Related papers: Axiomatic Differential Geometry III-1

We develop the theory of derived differential geometry in terms of bundles of curved $L_\infty[1]$-algebras, i.e. dg manifolds of positive amplitudes. We prove the category of derived manifolds is a category of fibrant objects. Therefore,…

In classical field theory, the composite fibred manifolds Y -> Z -> X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This work is devoted to connections on…

For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric…

Manifolds without boundary, and manifolds with boundary, are universally known in Differential Geometry, but manifolds with corners (locally modelled on [0,\infty)^k x R^{n-k}) have received comparatively little attention. The basic…

We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables to treat fluctuations of the metric and…

We discuss the differential algebras used in Connes' approach to Yang-Mills theories with spontaneous symmetry breaking. These differential algebras generated by algebras of the form functions $\otimes$ matrix are shown to be skew…

We try to convince the reader that the categorical version of differential geometry, called Synthetic Differential Geometry (SDG), offers valuable tools which can be applied to work with some unsolved problems of general relativity. We do…

We build a toy model of differential geometry on the real line, which includes derivatives of the second order. Such construction is possible only within the framework of noncommutative geometry. We introduce the metric and briefly discuss…

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their…

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…

A `discrete differential manifold' we call a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides us with a convenient framework for the formulation of…

We formalize the concept of sheaves of sets on a model site by considering variables thereof, or motifs, and we construct functorially defined derived algebraic stacks from them, thereby eliminating the necessity to choose derived…

We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.

This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130,…

This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system);…

We provide a complete geometric solution to the problem of differentiating simplicial manifolds, extending classical Lie theory and complementing existing homotopical and formal approaches within a unifying framework. First, we establish a…

This is the second in a series of two papers developing a moduli-theoretic framework for differential ideal sheaves associated with formally integrable, involutive systems of algebraic partial differential equations (PDEs). Building on…

These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…

Real valued homomorphisms on the algebra of smooth functions on a differential space are described. The concept of generators of this algebra is emphasized in this description.

We first introduce the Wigner-Weyl-Moyal formalism for a theory whose phase-space is an arbitrary Lie algebra. We also generalize to quantum Lie algebras and to supersymmetric theories. It turns out that the non-commutativity leads to a…