Related papers: Proper base change for separated locally proper ma…
Open discrete mappings with a modulus condition in metric spaces are considered. Some results related to local behavior of mappings as well as theorems about continuous extension to a boundary are proved.
Examples of differentiable mappings into real or complex topological vector spaces with specific properties are given, which illustrate the differences between differential calculus in the locally convex and the non-locally convex case. In…
For a smooth map between noetherian schemes, Verdier relates the top relative differentials of the map with the twisted inverse image functor `upper shriek'. We show that the associated traces for smooth proper maps can be rendered concrete…
As data grows in size and complexity, finding frameworks which aid in interpretation and analysis has become critical. This is particularly true when data comes from complex systems where extensive structure is available, but must be drawn…
We consider an algebraic variety X together with the choice of a subvariety Z. We show that any coherent sheaf on X can be constructed out of a coherent sheaf on the formal neighborhood of Z, a coherent sheaf on the complement of Z, and an…
We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological…
Comparative analysis of scalar fields is an important problem with various applications including feature-directed visualization and feature tracking in time-varying data. Comparing topological structures that are abstract and succinct…
The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable $\infty$-category $\mathcal{C}$ together with a collection of…
Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can…
This paper provides an overview of the applications of sheaf theory in deep learning, data science, and computer science in general. The primary text of this work serves as a friendly introduction to applied and computational sheaf theory…
In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham…
In [1] we defined a new kind of space called 'structured space' which locally resembles, near each of its points, some algebraic structure. We noted in the conclusion of the cited paper that the maps $f_s$ and $h$, which are of great…
By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras. In our recent article, we have extended de Vries duality to completely regular spaces by generalizing de Vries algebras…
Correspondences between k-tuples of points are key in multiple view geometry and motion analysis. Regular transformations are posed by homographies between two projective planes that serves as structural models for images. Such…
We develop relativistic causality theory in the setting of point-free topology by introducing a notion of causal coverage in ordered locales, generalising their canonical coverage relation to incorporate causal structure. This improves…
Continuous mappings between compact Hausdorff spaces can be studied using homomorphisms between algebraic structures (lattices, Boolean algebras) associated with the spaces. This gives us more tools with which to tackle problems about these…
Paradigms of bilinear maps f between locally convex spaces (like evaluation or composition) are not continuous, but merely hypocontinuous. We describe situations where, nonetheless, compositions of f with Keller C^n_c-maps (on suitable…
We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical…
Here we prove a Poincar\'e-Verdier duality theorem for the o-minimal sheaf cohomology with definably compact supports of definably normal, definably locally compact spaces in an arbitrary o-minimal structure.
Fix a noetherian scheme S. For any flat map f: X->Y of separated essentially-finite-type perfect S-schemes we define a canonical derived-category map c(f):\H(X)->f^!\H(Y), the fundamental class of f, where \H(Z) is the (pre-)Hochschild…