Related papers: Coarse geometry and its applications in solid stat…
We provide an introduction to the old-standing problem of isometric immersions. We combine a historical account of its multifaceted advances, which have fascinated geometers and analysts alike, with some of the applications in the…
The recent discovery of topological insulator (TI) offers new opportunities for the development of thermoelectrics, because many TIs (like Bi$_2$Te$_3$) are excellent thermoelectric (TE) materials. In this review, we will first describe the…
A short survey on applications of algebraic geometry in topological data analysis.
We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian…
The study of topology of energy bands in solid has always been interesting and fruitful. Historically, Thouless et al proposed the TKNN number or Chern number of the energy bands to explain the quantization of Hall conductance in the…
Topological insulators have become one of the most active research areas in condensed matter physics. This article reviews progress on the topic of electronic correlations effects in the two-dimensional case, with a focus on systems with…
The coherent states are viewed as a powerful tool in differential geometry. It is shown that some objects in differential geometry can be expressed using quantities which appear in the construction of the coherent states. The following…
Systems as diverse as mechanical structures assembled from elastic components, and photonic metamaterials enjoy a common geometrical feature: a sublattice symmetry. This property realizes a chiral symmetry first introduced to characterize a…
Topological materials have gained significant attention in condensed matter physics due to their unique electronic and transport properties. Three-dimensional (3D) topological materials are characterized by robust electronic states that are…
We use homotopy theory to extend the notion of strong and weak topological insulators to the non-stable regime (low numbers of occupied/empty energy bands). We show that for strong topological insulators in d spatial dimensions to be "truly…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
We survey recent applications of topology and singularity theory in the study of the algebraic complexity of concrete optimization problems in applied algebraic geometry and algebraic statistics.
To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on…
In this paper we give a brief overview of the geometry of involute gears, from a mathematical more than an engineering perspective. We also list some of the many variant geared mechanisms and discuss some of our 3D printed mechanisms.
In this monograph we lay the foundation for a theory of coarse groups and coarse actions. Coarse groups are group objects in the category of coarse spaces, and can be thought of as sets with operations that satisfy the group axioms "up to…
Topological insulators are a new class of insulators in which a bulk gap for electronic excitations is generated by strong spin orbit coupling. These novel materials are distinguished from ordinary insulators by the presence of gapless…
Two-dimensional photonic crystals made of six air holes on a core-shell dielectric material has been proposed to study the newly emerged photonic quantum spin Hall insulator. Specifically, radii modification of the air holes and core-shell…
We propose a surface-edge state theory for half quantized Hall conductance of surface states in topological insulators. The gap opening of a single Dirac cone for the surface states in a weak magnetic field is demonstrated. We find a new…
A salient feature of solid-state topological materials in two dimensions is the presence of conducting electronic edge states that are insensitive to scattering by disorder. Such unidirectional edge states have been explored in many…
The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the…