We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $\mathbf {R}^4$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$ -close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in $\mathbf {R}^4$ . The new proof is more closely related to techniques from the study of strictly positive scalar curvature.

For every $p\in (0,\infty )$ we associate to every metric space $(X,d_{X})$ a numerical invariant $\mathfrak{X}_{p}(X)\in [0,\infty ]$ such that if $\mathfrak{X}_{p}(X)<\infty$ and a metric space $(Y,d_{Y})$ admits a bi-Lipschitz embedding into $X$ then also $\mathfrak{X}_{p}(Y)<\infty$ . We prove that if $p,q\in (2,\infty )$ satisfy $q<p$ then $\mathfrak{X}_{p}(L_{p})<\infty$ yet $\mathfrak{X}_{p}(L_{q})=\infty$ . Thus, our new bi-Lipschitz invariant certifies that $L_{q}$ does not admit a bi-Lipschitz embedding into $L_{p}$ when $2<q<p<\infty$ . This completes the long-standing search for bi-Lipschitz invariants that serve as an obstruction to the embeddability of $L_{p}$ spaces into each other, the previously understood cases of which were metric notions of type and cotype, which however fail to certify the nonembeddability of $L_{q}$ into $L_{p}$ when $2<q<p<\infty$ . Among the consequences of our results are new quantitative restrictions on the bi-Lipschitz embeddability into $L_{p}$ of snowflakes of $L_{q}$ and integer grids in $\ell _{q}^{n}$ , for $2<q<p<\infty$ . As a byproduct of our investigations, we also obtain results on the geometry of the Schatten $p$ trace class $S_{p}$ that are new even in the linear setting.

We prove a generalisation of the Brill-Noether theorem for the variety of special divisors $W^r_d(C)$ on a general curve C of prescribed gonality. Our main theorem gives a closed formula for the dimension of $W^r_d(C)$ . We build on previous work of Pflueger, who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of Speyer on genus $1$ curves to arbitrary genus.

We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.

We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $\operatorname {\mathrm {GL}}_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.

For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification.

The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.

We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field ${\mathbb K}$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a $\mathbb {Z}$-model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. 371 no.2 (2018)]) when $\mathsf {char}\, {\mathbb K} =0$ or $\gg 0$, and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$. Some particular cases of these results play crucial roles in our proof [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.

We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .

We study the higher genus equivariant Gromov–Witten theory of the Hilbert scheme of $n$ points of $\mathbb{C}^{2}$. Since the equivariant quantum cohomology, computed by Okounkov and Pandharipande [Invent. Math. 179 (2010), 523–557], is semisimple, the higher genus theory is determined by an $\mathsf{R}$-matrix via the Givental–Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required $\mathsf{R}$-matrix by explicit data in degree $0$. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme $\mathsf{Hilb}^{n}(\mathbb{C}^{2})$ and the Gromov–Witten/Donaldson–Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the analytic continuation of the fundamental solution of the QDE of the Hilbert scheme of points determined by Okounkov and Pandharipande [Transform. Groups 15 (2010), 965–982]. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov–Witten theory of the symmetric product $\mathsf{Sym}^{n}(\mathbb{C}^{2})$ is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [Bryan and Graber, Algebraic Geometry–Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23–42; Coates et al., Geom. Topol. 13 (2009), 2675–2744; Coates & Ruan, Ann. Inst. Fourier (Grenoble) 63 (2013), 431–478].

In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces.

We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .

The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.

What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from $1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.

