We solve two open problems: first we prove a conjecture of Bondal and Van den Bergh, showing that the category $\mathbf{D}^{\mathrm{perf}}(X)$ is strongly generated whenever $X$ is a quasicompact, separated scheme, admitting a cover by open affine subsets $\mathrm{Spec}(R_i)$ with each $R_i$ of finite global dimension. We also prove that, for a noetherian scheme $X$ of finite type over an excellent scheme of dimension $\le 2$, the derived category $\mathbf{D}^b_{\mathrm{coh}}(X)$ is strongly generated. The known results in this direction all assumed equal characteristic; we have no such restriction.

The method is interesting in other contexts: our key lemmas turn out to give a simple proof that, if $f:X \longrightarrow Y$ is a separated morphism of quasicompact, quasiseparated schemes such that $\mathbf{R} f_*: \mathbf{D}_{\mathbf{qc}}(X)\longrightarrow \mathbf{D}_{\mathbf{qc}}(Y)$ takes perfect complexes to complexes of bounded-below Tor-amplitude, then $f$ must be of finite Tor-dimension.

The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When $n$ is less than the rank of the associated symmetric space, we show that the $n$-dimensional filling volume function of the lattice grows at the same rate as that of the associated symmetric space, and when $n$ is equal to the rank, we show that the $n$-dimensional filling volume function grows exponentially. This broadly generalizes a theorem of Lubotzky--Mozes--Raghunathan on length distortion in lattices and confirms conjectures of Thurston, Gromov, and Bux--Wortman.

We show that Martin's Maximum${}^{++}$ implies Woodin's $\mathbb{P}_{\mathrm{max}}$ axiom $(*)$. This answers a question from the 1990s and amalgamates two prominent axioms of set theory which were both known to imply that there are $ℵ_2$ many real numbers.

Let $\Gamma $ be a lattice in $\mathrm{SO}_0(n, 1)$. We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\Gamma$ is arithmetic. This answers a question of Reid for hyperbolic $n$-manifolds and, independently, McMullen for hyperbolic $3$-manifolds. We prove these results by proving a superrigidity theorem for certain representations of such lattices. The proof of our superrigidity theorem uses results on equidistribution from homogeneous dynamics, and our main result also admits a formulation in that language.

We study a partially linearized version of the relative trace formula for the arithmetic Gan--Gross--Prasad conjecture for the unitary group $\mathrm{U}(V)$. The linear factor in this relative trace formula provides an $\mathrm{SL}_2$-symmetry which allows us to prove by induction the arithmetic fundamental lemma over $\mathbb{Q}_p$ when $p$ is odd and $p\geq \mathrm{dim}\ V$.