Related papers: A Generalized Carpenter's Rule Theorem for Self-To…
We give a new proof of Cartan's fixed point theorem using topological fixed point theory. For an odd dimensional, simply connected and complete manifold having non-positive curvature, we further prove that every isometry with finite order…
Slender beam-like structures frequently occur in engineering applications and often interact at discrete locations through joints or connectors. Accurate modeling of such interactions is particularly challenging when different numerical…
In general a universal covering of a non connected topological group need not admit a topological group structure such that the covering map is a morphism of topological groups. This result is due to R.L. Taylor (1953). We generalise this…
The edge-of-the-wedge theorem in several complex variables gives the analytic continuation of functions defined on the poly upper half plane and the poly lower half plane, the set of points in $\mathbb{C}^d$ with all coordinates in the…
Continuing results from JCDCGGG 2016 and 2017, we solve several new cases of the simple foldability problem -- deciding which crease patterns can be folded flat by a sequence of (some model of) simple folds. We give new efficient algorithms…
Understanding the topology of decision regions is central to explaining the inner workings of deep neural networks. Prior empirical work has provided evidence that these regions are path connected. We study a stronger topological question:…
The generic chaining method provides a sharp description of the suprema of many random processes in terms of the geometry of their index sets. The chaining functionals that arise in this theory are however notoriously difficult to control…
This paper establishes a rigorous, quantitative link between the combinatorial complexity of a fractal partition and the intrinsic geometry of its interfaces. We introduce the concept of the \emph{Separation Dimension} ($\sepdim$), a novel…
A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to…
We derive a generalized Stokes' theorem, valid in any dimension and for arbitrary loops, even if self intersecting or knotted. The generalized theorem does not involve an auxiliary surface, but inherits a higher rank gauge symmetry from the…
Combinatorial gauge symmetry is a principle that allows us to construct lattice gauge theories with two key and distinguishing properties: a) only one- and two-body interactions are needed; and b) the symmetry is exact rather than emergent…
We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate…
It is proven that a connected graph is planar if and only if all its cocycles with at least four edges are "grounded" in the graph. The notion of grounding of this planarity criterion, which is purely combinatorial, stems from the intuitive…
In this paper, in the context of the ``Dessins d'enfants'' theory, we give a combinatorial criterion for a plane tree to cover a tree from the classes of "chains" or "stars''. Besides, we discuss some applications of this result which are…
The classical Thistlethwaite theorem for links can be phrased as asserting that the Kauffman bracket of a link can be obtained from an evaluation of the Bollob\'as-Riordan polynomial of a ribbon graph associated to one of the link's…
Recently we gave arguments that only two unique topologically different configurations of 7 equal all mutually touching round cylinders (the configurations being mirror reflections of each other) are possible in 3D, although a whole world…
We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schroedinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are…
A flip in a plane spanning tree $T$ is the operation of removing one edge from $T$ and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two…
We show that thin sheets under boundary confinement spontaneously generate a universal self-similar hierarchy of wrinkles. From simple geometry arguments and energy scalings, we develop a formalism based on wrinklons, the transition zone in…
Contact can be conceptualized as a set of constraints imposed on two bodies that are interacting with one another in some way. The nature of a contact, whether a point, line, or surface, dictates how these bodies are able to move with…