Abstract. We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type A. We prove that such sequences have length n + t, where n is the number of vertices and t is the number of 3-cycles in the quiver. Moreover, we develop a procedure that yields these minimal length maximal green sequences.

The canonical join complex of a semidistributive lattice is a simplicial complex whose faces are canonical join representations of elements of the semidistributive lattice. We give a combinatorial classification of the faces of the canonical join complex of the lattice of biclosed sets of segments supported by a tree, as introduced by the third author and McConville. We also use our classification to describe the elements of the shard intersection order of the lattice of biclosed sets. As a consequence, we prove that this shard intersection order is a lattice.

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We generalize the concept of rigid inner forms, defined by Kaletha in [Kal16], to the setting of a local function field F in order state the local Langlands conjectures for arbitrary connected reductive groups over F . To do this, we define for a connected reductive group G overfppf (F, u) for a certain canonically-defined profinite commutative group scheme u, building up to an analogue of the classical Tate-Nakayama duality theorem. We define a relative transfer factor for an endoscopic datum serving a connected reductive group G over F , and use rigid inner forms to extend this to an absolute transfer factor, enabling the statement of endoscopic conjectures relating stable virtual characters and ṡ-stable virtual characters for a semisimple ṡ associated to a tempered Langlands parameter.