We establish equidistribution with respect to the bifurcation measure of postcritically finite (PCF) maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set ${\mathcal{M}}_2$ (or generalized Mandelbrot set ${\mathcal{M}}_d$ for degree $d>2$), we classify all curves $C\subset {\mathbb{A}}^2$ defined over $\mathbb{C}$ with Zariski-dense subsets of points $(a,b)\in C$, such that both $z^d+a$ and $z^d+b$ are simultaneously PCF for a fixed degree $d\ge 2$. Our result is analogous to the famous result of André regarding plane curves which contain infinitely many points with both coordinates being complex multiplication parameters in the moduli space of elliptic curves and is the first complete case of the dynamical André–Oort phenomenon studied by Baker and DeMarco.

Let $W$ be a complex reflection group, and let $H_c\left(W\right)$ be the rational Cherednik algebra for $W$ depending on a parameter $c$. One can consider the category $\mathcal{O}$ for $H_c\left(W\right)$. We prove a conjecture of Rouquier that the categories $\mathcal{O}$ for $H_c\left(W\right)$ and $H_{c\text{'}}\left(W\right)$ are derived-equivalent, provided that the parameters $c,c\text{'}$ have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analogue of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse.

We give a complete description of the group of exact autoequivalences of the bounded derived category of coherent sheaves on a K3 surface of Picard rank $1$. We do this by proving that a distinguished connected component of the space of stability conditions is preserved by all autoequivalences and is contractible.

We prove the $L^p\left(bD\right)$-regularity of the Cauchy–Szegő projection (also known as the Szegő projection) for bounded domains $D\subset {\mathbb{C}}^n$ which are strongly pseudoconvex and whose boundary satisfies the minimal regularity condition of class $C^2$.

We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension $1$. Using a recent result of Datar and Székelyhidi, we effectively determine the existence of Kähler–Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kähler–Einstein Fano threefolds and Fano threefolds admitting a nontrivial Kähler–Ricci soliton.