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We resolve the long-standing problem of constructing the action of the operad of framed (stable) genus-0 curves on Hamiltonian Floer theory; this operad is equivalent to the framed E 2 operad. We formulate the construction in the following general context: we associate to each compact subset of a closed symplectic manifold a new chain-level model for symplectic cohomology with support, which we show carries an action of a model for the chains on the moduli space of framed genus 0 curves. This construction turns out to be strictly functorial with respect to inclusions of subsets, and the action of the symplectomorphism group. In the general context, we appeal to virtual fundamental chain methods to construct the operations over fields of characteristic 0, and we give a separate account, over arbitrary rings, in the special settings where Floer's classical transversality approach can be applied. We perform all constructions over the Novikov ring, so that the algebraic structures we produce are compatible with the quantitative information that is contained in Floer theory. Over fields of characteristic 0, our construction can be combined with results in the theory of operads to produce explicit operations encoding the structure of a homotopy BV algebra. In an appendix, we explain how to extend the results of the paper from the class of closed symplectic manifolds to geometrically bounded ones. [ABSTRACT FROM AUTHOR]

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring R and a module-finite R -algebra Λ, we study the set tors Λ (respectively, torf Λ) of torsion (respectively, torsionfree) classes of the category of finitely generated Λ-modules. We construct a bijection from torf Λ to ∏ p torf (κ (p) ⊗ R Λ) , and an embedding Φ t from tors Λ to T R (Λ) : = ∏ p tors (κ (p) ⊗ R Λ) , where p runs over all prime ideals of R. When Λ = R , these give classifications of torsionfree classes, torsion classes and Serre subcategories of mod R due to Takahashi, Stanley-Wang and Gabriel. To give a description of Im Φ t , we introduce the notion of compatible elements in T R (Λ) , and prove that all elements in Im Φ t are compatible. We give a sufficient condition on (R , Λ) such that all compatible elements belong to Im Φ t (we call (R , Λ) compatible in this case). For example, if R is semi-local and dim ⁡ R ≤ 1 , then (R , Λ) is compatible. We also give a sufficient condition in terms of silting Λ-modules. As an application, for a Dynkin quiver Q , (R , R Q) is compatible and we have a poset isomorphism tors R Q ≃ Hom poset (Spec R , C Q) for the Cambrian lattice C Q of Q. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This generalizes recent related results of Ghandehari-Lee-Ludwig-Spronk-Turowska [11] about the spectrum of Beurling-Fourier algebras on some Lie groups. In the case of discrete groups we show that the spectrum of Beurling-Fourier algebra is homeomorphic to the group itself. [ABSTRACT FROM AUTHOR]

Cluster algebra structures for Grassmannians and their (open) positroid strata are controlled by a Postnikov diagram D or, equivalently, a dimer model on the disc, as encoded by either a bipartite graph or the dual quiver (with faces). The associated dimer algebra A , determined directly by the quiver with a certain potential, can also be realised as the endomorphism algebra of a cluster-tilting object in an associated Frobenius cluster category. In this paper, we introduce a class of A -modules corresponding to perfect matchings of the dimer model of D and show that, when D is connected, the indecomposable projective A -modules are in this class. Surprisingly, this allows us to deduce that the cluster category associated to D embeds into the cluster category for the appropriate Grassmannian. We show that the indecomposable projectives correspond to certain matchings which have appeared previously in work of Muller–Speyer. This allows us to identify the cluster-tilting object associated to D , by showing that it is determined by one of the standard labelling rules constructing a cluster of Plücker coordinates from D. By computing a projective resolution of every perfect matching module, we show that Marsh–Scott's formula for twisted Plücker coordinates, expressed as a dimer partition function, is a special case of the general cluster character formula, and thus observe that the Marsh–Scott twist can be categorified by a particular syzygy operation in the Grassmannian cluster category. [ABSTRACT FROM AUTHOR]

In this paper we develop a new groupoid-based structure theory for the class of regular ⁎-semigroups. This class occupies something of a 'sweet spot' between the important classes of inverse and regular semigroups, and contains many natural examples. Some of the most significant families include the partition, Brauer and Temperley-Lieb monoids, among other diagram monoids. Our main result is that the category of regular ⁎-semigroups is isomorphic to the category of so-called 'chained projection groupoids'. Such a groupoid is in fact a triple (P , G , ε) , where: • P is a projection algebra (in the sense of Imaoka and Jones), • G is an ordered groupoid with object set P , and • ε : C → G is a special functor, where C is a certain natural 'chain groupoid' constructed from P. Roughly speaking: the groupoid G = G (S) remembers only the 'easy' products in a regular ⁎-semigroup S ; the projection algebra P = P (S) remembers only the 'conjugation action' of the projections of S ; and the functor ε = ε (S) tells us how G and P 'fit together' in order to recover the entire structure of S. In this way, we obtain the first completely general structure theorem for regular ⁎-semigroups. As a consequence of our main result, we give a new proof of the celebrated Ehresmann–Schein–Nambooripad Theorem, which establishes an isomorphism between the categories of inverse semigroups and inductive groupoids. Other applications will be given in future works. We consider several examples along the way, and pose a number of problems that we believe are worthy of further attention. [ABSTRACT FROM AUTHOR]

We relate the group structure of van der Kallen on orbit sets of unimodular rows ([26]) with values in a smooth algebra A over a field k with the motivic cohomotopy groups of X = Spec A with coefficients in A n ∖ 0 in the sense of [5]. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by A n + 1 ∖ 0 or, equivalently, by an A 1 -weakly equivalent quadric Q 2 n + 1 to that considered in [5] , defined by a quadric Q 2 n , by means of explicit morphisms Q 2 n + 1 → Q 2 n , Q 2 n × G m → Q 2 n + 1 of quadrics. [ABSTRACT FROM AUTHOR]