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A longstanding goal of biology is to identify the key genes and species that critically impact evolution, ecology, and health. Network analysis has revealed keystone species that regulate ecosystems and master regulators that regulate cellular genetic networks. Yet these studies have focused on pairwise biological interactions, which can be affected by the context of genetic background and other species present generating higher-order interactions. The important regulators of higher-order interactions are unstudied. To address this, we applied a new high-dimensional geometry approach that quantifies epistasis in a fitness landscape to ask how individual genes and species influence the interactions in the rest of the biological network. We then generated and also reanalyzed 5-dimensional datasets (two genetic, two microbiome). We identified key genes (e.g. the rbs locus and pykF) and species (e.g. Lactobacilli) that control the interactions of many other genes and species. These higher-order master regulators can induce or suppress evolutionary and ecological diversification by controlling the topography of the fitness landscape. Thus, we provide mathematical intuition and justification for exploration of biological networks in higher dimensions., Comment: 77 pages, various figures

We revisit the geometric description of cluster categories in type E in terms of colored diagonals in a polygon and generalize it to the case of m-cluster categories. As an application, we relate colored diagonals in a polygon to semi-standard Young tableaux, in type E_6,E_7,E_8. This provides a new compatibility description of semi--standard Young tableaux in Grassmannian cluster algebras in type E_6, E_8 and in a sub-cluster algebra of type E_7.

We study cluster tilting modules in mesh algebras of Dynkin type, providing a new proof for their existence. In all but one case, we show that these are precisely the maximal rigid modules, and that they are equivariant for a certain automorphism. We further study their mutation, providing an example of mutation in an abelian category which is not stably 2-Calabi-Yau, and explicitly describe the combinatorics., Comment: comments appreciated; the third version includes a discussion on the combinatorics of the mutations

Beerenwinkel et al.(2007) suggested studying fitness landscapes via regular subdivisions of convex polytopes. Building on their approach we propose cluster partitions and cluster filtrations of fitness landscapes as a new mathematical tool. In this way, we provide a concise combinatorial way of processing metric information from epistatic interactions. Using existing Drosophila microbiome data, we demonstrate similarities with and differences to the previous approach. As one outcome we locate interesting epistatic information where the previous approach is less conclusive.

In this paper, we describe a class of elements in the ring of $\mathrm{SL}(V)$-invariant polynomial functions on the space of configurations of vectors and linear forms of a 3-dimensional vector space $V.$ These elements are related to one another by an induction formula using Chebyshev polynomials. We also investigate the relation between these polynomials and G. Lusztig's dual canonical basis in tensor products of representations of $U_q(\mathfrak{sl}_3(\mathbb C)).$, Comment: 41 pages. Exposition shortened and clarified following the suggestions of the anonymous referees. Figures added. Comments are welcome

In this paper we give a geometric-combinatorial description of the cluster categories of type E. In particular, we give an explicit geometric description of all cluster tilting objects in the cluster category of type E_6. The model we propose here arises from combining two polygons, and it generalises the description of the cluster category of type A and D., Comment: Following the suggestion of the referee we significantely extended Section 3, 4 and 6. To appear in the Journal of Algebraic Combinatorics

The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a geometric interpretation of the triangulated structure of $m$-cluster categories. Furthermore, we characterize all the connected components arising from a cluster category when taking the $m$-th power of its Auslander-Reiten quiver.