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

A generalised notion of Kac-Moody algebra is defined using smooth maps from a compact real manifold $\mathcal{M}$ to a finite-dimensional Lie group, by means of complete orthonormal bases for a Hermitian inner product on the manifold and a…

We define the category of $G_2$-structures over a Riemannian 7-manifold $M$ and present an isomorphism between this category and a full subcategory of the category of octonion algebras over the ring of smooth real-valued functions…

We introduce a new geometric approach to a manifold equipped with a smooth density function that takes a torsion-free affine connection, as opposed to a weighted measure or Laplacian, as the fundamental object of study. The connection…

We develop algebraic geometry for general Segal's Gamma-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under Spec Z). The starting observation is that the category obtained by gluing…

We introduce the holonomy-diffeomorphism algebra, a C*-algebra generated by flows of vectorfields and the compactly supported smooth functions on a manifold. We show that the separable representations of the holonomy-diffeomorphism algebra…

We recast basic topological concepts underlying differential geometry using the language and tools of noncommutative geometry. This way we characterize principal (free and proper) actions by a density condition in (multiplier) C*-algebras.…

The paper presents the complete classification of Automorphic Lie Algebras based on $\mathfrak{sl}_n (\mathbb{C})$, where the symmetry group $G$ is finite and the orbit is any of the exceptional $G$-orbits in $\overline{\mathbb{C}}$. A key…

Let X be a singular affine normal variety with coordinate ring R and assume that there is an R-order admitting a stability structure such that the scheme of relevant semistable representations is smooth, then we construct a partial…

Quantum general relativity may be considered as generally covariant QFT on differentiable manifolds, without any a priori metric structure. The kinematically covariance group acts by general diffeomorphisms on the manifold and by…

We introduce a general scheme of constructing smooth subalgebras of C$^*$-algebras that are closed under the smooth calculus of self-adjoint elements. We illustrate the scheme with a number of examples.

The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that…

Many important theorems in differential topology relate properties of manifolds to properties of their underlying homotopy types -- defined e.g. using the total singular complex or the \v{C}ech nerve of a good open cover. Upon embedding the…

We study a special type of $E_\infty$-operads that govern strictly unital $E_\infty$-coalgebras (and algebras) over the ring of integers. Morphisms of coalgebras over such an operad are defined by using universal $E_\infty$-bimodules. Thus…

The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like…

We provide a framework for the construction of diffeomorphism invariant sheaves of nonlinear generalized functions spaces. As an application, global algebras of generalized functions for distributions on manifolds and diffeomorphism…

We study "higher-dimensional" generalizations of differential forms. Just as differential forms can be defined as the universal commutative differential algebra containing C^\infty(M), we can define differential gorms as the universal…

This paper lays the foundations for a unified framework for numerically and computationally applying methods drawn from a range of currently distinct geometrical approaches to statistical modelling. In so doing, it extends information…

We study the dg-algebra $\Omega ^\bullet_{A|\mathbb{R}}$ of algebraic de Rham forms of a real soft function algebra $A$, i.e., the algebra of global sections of a soft subsheaf of $C_X$, the sheaf of continuous functions on a space $X$. We…

We study the geometry and topology of Hilbert schemes of points on the orbifold surface [C^2/G], respectively the singular quotient surface C^2/G, where G is a finite subgroup of SL(2,C) of type A or D. We give a decomposition of the…

For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is…