Related papers: Rhombic tilings and Bott-Samelson varieties
We construct algebras from rhombohedral tilings of Euclidean space obtained as projections of certain cubical complexes. We show that these `Cubist algebras' satisfy strong homological properties, such as Koszulity and quasi-heredity,…
Motivated by the study of symplectic Lie algebroids, we study a describe a type of algebroid (called an $E$-tangent bundle) which is particularly well-suited to study of singular differential forms and their cohomology. This setting…
We apply the theory of Groebner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions T_N=T_{3k-1} and T_N=T_{3k} in a hexagonal…
The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and hermitian forms over fields of good characteristic. Explicit results are given for the partitions $(n)$,…
The two operations, deletion and contraction of an edge, on multigraphs directly lead to the Tutte polynomial which satisfies a universal problem. As observed by Brylawski in terms of order relations, these operations may be interpreted as…
Eventually after Dieudonn\'e-Grothendieck, we give intrinsic definitions of \'etale, lisse and non-ramifi\'e morphisms for general adic rings and general locally convex rings. And we investigate the corresponding \'etale-like, lisse-like…
The study of tilings is a major problem in many mathematical instances, which is studied in two main different approaches: when considering the existence (or obstructions to the existence) of a tiling with a given tile and the other…
Originally motivated by questions of P. Etingof related to growth rates of tensor powers in symmetric tensor categories, we obtain general bounds on the order of finite subgroups of ${\rm GL}(n,\mathbb{C})$ with restricted composition…
We introduce rational Dyck tilings, or $(a,b)$-Dyck tilings, and study them by the decomposition into $(1,1)$-Dyck tilings. This decomposition allows us to make use of combinatorial models for $(1,1)$-Dyck tilings such as the Hermite…
We introduce symplectic structures on "Lie pairs" of (real or complex) algebroids as studied by Chen, Stienon and the second author (From Atiyah classes to homotopy Leibniz algebras, arXiv:1204.1075), encompassing homogeneous symplectic…
We construct an explicit basis for the coordinate ring of the Bott-Samelson variety Z_i associated to G = GL(n) and an arbitrary sequence of simple reflections i. Our basis is parametrized by certain standard tableaux and generalizes the…
In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a direct description of this Batalin-Vilkovisky algebra in the…
We continue the~study of embeddings between different classes of Sobolev spaces of differential forms started in 2006 in a~paper by Gol$'$dshtein and Troyanov. As in this paper, our study is based on relations between $L_{q,p}$-cohomology…
We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the…
A method is described for constructing, with computer assistance, planar substitution tilings that have n-fold rotational symmetry. This method uses as prototiles the set of rhombs with angles that are integer multiples of pi/n, and…
We consider translational integer tilings by finite sets $A\subset\mathbb{Z}$. We introduce a new method based on \emph{splitting}, together with a new combinatorial interpretation of some of the main tools from our earlier work. We also…
Delzant's theorem for symplectic toric manifolds says that there is a one-to-one correspondence between certain convex polytopes in $\mathbb{R}^n$ and symplectic toric $2n$-manifolds, realized by the image of the moment map. I review proofs…
The perturbative expansion of tensorial field theories in Feynman graphs can be interpreted as weighted generating series of some piecewise linear varieties. This simple fact establishes a link between two a priori distinct fields: the…
In this paper we present a new version of the second author's factorization theorem for perfect matchings of symmetric graphs. We then use our result to solve four open problems of Propp on the enumeration of trimer tilings on the hexagonal…
We study the Ehrhart theory of hypersimplices of type C, as introduced by Lam and Postnikov for general crystallographic root systems. The $h^*$-polynomials of classical hypersimplices are known to relate to various Eulerian statistics on…