Generalizing a result of Furstenberg, we show that, for every infinite discrete group G, the Bernoulli flow $2^G$ is disjoint from every minimal G-flow. From this, we deduce that the algebra generated by the minimal functions $\mathfrak{A}(G)$ is a proper subalgebra of ${\ell }^{\mathrm{\infty }}(G)$ and that the enveloping semigroup of the universal minimal flow $M(G)$ is a proper quotient of the universal enveloping semigroup $\mathrm{\beta }G$. When G is countable, we also prove that, for any metrizable, minimal G-flow, there exists a free, minimal flow disjoint from it and that there exist continuum many mutually disjoint minimal, free, metrizable G-flows. Finally, improving a result of Frisch, Tamuz, and Vahidi Ferdowsi and answering a question of theirs, we show that if G is a countable group with infinite conjugacy classes, then it admits a free, minimal, proximal flow.

Let X be a locally symmetric space defined by a simple Chevalley group G and a congruence subgroup of $G(\mathbb{Q})$. In this generality, the Weyl law for X was proved by Lindenstrauss and Venkatesh. In the case where G is simply connected, we sharpen their result by giving a power-saving estimate for the remainder term.

We introduce a technique for proving all-genus wall-crossing formulas in the gauged linear sigma model as the stability parameter varies, without assuming factorization properties of the virtual class. Implementing this technique to the gauged linear sigma model associated to a complete intersection in weighted projective space, we obtain a uniform proof of the wall-crossing formula in both the geometric and the Landau–Ginzburg phase.

Let Γ be a Zariski-dense Kleinian Schottky subgroup of ${\mathrm{PSL}}_2(\mathbb{C})$. Let ${\mathrm{\Lambda }}_{\mathrm{\Gamma }}\subset \mathbb{C}$ be its limit set, endowed with a Patterson–Sullivan measure μ supported on ${\mathrm{\Lambda }}_{\mathrm{\Gamma }}$. We show that the Fourier transform $\hat{\mu }(\xi )$ enjoys polynomial decay as $|\xi |$ goes to infinity. As a corollary, all limit sets of Zariski-dense Kleinian groups have positive Fourier dimension. This is a ${\mathrm{PSL}}_2(\mathbb{C})$ version of the $\mathrm{PS}{L}_2(\mathbb{R})$ result of Bourgain and Dyatlov, and uses the decay of exponential sums based on Bourgain–Gamburd’s sum-product estimate on $\mathbb{C}$. These bounds on exponential sums require a delicate nonconcentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.