For natural numbers $n,r\in \mathbb{N}$ with $n\geqslant r$, the Kneser graph $K(n,r)$ is the graph on the family of $r$-element subsets of $\{1,\ldots ,n\}$ in which two sets are adjacent if and only if they are disjoint. Delete the edges of $K(n,r)$ with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as $r/n$ is bounded away from $1/2$, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős–Ko–Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.

We establish the compactness of the moduli space of noncollapsed Calabi–Yau spaces with mild singularities. Based on this compactness result, we develop a new approach to study the weak compactness of Riemannian manifolds.

The ‘square peg problem’ or ‘inscribed square problem’ of Toeplitz asks if every simple closed curve in the plane inscribes a (nondegenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ‘homological’ nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper, we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs $f$, $g:[t_{0},t_{1}]\rightarrow \mathbb{R}$ that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums $y_{1}+y_{2}+y_{3}$ for $y_{1},y_{2},y_{3}$ ranging in finite sets of real numbers.

We show that, by taking normalizations over certain auxiliary good reduction integral models, one obtains integral models of toroidal and minimal compactifications of PEL-type Shimura varieties which enjoy many features of the good reduction theory studied as in the earlier works of Faltings and Chai’s and the author’s. We treat all PEL-type cases uniformly, with no assumption on the level, ramifications, and residue characteristics involved.

Given a mixed Hodge module $\mathcal{N}$ and a meromorphic function $f$ on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module $\mathcal{N}\otimes \mathcal{E}^{f}$, which extends the construction of Esnault et al. ($E_{1}$-degeneration of the irregular Hodge filtration (with an appendix by Saito), J. reine angew. Math. (2015), doi:10.1515/crelle-2014-0118). We show the strictness of the push-forward filtered ${\mathcal{D}}$-module through any projective morphism ${\it\pi}:X\rightarrow Y$, by using the theory of mixed twistor ${\mathcal{D}}$-modules of Mochizuki. We consider the example of the rescaling of a regular function $f$, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss–Manin systems of $f$ in terms of the Harder–Narasimhan filtration of the Kontsevich bundles associated with $f$.

We prove a simple level-raising result for regular algebraic, conjugate self-dual automorphic forms on $\mathrm{GL}_n$. This gives a systematic way to construct irreducible Galois representations whose residual representation is reducible.

We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$ that was first considered by Petersen et al. [J. Algebra324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product $[k]\times [n-k]$ of two chains (also called Gelfand–Tsetlin polytope), and that it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). We then observe that this already implies the existence of a flag simplicial polytope generalizing the dual associahedron, whose Stanley–Reisner ideal is an initial ideal of the Grassmann–Plücker ideal, while previous constructions of such a polytope did not guarantee flagness nor reduced to the dual associahedron for $k=2$. On our way we provide general results about order polytopes and their triangulations. We call the simplicial complex the noncrossing complex, and the polytope derived from it the dual Grassmann associahedron. We extend results of Petersen et al. [J. Algebra324(5) (2010), 951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflect the relations between Grassmannians with different parameters, in particular the isomorphism$G_{k,n}\cong G_{n-k,n}$. Moreover, our approach allows us to show that the adjacency graph of the noncrossing complex admits a natural acyclic orientation that allows us to define a Grassmann–Tamari order on maximal noncrossing families. Finally, we look at the precise relation of the noncrossing complex and the weak separability complex of Leclerc and Zelevinsky [Amer. Math. Soc. Transl.181(2) (1998), 85–108]; see also Scott [J. Algebra290(1) (2005), 204–220] among others. We show that the weak separability complex is not only a subcomplex of the noncrossing complex as noted by Petersen et al. [J. Algebra324(5) (2010), 951–969] but actually its cyclically invariant part.

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate $L^{p}$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.

For a prime p and a field k of characteristic $p,$ we define Steenrod operations $P^{n}_{k}$ on motivic cohomology with $\mathbb {F}_{p}$ -coefficients of smooth varieties defined over the base field $k.$ We show that $P^{n}_{k}$ is the pth power on $H^{2n,n}(-,\mathbb {F}_{p}) \cong CH^{n}(-)/p$ and prove an instability result for the operations. Restricted to mod p Chow groups, we show that the operations satisfy the expected Adem relations and Cartan formula. Using these new operations, we remove previous restrictions on the characteristic of the base field for Rost’s degree formula. Over a base field of characteristic $2,$ we obtain new results on quadratic forms.

