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The aim of this paper is to study a poset isomorphism between two support \tau-tilting posets. We take several information from combinatorial properties of support \tau-tilting posets and give an analogue of Happel-Unger's reconstruction theorem. [ABSTRACT FROM AUTHOR]

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C^\ast-algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C^\ast-algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space. [ABSTRACT FROM AUTHOR]

The Witt ring of a complex flag variety describes the interesting – i.e. torsion – part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides a few well-studied examples such as full flag varieties and projective spaces, this class includes many flag varieties whose Witt rings were previously unknown, including many flag varieties of exceptional types. In particular, it includes all flag varieties of types G_2 and F_4. The results also extend to flag varieties over other algebraically closed fields. [ABSTRACT FROM AUTHOR]

We extend Nekrashevych's KK-duality for C^*-algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid (G,E) acting faithfully on a finite directed graph E, we associate two C^*-algebras, \mathcal {O}(G,E) and \widehat {\mathcal {O}}(G,E), to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in KK-theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author. [ABSTRACT FROM AUTHOR]

We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a 2\times n sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic. [ABSTRACT FROM AUTHOR]