Related papers: Configuration Spaces for the Working Undergraduate
This is a short introduction to affine and convex spaces, written especially for physics students. It summarizes different elementary presentations available in the mathematical literature, and blends analytic- and geometric-flavoured…
Machine learning often aims to produce latent embeddings of inputs which lie in a larger, abstract mathematical space. For example, in the field of 3D modeling, subsets of Euclidean space can be embedded as vectors using implicit neural…
Visualization researchers and visualization professionals seek appropriate abstractions of visualization requirements that permit considering visualization solutions independently from specific problems. Abstractions can help us design,…
We construct a topological space to study contextuality in quantum mechanics. The resulting space is a classifying space in the sense of algebraic topology. Cohomological invariants of our space correspond to physical quantities relevant to…
A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively. This map $\phi: B_k \rightarrow…
Shape grammars compute over shapes which are defined in the universe $U^*$. Shapes in the universe $U^*$ are analogous to line drawings that can be physically realized in the plane. Any shape is embedded or contained in an arrangement of…
The configuration category of a manifold is a topological category which we view as a Segal space, via the nerve construction. Our main result is that the unordered configuration category, suitably truncated, admits a finite presentation as…
Optimization of product performance repetitively introduces the need to make products adaptive in a more general sense. This more general idea is often captured under the term 'self-configuration'. Despite the importance of such capability,…
These lectures provide an introduction to the microscopic description of branes in curved backgrounds. After a brief reminder of the flat space theory, the basic principles and techniques of (rational) boundary conformal field theory are…
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…
A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real…
It is well known that not every combinatorial configuration admits a geometric realization with points and lines. Moreover, some of them do not even admit realizations with pseudoline arrangements, i.e., they are not topological. In this…
We consider generalizations of the familiar fifteen-piece sliding puzzle on the 4 by 4 square grid. On larger grids with more pieces and more holes, asymptotically how fast can we move the puzzle into the solved state? We also give a…
This paper begins by extending the notion of a combinatorial configuration of points and lines to a combinatorial configuration of points and planes that we refer to as configurations of order $2$. We then proceed to investigate a further…
Graph-structured data are widespread in real-world applications, such as social networks, recommender systems, knowledge graphs, chemical molecules etc. Despite the success of Euclidean space for graph-related learning tasks, its ability to…
Urban planning refers to the efforts of designing land-use configurations. Effective urban planning can help to mitigate the operational and social vulnerability of a urban system, such as high tax, crimes, traffic congestion and accidents,…
We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell…
We discuss various aspects of "braid spaces'' or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of…
These are some informal notes concerning topological vector spaces, with a brief overview of background material and basic notions, and emphasis on examples related to classical analysis.
Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented…