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This paper introduces topological data analysis. Starting from notions of a metric space and some elementary graph theory, we take example sets of data and demonstrate some of their topological properties. We discuss simplicial complexes…
Topological states of matter are robust quantum phases, characterised by propagating or localised edge states in an insulating bulk. Topological boundary states can be triggered by various mechanisms, for example by strong spin-orbit…
We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was…
The first part of this article is a short and selective survey of developments in differential and algebraic geometry from the 1980's involving enumerative questions and nonlinear elliptic partial differential equations. In the second part…
The irreducible complexity of natural phenomena has led Graph Neural Networks to be employed as a standard model to perform representation learning tasks on graph-structured data. While their capacity to capture local and global patterns is…
Let $k$ be a field that is finitely generated over its prime field. In Grothendieck's anabelian letter to Faltings, he conjectured that sending a $k$-scheme to its \'{e}tale topos defines a fully faithful functor from the localization of…
This paper is first-line research expanding GANs into graph topology analysis. By leveraging the hierarchical connectivity structure of a graph, we have demonstrated that generative adversarial networks (GANs) can successfully capture…
Topology captures the essence of what remains unchanged under a transformation. In this study, a topological invariant called super conformality was found to preserve a critical angle between the constitutive plane and its second order…
We study time- and parameter-dependent ordinary differential equations in the geometric setting of vector fields and their flows. Various degrees of regularities in state are considered, including Lipschitz, finitely diferentiable, smooth,…
We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield F of "max-plus…
In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible…
Let E be a locally solid vector lattice. In this paper, we consider two particular vector subspaces of the space of all order bounded operators on E. With the aid of two appropriate topologies, we show that under some conditions, they…
In order to make the fundamental group, one of the most well known invariants in algebraic topology, more useful and powerful some researchers have introduced and studied various topologies on the fundamental group from the beginning of the…
Tate gave a famous construction of the residue symbol on curves by using some non-commutative operator algebra in the context of algebraic geometry. We explain Beilinson's multidimensional generalization, which is not so well-documented in…
Modern methods of graph theory describe a graph up to isomorphism, which makes it difficult to create mathematical models for visualizing graph drawings on a plane. The topological drawing of the planar part of a graph allows representing…
We discuss analogies between the etale site of arithmetic schemes and the algebraic topology of dynamical systems. The emphasis is on Lefschetz numbers. We also discuss similarities between infinite primes in arithmetic and fixed points of…
A new technique is proposed to classify a topological field in abelian lattice gauge theories. We perform the classification by regarding the topological field as a local composite field of the gauge field tensor instead of the vector…
It is shown that the SO(3) gauge field configurations can be completely characterised by certain gauge invariant vector fields. The singularities of these vector fields describe the topological aspects of the gauge field configurations. The…
We are dealing with the problem of space layout planning here. We present an architectural conceptual CAD approach. Starting with design specifications in terms of constraints over spaces, a specific enumeration heuristics leads to a…
Complex networks, which are the abstractions of many real-world systems, present a persistent challenge across disciplines for people to decipher their underlying information. Recently, hyperbolic geometry of latent spaces has gained…