Related papers: Algebraic Geometry over $C^\infty$-rings

We investigate conditions on a graph $C^*$-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth $(1,\infty)$-summable semfinite…

We consider, for each smooth manifold $M$, the set $\mathbb{M}$ comprised by all the primary ideals of $\mathcal{C}^\infty(M)$ which are closed and whose radical is maximal. The classical Lie theory of jets (jets of submanifolds) must be…

The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real…

In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations,…

We introduce finitely $C^\infty$-generated algebras, which can be treated as `algebras of functions' on non-commutative $C^\infty$-differentiable spaces. Our approach uses the category of projective limits of real Banach algebras of…

We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains…

We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The…

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold $\mathbb{L}$, there exists a global trivialization of the tangent bundle, which defines a map…

A {\em $k$-trinitary algebra} is any subalgebra of the space of smooth functions $f: M \to {\mathbb R}$ that is distinguished in this space by $k$ independent conditions of the form $f(x_i) = f(\tilde x_i) = f(\hat x_i)$, where $x_i, \tilde…

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 shall introduce the notion of $C^\infty$ logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex…

This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of…

This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces $\mathbb R^N,\mathbb C^N$ have no free analogues in the operator algebra sense, but the corresponding unit spheres…

In the present work we carry on the study of the order theory for ($\mathcal{C}^{\infty}-$-reduced) $\mathcal{C}^{\infty}-$-rings initiated in \cite{rings1} (see also \cite{BM2}). In particular, we apply some results of the order theory of…

This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…

We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal…

This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In this paper, we lay the foundations for this study by introducing the…

This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category,…

Let $A$ and $B$ be $C^*$-algebras with $A\subseteq M(B)$. Exploiting the duality between sober spaces and spatial locales, and the adjunction between restriction and induction for ideals in $A$ and $B$, we identify conditions that allow to…

We extend the classical notion of standardly stratified $k$-algebra (stated for finite dimensional $k$-algebras) to the more general class of rings, possibly without $1,$ with enough idempotents. We show that many of the fundamental…