In this paper, we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra $L(E,\omega )$ of a row-finite vertex weighted graph $(E,\omega )$ is $*$-isomorphic to the lower Leavitt path algebra of a certain bipartite separated graph $(E(\omega ),C(\omega ))$. For a general locally finite weighted graph $(E, \omega )$, we show that a certain quotient $L_1(E,\omega )$ of $L(E,\omega )$ is $*$-isomorphic to an upper Leavitt path algebra of another bipartite separated graph $(E(w)_1,C(w)^1)$. We furthermore introduce the algebra ${L^{\mathrm {ab}}} (E,w)$, which is a universal tame $*$-algebra generated by a set of partial isometries. We draw some consequences of our results for the structure of ideals of $L(E,\omega )$, and we study in detail two different maximal ideals of the Leavitt algebra $L(m,n)$.

We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero–Moser space induced by an element of finite order of the normalizer of the associated complex reflection group. We give a parametrization à la Harish-Chandra of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero–Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper, C. Bonnafé [‘Calogero–Moser spaces vs unipotent representations’, Pure Appl. Math. Q., to appear, Preprint, 2021, arXiv:2112.13684].

We prove an incidence result counting the k-rich $\delta $-tubes induced by a well-spaced set of $\delta $-atoms. Our result coincides with the bound that would be heuristically predicted by the Szemerédi–Trotter theorem and holds in all dimensions $d \geq 2$. We also prove an analogue of Beck’s theorem for $\delta $-atoms and $\delta $-tubes as an application of our result.

We consider a subshift of finite type on q symbols with a union of t cylinders based at words of identical length p as the hole. We explore the relationship between the escape rate into the hole and a rational function, $r(z)$, of correlations between forbidden words in the subshift with the hole. In particular, we prove that there exists a constant $D(t,p)$ such that if $q>D(t,p)$, then the escape rate is faster into the hole when the value of the corresponding rational function $r(z)$ evaluated at $D(t,p)$ is larger. Further, we consider holes which are unions of cylinders based at words of identical length, having zero cross-correlations, and prove that the escape rate is faster into the hole with larger Poincaré recurrence time. Our results are more general than the existing ones known for maps conjugate to a full shift with a single cylinder as the hole.

Homotopy theory folklore tells us that the sheaf defining the cohomology theory $\operatorname {\mathrm {Tmf}}$ of topological modular forms is unique up to homotopy. Here we provide a proof of this fact, although we claim no originality for the statement. This retroactively reconciles all previous constructions of $\operatorname {\mathrm {Tmf}}$.

We show that the properties of being rationally K-stable passes from the fibres of a continuous $C(X)$-algebra to the ambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) K-stable provided the underlying C*-algebra is (rationally) K-stable, and the action has finite Rokhlin dimension with commuting towers.