Related papers: Axiomatic Differential Geometry III-3
The study of symmetries in the realm of manifolds can be approached in two different ways. On one hand, Killing vector fields on a (pseudo-)Riemannian manifold correspond to the directions of local isometries within it. On the other hand,…
The Weil correspondence states that the datum of a Seiberg-Witten differential is equivalent to an algebraic group extension of the integrable system associated to the Seiberg-Witten geometry. Remarkably this group extension represents…
We discuss the nature of structure-preserving maps of varies function algebras. In particular, we identify isomorphisms between special Colombeau algebras on manifolds with invertible manifold-valued generalized functions in the case of…
We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkl operator associated with the Weyl group of the exceptional root system $G_2$. For this symmetry algebra, we give both an abstract definition and an explicit…
In a previous work we proved the uniqueness and functoriality of primary unfoldings on simple Thom-Mather spaces, which is a functor to the category of smooth manifolds. In this article we extend these results for any stratified Thom-Mather…
We develop sheaf theory in the context of difference algebraic geometry. We introduce categories of difference sheaves and develop the appropriate cohomology theories. As specializations, we get difference Galois cohomology, difference…
Recent developments in motivic homotopy theory, especially the construction of norm functors by Bachmann and Hoyois, rely heavily on the machinery of infinite categories. In this paper, we take a purely geometric and elementary approach via…
For the base field of complex numbers we discuss the relationship between categories of coherent sheaves on compact Riemann surfaces and categories of coherent sheaves on weighted smooth projective curves. This is done by bringing back to…
Previous work established a connection between the geometric invariant theory of the third exterior power of a 9-dimensional complex vector space and the moduli space of genus 2 curves with some additional data. We generalize this…
This third part of the series is a brief comment to certain aspects of the theory of classical $r$-matrix and bihamiltonian formalism, which motivations lie in constructions of the previous two parts.
The aim of this note is threefold. The first is to obtain a simple characterization of relative constructible sheaves when the parameter space is projective. The second is to study the relative Fourier-Mukai for relative constructible…
Rosenfeld postulated ``generalized'' projective planes, which exploit a correspondence between rank-one idempotents of Jordan algebras $\mathfrak{J}_3(\mathbb{A})$ and points of projective planes $\mathbb{A}P^2$. The isometry groups of the…
Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance…
In this paper, we study the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras.
A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples,…
The aim of this paper is to generalize Grothendieck's theory of smooth functors in order to include within this framework the theory of fibered categories. We obtain in particular a new characterization of fibered categories.
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,…
Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Fr\"olicher spaces, another generalisation of smooth structures. We then…
Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate…
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the…