Generalized Property R and the Schoenflies Conjecture

There is a relation between the generalized Property R Conjecture and the Schoenflies Conjecture that suggests a new line of attack on the latter. The approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices to prove the genus 3 case, even in the absence of new progress on the generalized Property R Conjecture.Comment: 29 pages, 8 figure

HexaMesh: Scaling to Hundreds of Chiplets with an Optimized Chiplet Arrangement

2.5D integration is an important technique to tackle the growing cost of manufacturing chips in advanced technology nodes. This poses the challenge of providing high-performance inter-chiplet interconnects (ICIs). As the number of chiplets grows to tens or hundreds, it becomes infeasible to hand-optimize their arrangement in a way that maximizes the ICI performance. In this paper, we propose HexaMesh, an arrangement of chiplets that outperforms a grid arrangement both in theory (network diameter reduced by 42%; bisection bandwidth improved by 130%) and in practice (latency reduced by 19%; throughput improved by 34%). MexaMesh enables large-scale chiplet designs with high-performance ICIs

Genus two 3-manifolds are built from handle number one pieces

Let M be a closed, irreducible, genus two 3-manifold, and F a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in M. Then each component manifold M_i of M-F has handle number at most one, i.e. admits a Heegaard splitting obtained by attaching a single 1-handle to one or two components of boundary M_i. This result also holds for a decomposition of M along a maximal collection of incompressible tori.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-38.abs.htm

Teaching Einsteinian Physics at Schools: Part 1, Models and Analogies for Relativity

The Einstein-First project aims to change the paradigm of school science teaching through the introduction of modern Einsteinian concepts of space and time, gravity and quanta at an early age. These concepts are rarely taught to school students despite their central importance to modern science and technology. The key to implementing the Einstein-First curriculum is the development of appropriate models and analogies. This paper is the first part of a three-paper series. It presents the conceptual foundation of our approach, based on simple physical models and analogies, followed by a detailed description of the models and analogies used to teach concepts of general and special relativity. Two accompanying papers address the teaching of quantum physics (Part 2) and research outcomes (Part 3)

The effect of atomic-scale defects and dopants on graphene electronic structure

Graphene, being one-atom thick, is extremely sensitive to the presence of adsorbed atoms and molecules and, more generally, to defects such as vacancies, holes and/or substitutional dopants. This property, apart from being directly usable in molecular sensor devices, can also be employed to tune graphene electronic properties. Here we briefly review the basic features of atomic-scale defects that can be useful for material design. After a brief introduction on isolated $p_z$ defects, we analyse the electronic structure of multiple defective graphene substrates, and show how to predict the presence of microscopically ordered magnetic structures. Subsequently, we analyse the more complicated situation where the electronic structure, as modified by the presence of some defects, affects chemical reactivity of the substrate towards adsorption (chemisorption) of atomic/molecular species, leading to preferential sticking on specific lattice positions. Then, we consider the reverse problem, that is how to use defects to engineer graphene electronic properties. In this context, we show that arranging defects to form honeycomb-shaped superlattices (what we may call "supergraphenes") a sizeable gap opens in the band structure and new Dirac cones are created right close to the gapped region. Similarly, we show that substitutional dopants such as group IIIA/VA elements may have gapped quasi-conical structures corresponding to massive Dirac carriers. All these possible structures might find important technological applications in the development of graphene-based logic transistors.Comment: 16 pages, 14 figures, "Physics and Applications of Graphene - Theory" - Chapter 3, http://www.intechweb.org/books/show/title/physics-and-applications-of-graphene-theor

The thick-thin decomposition and the bilipschitz classification of normal surface singularities

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts. By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of $(X,0)$ in terms of its topology and a finite list of numerical bilipschitz invariants.Comment: Minor corrections. To appear in Acta Mathematic