Let $P_{1},\ldots ,P_{k}:\mathbb{Z}\rightarrow \mathbb{Z}$ be polynomials of degree at most $d$ for some $d\geqslant 1$, with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$, there exists integers $n,m$ such that $n+P_{1}(m),\ldots ,n+P_{k}(m)$ are all not divisible by $p$. We show that there exist infinitely many natural numbers $n,m$ such that $n+P_{1}(m),\ldots ,n+P_{k}(m)$ are simultaneously prime, generalizing a previous result of the authors, which was restricted to the special case $P_{1}(0)=\cdots =P_{k}(0)=0$ (though it allowed for the top degree coefficients to coincide). Furthermore, we obtain an asymptotic for the number of such prime pairs $n,m$ with $n\leqslant N$ and $m\leqslant M$ with $M$ slightly less than $N^{1/d}$. This asymptotic is already new in general in the homogeneous case $P_{1}(0)=\cdots =P_{k}(0)=0$. Our arguments rely on four ingredients. The first is a (slightly modified) generalized von Neumann theorem of the authors, reducing matters to controlling certain averaged local Gowers norms of (suitable normalizations of) the von Mangoldt function. The second is a more recent concatenation theorem of the authors, controlling these averaged local Gowers norms by global Gowers norms. The third ingredient is the work of Green and the authors on linear equations in primes, allowing one to compute these global Gowers norms for the normalized von Mangoldt functions. Finally, we use the Conlon–Fox–Zhao densification approach to the transference principle to combine the preceding three ingredients together. In the special case $P_{1}(0)=\cdots =P_{k}(0)=0$, our methods also give infinitely many $n,m$ with $n+P_{1}(m),\ldots ,n+P_{k}(m)$ in a specified set primes of positive relative density $\unicode[STIX]{x1D6FF}$, with $m$ bounded by $\log ^{L}n$ for some $L$ independent of the density $\unicode[STIX]{x1D6FF}$. This improves slightly on a result from our previous paper, in which $L$ was allowed to depend on $\unicode[STIX]{x1D6FF}$.

We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma $ in $\mathbb R^d$ , $d\ge 3$ . Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran–Guo–Hickman–Seeger [1]. Besides, we show the sharp local smoothing estimates on a range of p for every $d\ge 3$ . As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $\gamma $ for $d\ge 4$ .

A finite set of integers A tiles the integers by translations if $\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $A\oplus B=\mathbb {Z}_M$ for some $M\in \mathbb {N}$ and $B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B.

In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group $\mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].

Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline {K}}$ has infinitely many rational curves or X has infinitely many unirational specialisations.

Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.

We construct families of translationally invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lower-bounded by a constant. Moreover, this constant can be taken equal to the operator norm of the local operator that generates the Hamiltonian (the local interaction strength). The result still holds true if one restricts to arbitrarily small quantum perturbations of classical Hamiltonians. The proof combines a robustness analysis of Robinson’s aperiodic tiling, together with tools from quantum information theory: the quantum phase estimation algorithm and the history state technique mapping Quantum Turing Machines to Hamiltonians.

In this paper, we completely resolve the well-known problem of Erdős and Sauer from 1975 which asks for the maximum number of edges an n-vertex graph can have without containing a k-regular subgraph, for some fixed integer $k\geq 3$. We prove that any n-vertex graph with average degree at least $C_k\log \log n$ contains a k-regular subgraph. This matches the lower bound of Pyber, Rödl and Szemerédi and substantially improves an old result of Pyber, who showed that average degree at least $C_k\log n$ is enough.

Our method can also be used to settle asymptotically a problem raised by Erdős and Simonovits in 1970 on almost regular subgraphs of sparse graphs and to make progress on the well-known question of Thomassen from 1983 on finding subgraphs with large girth and large average degree.

The affine Deligne–Lusztig variety $X_w(b)$ in the affine flag variety of a reductive group ${\mathbf G}$ depends on two parameters: the $\sigma $ -conjugacy class $[b]$ and the element w in the Iwahori–Weyl group $\tilde {W}$ of ${\mathbf G}$ . In this paper, for any given $\sigma $ -conjugacy class $[b]$ , we determine the nonemptiness pattern and the dimension formula of $X_w(b)$ for most $w \in \tilde {W}$ .

We show the validity of two special cases of the four-dimensional minimal model program (MMP) in characteristic $p>5$ : for contractions to ${\mathbb {Q}}$ -factorial fourfolds and in families over curves (‘semistable MMP’). We also provide their mixed characteristic analogues. As a corollary, we show that liftability of positive characteristic threefolds is stable under the MMP and that liftability of three-dimensional Calabi–Yau varieties is a birational invariant. Our results are partially contingent upon the existence of log resolutions.