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We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was…
We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an $SU(3)\times Spin(6+n)$ structure within…
These notes provide a pedagogical introduction to the role of transversality theory in the analysis of statistical degeneracies within the framework of distributional statistical models. The classical question of when a statistical model is…
Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and…
We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral…
We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten- and eleven-dimensional supergravity. In all cases the reduction manifold admits a "generalised parallelisation" with a…
Hadwiger's theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent…
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 present a short review of the group-geometric approach to supergravity theories, from the point of view of recent developments. The central idea is the unification of usual diffeomorphisms, gauge symmetries and supersymmetries into…
We consider compatibility conditions between Poisson and Riemannian structures on smooth manifolds by means of a contravariant partially complex structure, or $f$-structure, introducing the notion of (almost) K\"ahler--Poisson manifolds. In…
Inspired by superstring field theory, we study differential, integral, and inverse forms and their mutual relations on a supermanifold from a sheaf-theoretical point of view. In particular, the formal distributional properties of integral…
We compute the essential dimension of the functors Forms_{n,d} and Hypersurf_{n, d} of equivalence classes of homogeneous polynomials in n variables and hypersurfaces in P^{n-1}, respectively, over any base field k of characteristic 0. Here…
We give an intrinsic definition of the special geometry which arises in global N=2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special Kahler…
Although scalar curvature is the simplest curvature invariant, our understanding of scalar curvature has not matured to the same level as Ricci or sectional curvature. Despite this fact, many rigidity phenomenon have been established which…
Shape invariance is a powerful solvability condition, that allows for complete knowledge of the energy spectrum, and eigenfunctions of a system. After a short introduction into the deformation quantization formalism, this paper explores the…
This paper determined the components of the generalized curvature tensor for the class of Kenmotsu type and established the mentioned class is {\eta}-Einstein manifold when the generalized curvature tensor is flat; the converse holds true…
The aim of this paper is to introduce a new class of softly normal called softly $\pi g\widehat{D}$ -normality by using $\pi g\widehat{D}$ -open sets and obtained several properties of such a space. We discuss many properties of this new…
We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to…
We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive…
By generalizing and extending some of the earlier results derived by Manin and Merkulov, a twistor description is given of four-dimensional N-extended (gauged) self-dual supergravity with and without cosmological constant. Starting from the…