Related papers: Topological proofs of results on large fields
Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
The topology of any complex system is key to understanding its structure and function. Fundamentally, algebraic topology guarantees that any system represented by a network can be understood through its closed paths. The length of each path…
Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis…
The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are…
Fix a field $K$. We show that $K$ is large if and only if some elementary extension of $K$ is the fraction field of a henselian local domain which is not a field. The proof uses a new result about the \'etale-open topology over $K$: if $K$…
The topological classification of all known non-magnetic crystalline compounds is now complete, revealing thousands of new candidate topological materials waiting to be explored in the lab.
Even though flt is a number theoretic result we prove that the result depends on the topological as well as the field structure of the underlying space.
We discuss topological versions of the closed graph theorem, where continuity is inferred from near continuity in tandem with suitable conditions on source or target spaces. We seek internal characterizations of spaces satisfying a closed…
Let $K$ be a field. The \'etale open topology on the $K$-points $V(K)$ of a $K$-variety $V$ was introduced in our previous work. The \'etale open topology is non-discrete if and only if $K$ is large. If $K$ is separably, real, $p$-adically…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
We survey recent developments in the theory of achievement sets and present a substantial collection of open problems.
We give a short, topological proof that all graphs admit tree-decompositions displaying their topological ends.
Some of the basic concepts of topology are explored through known physics problems. This helps us in two ways, one, in motivating the definitions and the concepts, and two, in showing that topological analysis leads to a clearer…
We survey recent developments on mapping class groups of surfaces of infinite topological type.
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
We give a reformuation of the Tate conjecture for a surface over a finite field in terms of suitable affine open subsets. We then present three attempts to prove this reformulation, each of them falling short. Interestingly, the last two…
Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data…
Basic pairs and their morphisms are the most elementary framework in which standard topological notions can be defined. We present here a new interpretation of topological concepts as those which can be communicated faithfully between the…
We study a class of tame $\mathcal{L}$-theories $T$ of topological fields and their $\mathcal{L}_\delta$-extension $T_{\delta}^*$ by a generic derivation $\delta$. The topological fields under consideration include henselian valued fields…
The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data…