Related papers: Skeleta in non-Archimedean and tropical geometry

Let $A$ be an abelian variety with totally degenerate reduction over a non-Archimedean field. We describe the moduli space of semihomogeneous vector bundles on $A$ from the perspective of non-Archimedean uniformization and show that the…

We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of To\"{e}n--Vaqui\'{e}--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category…

We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev…

This paper provides an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff in the case of…

In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean field, and let $X$ be a closed subscheme of a toric variety over $K$. We define the…

Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model $\mathscr X$ of $X$ plus a suitable horizontal…

In this paper we prove the existence of an algebraic model for quasi-coherent sheaves on certain non-connective geometric stacks arising in stable homotopy theory and spectral algebraic geometry using the machinery of adapted homology…

We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adele class space of the field of…

This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean…

Let K be an algebraically closed, complete nonarchimedean field and let X be a smooth K-curve. In this paper we elaborate on several aspects of the structure of the Berkovich analytic space X^an. We define semistable vertex sets of X^an and…

In this text, we outline a theory of schemes associated with a site, which generalizes a variety of geometries, such as manifolds, schemes, analytic spaces, simplicial complexes, and more. We present an abstract process of gluing model…

We propose a derived version of non-archimedean analytic geometry. Intuitively, a derived non-archimedean analytic space consists of an ordinary non-archimedean analytic space equipped with a sheaf of derived rings. Such a naive definition…

The collection of open sets of a topological space forms a Heyting algebra, which leads to the idea of a Heyting algebra as a generalized topological space. In fact, a sober topological space may be reconstructed from its locale of open…

We show that the non-commutative geometric approach to the Riemann zeta function has an algebraic geometric incarnation: the "Arithmetic Site". This site involves the tropical semiring viewed as a sheaf on the topos which is the dual of the…

We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT).…

We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of…

We develop the geometric and homological framework for non-commutative $n$-ary $\Gamma$-semirings by constructing a sheaf and derived theory over their non-commutative $\Gamma$-spectrum. Starting with a non-commutative $n$-ary…

We define a notion of a connectivity structure on an $\infty$-category, analogous to a $t$-structure but applicable in unstable contexts -- such as spaces, or algebras over an operad. This allows us to generalize notions of n-skeleta,…

In this work, we introduce a novel approach based on algebraic topology to enhance graph convolution and attention modules by incorporating local topological properties of the data. To do so, we consider the framework of sheaf neural…

We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category…