We consider bounds on the number of semiclassical resonances in neighbourhoods of the size of the semiclassical parameter, $h$, around energy levels at which the flow is hyperbolic. We show that the number of resonances is bounded by $h^{-\nu }$, where $2\nu +1$ is essentially the dimension of the trapped set on the energy surface. We note that in a confined setting, this dimension is equal to $2n-1$, where $n$ is the dimension of the physical space and the bound, $h^{1-n}$, corresponds to the optimal bound on the number of eigenvalues. Although no lower bounds of this type are rigorously known in the setting of semiclassical differential operators, the corresponding bound is optimal for certain models based on open quantum maps (see [26])

We show that any continuous measure $\nu$ in the class of a generalized Gibbs stream on the boundary of a CAT($-\kappa$) group $G$ arises as a harmonic measure for a random walk on $G$. Under an additional mild hypothesis on $G$ and for $\nu$, Hölder equivalent to a Gibbs measure, we show that $(\partial G,\nu )$ arises as a Poisson boundary for a random walk on $G$. We also prove a new approximation theorem for general metric measure spaces giving quite flexible conditions for a set of functions to be a positive basis for the cone of positive continuous functions

For a locally compact group, the property of rapid decay (property RD) gives a control on the convolutor norm of any compactly supported function in terms of its $L^2$-norm and the diameter of its support. We characterize the Lie groups that have property RD

We construct four families of Artin-Schelter (AS) regular algebras of global dimension $4$. This is a complete list of AS regular algebras of global dimension $4$ which are generated by two elements of degree $1$ and whose Ext-algebra satisfies certain “generic” conditions. These algebras are also strongly Noetherian, Auslander regular, and Cohen-Macaulay. One of the main tools is Keller's higher-multiplication theorem on $A_{\infty }$-Ext-algebras