Related papers: Conical calculus on schemes and perfectoid spaces …
This article is a survey of recent work of the authors developing a new approach to quantization based on the equivariance with respect to some Lie group of symmetries. Examples are provided by conformal and projective differential…
We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schr\"{o}dinger equation in…
In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified K\"ahler space, and we make explicit the…
In this paper the analogy between differential forms arising from integrals in additive calculus and forms arising from the integrals in product calculus is investigated. It is found that with an appropriate definition of scalar…
We outline an algorithm to recover the canonical (or, coarsest) stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively…
We covariantize calculations over the manifold of phase space, establishing Stokes' theorem for differential cross sections and providing new definitions of familiar observable properties like infrared and collinear safety. Through the…
A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…
We compute the cohomology ring of a generalised type of configuration space of points in $\mathbb{R}^r$. This configuration space is indexed by a graph. In the case the graph is complete the result is known and it is due to Arnold and…
We introduce singular homology for non-Archimedean analytic spaces using a cosimplicial perfectoid space as a Galois representation. We define an integration along a cycle, which gives a pairing with the singular homology and the space of…
This document introduces a generalization of calculus that treats both continuous and discrete variables on an equal footing. This generalization of calculus was developed independently of the "Calculus on Time Scales" literature but may be…
The standard definition of integration of differential forms is based on local coordinates and partitions of unity. This definition is mostly a formality and not used used in explicit computations or approximation schemes. We present a…
We give an example of proper smooth fourfold over a perfect field k of characteristic p > 0 with asymmetric Hodge--Witt numbers in total degree 3. Our example is sharp both in terms of dimension and total degree. We arrive at our example by…
In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a…
We propose definitions of hypercomplex analytic spaces and hypercomplex schemes. We show that such a hypercomplex space is canonically associated to the quotient of a hypercomplex manifold by a finite group action.
We introduce higher analytic geometry, a novel framework extending Lurie's derived complex analytic spaces. This theory generalizes classical complex analytic geometry, enabling the study of derived K\"ahler spaces with non-trivial higher…
In this paper, we describe a general theory of "spaces with structure sheaves." Specializations of this theory include the classical theory of schemes, the theory of Deligne-Mumford stacks, and their derived generalizations.
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of…
In this paper we solve the problem of analytic classification of plane curves singularities with two branches by presenting their normal forms. This is accomplished by means of a new analytic invariant that relates vectors in the tangent…
The main objects of study are adic spaces with logarithmic structures. After establishing the basic definitions, we analyze the Kummer \'etale and pro-Kummer \'etale topologies on log adic spaces. In particular, we show that log adic spaces…
We study complexes of stable $\infty$-categories, referred to as categorical complexes. As we demonstrate, examples of such complexes arise in a variety of subjects including representation theory, algebraic geometry, symplectic geometry,…