 26

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Cluster algebras with Grassmann variables
Valentin Ovsienko and MichaeL Shapiro
2019, 26: 1-15 doi: 10.3934/era.2019.26.001 +[Abstract](525) +[HTML](208) +[PDF](390.98KB)

We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of "extended quivers," which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This paper is a step towards understanding the notion of cluster superalgebra.

Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows
Livio Flaminio and Giovanni Forni
2019, 26: 16-23 doi: 10.3934/era.2019.26.002 +[Abstract](283) +[HTML](201) +[PDF](335.03KB)

We derive, from the work of M. Ratner on joinings of time-changes of horocycle flows and from the result of the authors on its cohomology, the property of orthogonality of powers for non-trivial smooth time-changes of horocycle flows on compact quotients. Such a property is known to imply P. Sarnak's Möbius orthogonality conjecture, already known for horocycle flows by the work of J. Bourgain, P. Sarnak and T. Ziegler.

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