Related papers: Crystals and D-modules
We develop some basic facts on deformations of exterior differential ideals on a smooth complex algebraic variety. With these tools we study deformations of several types of differential ideals, leading to several irreducible components of…
Crystal structures can be viewed as assemblies of space-filling polyhedra, which play a critical role in determining material properties such as ionic conductivity and dielectric constant. However, most conventional crystal structure…
We introduce Backstr\"om pairs and Backstr\"om rings, study their derived categories and construct for them a sort of categorical resolutions. For the latter we define the global dimension, construct a sort of semi-orthogonal decomposition…
For every non-exceptional affine Lie algebra, we explicitly construct a positive geometric crystal associated with a fundamental representation. We also show that its ultra-discretization is isomorphic to the limit of certain perfect…
The relation of crystal bases with $q$-identities is discussed, and some new results on crystals and $q$-identities associated with the affine Lie algebra $C_n^{(1)}$ are presented.
Crystallographic tilings of the Euclidean space $\mathbb{E}^n$ are defined as simple tilings whose group of isometric automorphisms is crystallographic. To classify crystallographic tilings by their automorphism groups it is necessary to…
In the present paper we examine the relationship between several type $A$ cluster theories and structures. We define a 2D geometric model of a cluster theory, which generalizes cluster algebras from surfaces, and encode several existing…
We argue for a convergence of crystallography, materials science and biology, that will come about through asking materials questions about biology and biological questions about materials, illuminated by considerations of information. The…
Geometric discretisation draws analogies between discrete objects and operations on a complex with continuum ones on a manifold. We generalise the theory to the cubic case and incorporate metric, by adding volume factors to our discrete…
Generalized cycles can be thought of as the extension of form-cycle duality between holomorphic forms and cycles, to meromorphic forms and generalized cycles. They appeared as an ubiquitous tool in the study of spectral curves and…
Constellations are partial algebras that are one-sided generalisations of categories. It has previously been shown that the category of inductive constellations is isomorphic to the category of left restriction semigroups. Here we consider…
This is a survey on the usage of the module theoretic notion of a "retractable module" in the study of algebras with actions. We explain how classical results can be interpreted using module theory and end the paper with some open…
We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.
We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample…
We elaborate on the notion of generalized tomograms, both in the classical and quantum domains. We construct a scheme of star-products of thick tomographic symbols and obtain in explicit form the kernels of classical and quantum generalized…
This paper examines the concept of gluing, placing it within its most general categorical context and tracing its foundational role in the broader architecture of algebraic geometry.
In this paper we introduce the category of stratified Pro-modules and the notion of induced object in this category. We propose a translation of a Morihiko Saito equivalence result using the dual language of Pro-objects. So we prove an…
Crystal structure modeling with graph neural networks is essential for various applications in materials informatics, and capturing SE(3)-invariant geometric features is a fundamental requirement for these networks. A straightforward…
We discuss an extension of classical combinatorics theory to the case of spatially distributed objects.
The morphologies of two-dimensional (2D) crystals, nucleated, grown, and integrated within 2D elastic fluids, for instance in giant vesicle membranes, are dictated by an interplay of mechanics, permeability, and thermal contraction.…