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Cluster structures have been established on numerous algebraic varieties. These lectures focus on the Grassmannian variety and explain the cluster structures on it. The tools include dimer models on surfaces, associated algebras, and the study of associated module categories.

This if the final paper in the series Continuous Quivers of Type A. In this part, we generalize existing geometric models of type A cluster structures for the new E-clusters introduced in part (III). We also introduce an isomorphism of cluster theories and a weak equivalence of cluster theories. Examples of both are given. We use these geometric models and isomorphisms of cluster theories to begin classifying continuous type A cluster theories. We also introduce a continuous generalization of mutation. This encompasses mu- tation and (infinite) sequences of mutation. Then we link continuous mutation to our earlier geometric models. Finally, we introduce the space of mutations which generalizes the exchange graph of a cluster structure, and show that paths in this space are continuous mutations.

In a previous paper the authors have attached to each Dynkin quiver an associative algebra. The definition is categorical and the algebra is used to construct desingularizations of arbitrary quiver Grassmannians. In the present paper we prove that this algebra is isomorphic to an algebra constructed by Hernandez-Leclerc defined combinatorially and used to describe certain graded Nakajima quiver varieties. This approach is used to get an explicit realization of the orbit closures of representations of Dynkin quivers as affine quotients., Comment: 12 pages

This paper contains the material discussed in the series of three lectures that I gave during the workshop of the ICRA 2018 in Prague. I will introduce the reader to some of the techniques used in the study of the geometry of quiver Grassmannians. The notes are quite elementary and thought for phd students or young researchers. I assume that the reader is familiar with the representation theory of quivers., To appear in the proceedings of the workshop and International Conference on Representations of Algebras (ICRA) held in Prague in 2018. The proceeding will be published by AMS in the series Contemporary Mathematics

We consider the Grassmannian X = G r n − k ( C n ) and describe a ‘mirror dual’ Landau-Ginzburg model ( X ˇ ∘ , W q : X ˇ ∘ → C ) , where X ˇ ∘ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian X ˇ , and we express W succinctly in terms of Plucker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to one proposed for homogeneous spaces in a previous work by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990's by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to W q a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of X. We also prove a T-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology D-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's J-function, which was conjectured in a 1998 paper by Batyrev, Ciocan-Fontanine, Kim and van Straten.

Motivated by the theory of cluster algebras, F. Chapoton, S. Fomin and A. Zelevinsky associated to each finite type root system a simple convex polytope called \emph{generalized associahedron}. They provided an explicit realization of this polytope associated with a bipartite orientation of the corresponding Dynkin diagram. In the first part of this paper, using the parametrization of cluster variables by their $g$-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we generalize the original construction to any orientation. In the second part we show that our construction agrees with the one given by C. Hohlweg, C. Lange, and H. Thomas in the setup of Cambrian fans developed by N. Reading and D. Speyer., 31 pages, 2 figures. Changelog: 20111106: initial version 20120403: fixed errors in figures 20120827: revised version

In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the $A_n$ case while providing a novel interpretation for the $B_n$, $C_n$, and $D_n$ cases., Dans cet article nous donnons une interprétation combinatoire en termes de théorie des graphes pour les variables de clusters qui apparaissent dans la plupart des algèbres à clusters de type fini. En particulier, nous décrivons une famille de graphes tels qu'une énumération pondérée de leurs matchings parfaits encode le numérateur du polynôme de Laurent associé, tandis que les décompositions du graphe correspondent au dénominateur. Ceci complète les récents travaux de Schiffler et Carroll-Price qui donnent une formule pour le développement d'une variable de cluster dans le cas $A_n$, tout en fournissant une nouvelle interprétation dans les cas $B_n$, $C_n$ et $D_n$.

This paper, arising from population studies, develops clustering algorithms for identifying patterns in data. Based on the concept of geometric variability, we have developed one polythetic-divisive and three agglomerative algorithms. The effectiveness of these procedures is shown by relating them to classical clustering algorithms. They are very general since they do not impose constraints on the type of data, so they are applicable to general (economics, ecological, genetics…) studies. Our major contributions include a rigorous formulation for novel clustering algorithms, and the discovery of new relationship between geometric variability and clustering. Finally, these novel procedures give a theoretical frame with an intuitive interpretation to some classical clustering methods to be applied with any type of data, including mixed data. These approaches are illustrated with real data on Drosophila chromosomal inversions.