We study the Chow ring of the moduli stack $\mathfrak {M}_{g,n}$ of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space $\overline {\mathcal {M}}_{g,n}$ of stable curves. This paper provides foundations for the paper [BS21].

In the appendix (jointly with J. Skowera), we develop the theory of a proper, but not necessary projective, pushforward of algebraic cycles. The proper pushforward is necessary for the construction of the tautological rings of $\mathfrak {M}_{g,n}$ and is important in its own right. We also develop operational Chow groups for algebraic stacks.

Let $\mathsf {C}$ be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to $\mathsf {C}$ . We prove that the decorated double Bott–Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras.

We explicitly describe the Donaldson–Thomas transformations on double Bott–Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock–Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson–Thomas transformations on a family of double Bott–Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov’s periodicity conjecture in the cases of $\Delta \square \mathrm {A}_r$ .

When $\mathsf {C}$ is of type $\mathrm {A}$ , the double Bott–Samelson cells are isomorphic to Shende–Treumann–Zaslow’s moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their $\mathbb {F}_q$ -points we obtain rational functions that are Legendrian link invariants.

We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let $M_{N}$ be a deterministic $N\times N$ matrix, and let $G_{N}$ be a complex Ginibre matrix. We consider the matrix ${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$, where $\unicode[STIX]{x1D6FE}>1/2$. With $L_{N}$ the empirical measure of eigenvalues of ${\mathcal{M}}_{N}$, we provide a general deterministic equivalence theorem that ties $L_{N}$ to the singular values of $z-M_{N}$, with $z\in \mathbb{C}$. We then compute the limit of $L_{N}$ when $M_{N}$ is an upper-triangular Toeplitz matrix of finite symbol: if $M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$ where $\mathfrak{d}$ is fixed, $a_{i}\in \mathbb{C}$ are deterministic scalars and $J$ is the nilpotent matrix $J(i,j)=\mathbf{1}_{j=i+1}$, then $L_{N}$ converges, as $N\rightarrow \infty$, to the law of $\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$ where $U$ is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when $\mathfrak{d}=1$, also of independent and identically distributed entries on the diagonals in $M_{N}$.

Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov–Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov–Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.

A celebrated result of Rödl and Ruciński states that for every graph $F$, which is not a forest of stars and paths of length 3, and fixed number of colours $r\geqslant 2$ there exist positive constants $c,C$ such that for $p\leqslant cn^{-1/m_{2}(F)}$ the probability that every colouring of the edges of the random graph $G(n,p)$ contains a monochromatic copy of $F$ is $o(1)$ (the ‘0-statement’), while for $p\geqslant Cn^{-1/m_{2}(F)}$ it is $1-o(1)$ (the ‘1-statement’). Here $m_{2}(F)$ denotes the 2-density of $F$. On the other hand, the case where $F$ is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in $G(n,p)$. Recently, the natural extension of the 1-statement of this theorem to $k$-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rödl and Schacht. In particular, they showed an upper bound of order $n^{-1/m_{k}(F)}$ for the 1-statement, where $m_{k}(F)$ denotes the $k$-density of $F$. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of threshold exists if $k\geqslant 4$: there are $k$-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour class. Along the way we obtain a general bound on the 1-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Spöhel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.

Let $L$ be a countable language. We say that a countable infinite $L$ -structure ${\mathcal{M}}$ admits an invariant measure when there is a probability measure on the space of $L$ -structures with the same underlying set as ${\mathcal{M}}$ that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of ${\mathcal{M}}$ . We show that ${\mathcal{M}}$ admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in $\text{Aut}({\mathcal{M}})$ of an arbitrary finite tuple of ${\mathcal{M}}$ fixes no additional points. When ${\mathcal{M}}$ is a Fraïssé limit in a relational language, this amounts to requiring that the age of ${\mathcal{M}}$ have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

We construct a hyperbolic group with a hyperbolic subgroup for which inclusion does not induce a continuous map of the boundaries.

It has been a long-standing question whether every amenable operator algebra is isomorphic to a (necessarily nuclear) $\mathrm{C}^*$-algebra. In this note, we give a nonseparable counterexample. Finding out whether a separable counterexample exists remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in $\mathrm{C}^*$-algebras and show that our method cannot produce a separable counterexample.

In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the corresponding Nikolskii inequality for the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.

We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.