We prove a conjecture due to Dadush, showing that if $\mathcal{L} \subset \mathbb{R}^n$ is a lattice such that $\det(\mathcal{L}') \ge 1$ for all sublattices $\mathcal{L}' \subseteq \mathcal{L}$, then$\sum_{\mathbf{y}\in \mathcal{L}} e^{-\pi t^2 \|\mathbf{y} \|^2} \le 3/2$,where $t := 10(\log n + 2)$.From this we derive bounds on the number of short lattice vectors, which can be viewed as a partial converse to Minkowski's celebrated first theorem. We also derive a bound on the covering radius.

We establish the flat cohomology version of the Gabber--Thomason purity for étale cohomology: for a complete intersection Noetherian local ring $(R, \mathfrak{m})$ and a commutative, finite, flat $R$-group $G$, the flat cohomology $H^i_\mathfrak{m}(R, G)$ vanishes for $i \lt \mathrm{dim}(R)$. For small $i$, this settles conjectures of Gabber that extend the Grothendieck--Lefschetz theorem and give purity for the Brauer group for schemes with complete intersection singularities. For the proof, we reduce to a flat purity statement for perfectoid rings, establish $p$-complete arc descent for flat cohomology of perfectoids, and then relate to coherent cohomology of $\mathbb{A}_{\mathrm{Inf}}$ via prismatic Dieudonné; theory. We also present an algebraic version of tilting for étale cohomology, use it to reprove the Gabber--Thomason purity, and exhibit general properties of fppf cohomology of (animated) rings with finite, locally free group scheme coefficients, such as excision, agreement with fpqc cohomology, and continuity.

In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for $n\ge 3$, asked if the same statement holds in dimension two. We show that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple. This settles what is known as the "simplicity conjecture" in the affirmative. In fact, we prove the a priori stronger statement that this group is not perfect.

Our general strategy is partially inspired by suggestions of Fathi and the approach of Oh towards the simplicity question. In particular, we show that infinite twist maps, studied by Oh, are not finite energy homeomorphisms, which resolves the "infinite twist conjecture" in the affirmative; these twist maps are now the first examples of Hamiltonian homeomorphisms that can be said to have infinite energy. Another consequence of our work is that various forms of fragmentation for volume-preserving homeomorphisms that hold for higher dimensional balls fail in dimension two.

A central role in our arguments is played by spectral invariants defined via periodic Floer homology. We establish many new properties of these invariants that are of independent interest. For example, we prove that these spectral invariants extend continuously to area-preserving homeomorphisms of the disc, and we also verify for certain smooth twist maps a conjecture of Hutchings concerning recovering the Calabi invariant from the asymptotics of these invariants.

We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0\lt q\lt 1$. Consequently, we can sample random spanning forests in a graph and estimate the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has edge expansion at least $1$.

Our algorithm and proof build on the recent results of Dinur, Kaufman, Mass and Oppenheim who show that a high-dimensional walk on a weighted simplicial complex mixes rapidly if for every link of the complex, the corresponding localized random walk on the $1$-skeleton is a strong spectral expander. One of our key observations is that a weighted simplicial complex $X$ is a $0$-local spectral expander if and only if a naturally associated generating polynomial $p_X$ is strongly log-concave. More generally, to every pure simplicial complex $X$ with positive weights on its maximal faces, we can associate a multiaffine homogeneous polynomial $p_X$ such that the eigenvalues of the localized random walks on $X$ correspond to the eigenvalues of the Hessian of derivatives of $p_X$.

In this paper we establish a congruence on the degree of the map from a component of a Hurwitz space of covers of elliptic curves to the moduli stack of elliptic curves. Combinatorially, this can be expressed as a congruence on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. Building on the work of Bourgain, Gamburd, and Sarnak, we apply this congruence to show that for all but finitely many primes $p$, the group of Markoff automorphisms acts transitively on the non-zero $\mathbb{F}_p$-points of the Markoff equation $x^2 + y^2 + z^2 - 3xyz = 0$. This yields a strong approximation property for the Markoff equation, the finiteness of congruence conditions satisfied by Markoff numbers, and the connectivity of a certain infinite family of Hurwitz spaces of $\mathrm{SL}_2(\mathbb{F}_p)$-covers of elliptic curves. With possibly finitely many exceptions, this resolves a conjecture of Bourgain, Gamburd, and Sarnak, first posed by Baragar in 1991, and a question of Frobenius, posed in 1913. Since their methods are effective, this reduces the conjecture to a finite computation.

We construct invariants of birational maps with values in the Kontsevich--Tschinkel group and in the truncated Grothendieck groups of varieties. These invariants are morphisms of groupoids and are well-suited to investigating the structure of the Grothendieck ring and L-equivalence. Building on known constructions of L-equivalence, we prove new unexpected results about Cremona groups.