Related papers: On Gluing Data, Finite Ringed Spaces and schemes
We describe applications of the gluing formalism discussed in the companion paper. When a $d$-dimensional local theory $\text{QFT}_d$ is supersymmetric, and if we can find a supersymmetric polarization for $\text{QFT}_d$ quantized on a…
A new framework is proposed for analyzing staggered-grid finite difference finite volume methods on unstructured meshes. The new framework employs the concept of external approximation of function spaces, and gauge convergence of numerical…
Based on the simple and well understood concept of subfields in a finite field, the technique called `field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field…
We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We…
Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted…
The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined as differential graded categories of a…
Imposing additional constraints on low-rank optimization has garnered growing interest. However, the geometry of coupled constraints hampers the well-developed low-rank structure and makes the problem intricate. To this end, we propose a…
In this research oriented manuscript, foundational aspects of rigid geometry are discussed, putting emphasis on birational side of formal schemes and topological feature of rigid spaces. Besides the rigid geometry itself, topics include the…
We give new and rather general gluing theorems for anti-self-dual (ASD) conformal structures, following the method suggested by Floer. The main result is a gluing theorem for pairs of conformally ASD manifolds `joined' across a common piece…
Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have…
We develop a geometric framework for generalized Milnor classifying spaces in the setting of diffeological spaces and infinite-dimensional geometry. Starting from Milnor's construction, we introduce spherical and projective models endowed…
Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring $R$, $\,R$-modules built from the rings of functions on principal affine open subschemes in…
Let X be a stratified space on which the Juteau-Mautner-Williamson theory of parity sheaves is available. We develop a "nearby cycles formalism" in the framework of the homotopy category of parity sheaves on X, also known as the mixed…
We study maps from a 2D world-sheet to a 2D target space which include folds. The geometry of folds is discussed and a metric on the space of folded maps is written down. We show that the latter is not invariant under area preserving…
In a previous paper, we showed how certain orientations of the edges of a graph G embedded in a closed oriented surface S can be understood as discrete spin structures on S. We then used this correspondence to give a geometric proof of the…
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic…
We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings,…
In this review, we give a pedagogical introduction to a systematic framework for constructing and analyzing supersymmetric field theories on curved spacetime manifolds. The framework is based on the use of off-shell supergravity background…
Many objects in nature and industry are wrapped in a thin sheet to enhance their chemical, mechanical, or optical properties. There are similarly a variety of methods for wrapping, from pressing a film onto a hard substrate, to using…