Related papers: A complexity theory of constructible functions and…
Statistical learning theory chiefly studies restricted hypothesis classes, particularly those with finite Vapnik-Chervonenkis (VC) dimension. The fundamental quantity of interest is the sample complexity: the number of samples required to…
A closed expression is given for the generating function of (virtual) Poincar\'e polynomials of moduli spaces of semi-stable sheaves on the projective plane $\mathbb{P}^2$ with arbitrary rank $r$ and Chern classes. This generating function…
Sheaves are objects of a local nature: a global section is determined by how it looks locally. Hence, a sheaf cannot describe mathematical structures which contain global or nonlocal geometric information. To fill this gap, we introduce the…
We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
In Chapter 3 of his Notes on constructive mathematics, Martin-L{\"o}f describes recursively constructed ordinals. He gives a constructively acceptable version of Kleene's computable ordinals. In fact, the Turing definition of computable…
In classical statistical learning theory, one of the most well studied problems is that of binary classification. The information-theoretic sample complexity of this task is tightly characterized by the Vapnik-Chervonenkis (VC) dimension. A…
An instance of the Valued Constraint Satisfaction Problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite…
In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes $A \not= B$ especially when $A$ is known to be a subset of $B$. In this paper we…
Verifiable delay functions (VDF) are functions that take a specified number of sequential steps to be evaluated but can be verified efficiently. In this paper, we introduce a new complexity class that contains all the VDFs. We show that…
Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit,…
It is usually not straightforward to work with the category of perverse sheaves on a variety using only its definition as a heart of a $t$-structure. In this paper, the category of perverse sheaves on a smooth toric variety with its orbit…
There have recently been several developments in synthetic mathematics using extensions of dependent type theory with univalence and higher inductive types: simplicial homotopy type theory, synthetic algebraic geometry and synthetic Stone…
We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from…
Building on Beilinson's work, ``constructible sheaves are holonomic,'' we introduce the notion of holonomicity for \'etale sheaves, without assuming a priori constructibility. Over a perfect base field, we establish the converse of…
We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The…
We study the compatibility with proper push-forward of the characteristic cycles of a constructible complex on a smooth variety over a perfect field.
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…
We develop a theory of weighted colimits in the framework of weakly bienriched $\infty$-categories, an extension of Lurie's notion of enriched $\infty$-categories. We prove an existence result for weighted colimits, study weighted colimits…
Given an open-closed decomposition of the stratifying poset, we construct a new semi-orthogonal decomposition of the $\infty$-category of constructible sheaves on a stratified space admitting an exit-path $\infty$-category. From this we…