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Derived brackets as introduced and studied by Kosmann-Schwarzbach and Voronov are a powerful tool for describing and understanding infinitesimal symmetry actions relevant in physics. Roytenberg and Weinstein showed that this continues to…
From descent theory to higher geometry, the idea of gluing has been embedded in many elegant and powerful techniques, proving instrumental for the solution of many problems. In this paper, we introduce a framework that allows to link…
We discuss a discretisation of the de Rham-Hodge theory in the two-dimensional case based on a discrete exterior calculus framework. We present discrete analogues of the Hodge-Dirac and Laplace operators in which key geometric aspects of…
Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete…
In this paper we discuss generalizations of discrete torsion to noninvertible symmetries in 2d QFTs. One point of this paper is to explain that there are two complementary generalizations. Both generalizations are counted by $H^2(G,U(1))$…
We study the relationship between singularity categories and relative singularity categories and discuss constructions of differential graded algebras of relative singularity categories. As consequences, we obtain structural results, which…
We characterise the quotient surface graphs arising from symmetric contact systems of line segments in the plane and also from symmetric pointed pseudotriangulations in the case where the group of symmetries is generated by a translation or…
We introduce a novel class of labeled directed acyclic graph (LDAG) models for finite sets of discrete variables. LDAGs generalize earlier proposals for allowing local structures in the conditional probability distribution of a node, such…
We develop a new perspective on principal bundles with connection as morphisms from the tangent bundle of the underlying manifold to a classifying dg-Lie groupoid. This groupoid can be identified with a lift of the inner homomorphisms…
We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming…
For a simplicial manifold we construct the differential geometry structure and use it to investigate linear connections, metric and gravity. We discuss and compare three main approaches and calculate the resulting gravity action…
In this paper, we discuss several concepts of the modern theory of discrete integrable systems, including: - Time discretization based on the notion of B\"acklund transformation; - Symplectic realizations of multi-Hamiltonian structures; -…
We consider general integrable systems on graphs as discrete flat connections with the values in loop groups. We argue that a certain class of graphs is of a special importance in this respect, namely quad-graphs, the cellular…
We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their…
The discrete symmetry transformations of the N=2 supersymmetric (n,m)-GNLS hierarchy are constructed. Their bosonic limit is analyzed and new discrete symmetries of the modified GNLS hierarchy are derived. The explicit relations connecting…
The feed-forward relationship naturally observed in time-dependent processes and in a diverse number of real systems -such as some food-webs and electronic and neural wiring- can be described in terms of so-called directed acyclic graphs…
A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions…
In this paper, the recently developed differential homotopy approach is applied to the problem of disentangling dynamical and topological fields of the $3d$ higher-spin gauge theory at the linear level. This formalism allows us to reproduce…
We prove that any derived equivalence between derived-discrete algebras of finite global dimension is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex.
Double coverings of the orthogonal groups of the real and complex spaces are considered. The relation between discrete transformations of these spaces and fundamental automorphisms of Clifford algebras is established, where an isomorphism…