Spontaneous symmetry breaking while preserving flow ability is a remarkable feature of nematic liquid crystals. When a nematic liquid crystal coexists with a solid, the surface field of the solid tends to anchor the director direction on the surface: anchoring effects. If geometrical frustration between nematic ordering and anchoring is strong enough, stable topological defects are formed. Defe...

Topological defects (TDs) appear almost unavoidably in continuous symmetry breaking phase transitions. The topological origin makes their key features independent of systems' microscopic details; therefore TDs display many universalities. Because of their strong impact on numerous material properties and their significant role in several technological applications it is of strong interest to fi...

We prove two topological theorems in physical chemistry. Namely, we introduce a hybrid of transverse and tangential measures on train tracks to prove sphericity of various simplicial complexes which arise from certain idealized models of physical chemistry. These complexes are at once identified with Thurston’s space of projective geodesic laminations on an ideal polygon and with the analogue o...

A geometric quantization using the topological recursion is established for the compactified cotangent bundle of a smooth projective curve of an arbitrary genus. In this quantization, the Hitchin spectral curve of a rank 2 meromorphic Higgs bundle on the base curve corresponds to a quantum curve, which is a Rees D-module on the base. The topological recursion then gives an all-order asymptotic ...

We survey two of the many aspects of the standard braid order, namely its set theoretical roots, and the known connections with knot theory, including results by Netsvetaev, Malyutin, and Ito, and very recent work in progress by Fromentin and Gebhardt. It has been known since 1992 [7, 8] that Artin’s braid groups Bn are leftorderable, by an ordering that has several remarkable properties. In pa...

We use topological models of intuitionistic analysis to answer some of the recent questions of Kadison [7] concerning diagonalization of matrices of continuous functions. Let X be a compact Hausdorff space, and let %(X) be the ring of continuous real functions on X. In the topological interpretation of intuitionistic analysis over X [3], %(X) represents the internal set of (Dedekind) reals. Thu...

The subject of topological materials has attracted immense attention in condensed-matter physics because they host new quantum states of matter containing Dirac, Majorana, or Weyl fermions. Although Majorana fermions can only exist on the surface of topological superconductors, Dirac and Weyl fermions can be realized in both 2D and 3D materials. The latter are semimetals with Dirac/Weyl cones e...