We develop a purely combinatorial theory of limit linear series on metric graphs. This will be based on the formalisms of hypercube rank functions and slope structures. We provide a full classification of combinatorial limit linear series of rank one, and discuss connections to other concepts in tropical algebra and combinatorial algebraic geometry.

We construct, for all g\ge 2 and n\ge 0 , a spectral sequence of rational S_{n} -representations which computes the S_{n} -equivariant rational cohomology of the tropical moduli spaces of curves \Delta_{g,n} in terms of compactly supported cohomology groups of configuration spaces of n points on graphs of genus g . Using the canonical S_{n} -equivariant isomorphisms \widetilde{H}^{i-1}(\Delta_{g,n};\mathbb{Q}) \cong W_{0} H^{i}_{c}(\mathcal{M}_{g,n};\mathbb{Q}) , we calculate the weight 0 , compactly supported rational cohomology of the moduli spaces \mathcal{M}_{g,n} in the range g=3 and n\le 9 , with partial computations available for n\le 13 .

We explicitly realize an internal action of the symplectic cactus group, recently defined by Halacheva for any complex, reductive, finite-dimensional Lie algebra, on crystals of Kashiwara–Nakashima tableaux. Our methods include a symplectic version of jeu de taquin due to Sheats and Lecouvey, symplectic reversal, and virtualization due to Baker. As an application, we define and study a symplectic version of the Berenstein–Kirillov group and show that it is a quotient of the symplectic cactus group. In addition two relations for symplectic Berenstein–Kirillov group are given that do not follow from the defining relations of the symplectic cactus group.

In this paper we study stable finiteness of ample groupoid algebras with applications to inverse semigroup algebras and Leavitt path algebras, recovering old results and proving some new ones. In addition, we develop a theory of (faithful) traces on ample groupoid algebras, mimicking the C^{\ast} -algebra theory but taking advantage of the fact that our functions are simple and so do not have integrability issues, even in the non-Hausdorff setting. The theory of traces is closely connected with the theory of invariant means on Boolean inverse semigroups. It turns out that for Hausdorff ample groupoids with compact unit space, having a stably finite algebra over some commutative ring implies the existence of a tracial state on its reduced C^{*} -algebra. We include an appendix on stable finiteness of more general semigroup algebras, improving on an earlier result of Munn, which is independent of the rest of the paper.

We provide a complete classification of the singularities of cluster algebras of finite cluster type. This extends our previous work on the case of trivial coefficients. Additionally, we classify the singularities of cluster algebras of rank two.

We prove a conjecture of Chapoton from 2010 stating that the pre-Lie operad, as a Lie algebra in the symmetric monoidal category of linear species, is freely generated by the free operad on the species of cyclic Lie elements.

In this paper we prove that an RGD-system over \mathbb{F}_{2} with prescribed commutation relations exists if and only if the commutation relations are Weyl-invariant and can be realized in the group U_{+} . This result gives us a machinery to produce new examples of RGD-systems with complicated commutation relations. We also discuss some applications of this result.

We provide a new branching rule from the general linear group \mathrm{GL}_{2n}(\mathbb{C}) to the symplectic group \mathrm{Sp}_{2n}(\mathbb{C}) by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a fixed shape to a disjoint union of several copies of sets of symplectic tableaux of various shapes. The algorithm arises from representation theory of a quantum symmetric pair of type A\mathrm{II}_{2n-1} , which is a q -analogue of the classical symmetric pair (\mathfrak{gl}_{2n}(\mathbb{C}), \mathfrak{sp}_{2n}(\mathbb{C})) .

Motivated by the study of profinite topology in branch groups, we prove a structural result about their finitely generated subgroups. More precisely, we show that finitely generated subgroups of a branch group with the subgroup induction property have a block structure, which roughly means that, up to a finite index, they are products of finite index subgroups, embedded in the group in a way that is coherent with its branch action on the rooted tree.

In this article, we consider tame \mathrm{SL}_{3} -frieze patterns that arise by specializing a cluster of Plücker variables in the coordinate ring of the Grassmannian \mathscr{G}(3,n) to 1 . We show how to calculate arbitrary entries of such frieze patterns from the cluster in question. Let \mathscr{F} be such a cluster. We study the set \mathscr{F}_{x} of cluster variables in \mathscr{F} that share a given index x and derive a structure theorem for \mathscr{F}_{x} . These sets prove central to calculating the first and last non-trivial rows of the frieze pattern. After that, simple recursive formulas can be used to calculate all remaining entries.