G. Masbaum | Searcharxiv

arXiv2025

Denjoy's anachronistic topological viewpoint on Aubry transition

The Aubry transition is a phase transition between two types of incommensurate states, originally described as a transition by ``breaking of analyticity''. Here we present Denjoy's (anachronistic) viewpoint, who almost hundred years ago described certain mathematical properties of circle homeomorphisms with irrational rotation numbers. The connection between the two lies in the existence of a change of variables from the incommensurate ground state variables to new simple phase variables that rotate by a constant irrational angle. This confers a cyclic order, an essential property of models with the Aubry transition. Denjoy's description indicates that there are two types of cyclic order, distinguished by the regular or singular nature of the change of variables or, in mathematical terms, by the distinction between topological conjugacy versus semiconjugacy. This allows rephrasing the breaking of analyticity as a breaking of topological conjugacy. We illustrate this description with numerical calculations on the Frenkel-Kontorova model.

cond-mat.other↗

arXiv2015

Frattini and related subgroups of Mapping Class Groups

Let $Γ_{g,b}$ denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus $g$ with $b$ punctures. For a group $G$ let $Φ_f(G)$ denote the intersection of all maximal subgroups of finite index in $G$. Motivated by a question of Ivanov as to whether $Φ_f(G)$ is nilpotent when $G$ is a finitely generated subgroup of $Γ_{g,b}$, in this paper we compute $Φ_f(G)$ for certain subgroups of $Γ_{g,b}$. In particular, we answer Ivanov's question in the affirmative for these subgroups of $Γ_{g,b}$.

math.GT↗

arXiv2004

Problems on invariants of knots and 3-manifolds

This is a list of open problems on invariants of knots and 3-manifolds with expositions of their history, background, significance, or importance. This list was made by editing open problems given in problem sessions in the workshop and seminars on `Invariants of Knots and 3-Manifolds' held at Kyoto in 2001.

math.GT↗