A family of permutations $\mathcal{F}\subset S_n$ is said to be t-intersecting if any two permutations in $\mathcal{F}$ agree on at least t points. It is said to be $(t-1)$-intersection-free if no two permutations in $\mathcal{F}$ agree on exactly $t-1$ points. If $S,T\subset \{1,2,\dots ,n\}$ with $|S|=|T|$, and $\mathrm{\pi }:S\to T$ is a bijection, then the π-star in $S_n$ is the family of all permutations that agree with π on S. An s-star is a π-star such that π is a bijection between sets of size s.

Friedgut and Pilpel, and independently the first author, showed that if $\mathcal{F}\subset S_n$ is t-intersecting, and n is sufficiently large depending on t, then $|\mathcal{F}|\le (n-t)!$; this proved a conjecture of Deza and Frankl from 1977.

Here, we prove a considerable strengthening of the Deza–Frankl conjecture, namely, that if n is sufficiently large depending on t, and $\mathcal{F}\subset S_n$ is $(t-1)$-intersection-free, then $|\mathcal{F}|\le (n-t)!$, with equality iff $\mathcal{F}$ is a t-star.

The main ingredient of our proof is a “junta approximation” result, namely, that any $(t-1)$-intersection-free family of permutations is essentially contained in a t-intersecting junta (a “junta” being a union of boundedly many $O(1)$-stars). The proof of our junta approximation result relies, in turn, on (i) a weak regularity lemma for families of permutations (which outputs a junta whose stars are intersected by $\mathcal{F}$ in a weakly pseudorandom way), (ii) a combinatorial argument that “bootstraps” the weak notion of pseudorandomness into a stronger one, and finally (iii) a spectral argument for highly pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature and one being algebraic. The connection we demonstrate between these notions of pseudorandomness may find further applications.

To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian $\mathrm{Gr}(2,4)$ and four shapes of lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement, and the second two essentially arise from the accidental local isomorphism between $\mathrm{SO}(4)$ and $\mathrm{SO}(3)\times \mathrm{SO}(3)$. As an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings, we prove simultaneous equidistribution of all of these objects under two splitting conditions.

We show that any closed connected hypersurface in ${\mathbb{R}}^4$ with entropy less than or equal to that of the round cylinder is smoothly isotopic to the standard three-sphere.

A Shimura variety of Hodge type is a moduli space for abelian varieties equipped with a certain collection of Hodge cycles. We show that the Newton strata on such varieties are nonempty provided that the corresponding group G is quasisplit at p, confirming a conjecture of Fargues and Rapoport in this case. Under the same condition, we conjecture that every mod p isogeny class on such a variety contains the reduction of a special point. This is a refinement of Honda–Tate theory. We prove a large part of this conjecture for Shimura varieties of PEL type. Our results make no assumption on the availability of a good integral model for the Shimura variety. In particular, the group G may be ramified at p.