The primary goal of this paper is to identify syntomic complexes with the p-adic étale Tate twists of Geisser–Sato–Schneider on regular p-torsion-free schemes. Our methods apply naturally to a broader class of schemes that we call ‘F-smooth’. The F-smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes.

We define, for each quasisyntomic ring R (in the sense of Bhatt et al., Publ. Math. IHES 129 (2019), 199–310), a category $\mathrm {DM}^{\mathrm {adm}}(R)$ of admissible prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to $\mathrm {DM}^{\mathrm {adm}}(R)$ . We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.

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.

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 $d\ge 4$ .

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.

We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$ , where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.

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 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.

The aims of this paper are to answer several conjectures and questions about the multiplier spectrum of rational maps and giving new proofs of several rigidity theorems in complex dynamics by combining tools from complex and non-Archimedean dynamics.

A remarkable theorem due to McMullen asserts that, aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston’s rigidity theorem for post-critically finite maps, in which Teichmüller theory is an essential tool. We will give a new proof of McMullen’s theorem (and therefore a new proof of Thurston’s theorem) without using quasiconformal maps or Teichmüller theory.

We show that, aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalizes the aforementioned McMullen’s theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas also yield a simple proof of a rigidity theorem due to Zdunik.

We show that a rational map is exceptional if and only if one of the following holds: (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field, and (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor.

This work studies the average complexity of solving structured polynomial systems that are characterised by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that computes, with high probability, an approximate zero of a polynomial system given only as black-box evaluation program. Secondly, we introduce a universal model of random polynomial systems with prescribed evaluation complexity L. Combining both, we show that we can compute an approximate zero of a random structured polynomial system with n equations of degree at most ${D}$ in n variables with only $\operatorname {poly}(n, {D}) L$ operations with high probability. This exceeds the expectations implicit in Smale’s 17th problem.

We prove two theorems about the Malcev Lie algebra associated to the Torelli group of a surface of genus g: Stably, it is Koszul and the kernel of the Johnson homomorphism consists only of trivial $\mathrm {Sp}_{2g}(\mathbb {Z})$ -representations lying in the centre.

We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno–Drinfeld type theorem on type $\mathrm {B}$ braid group representations defined by the monodromy of KZ-equations and by the Balagović–Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.

Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq 2$ . We develop a new method for integrating over the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$ . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g.

Given $\phi \in \mathbb {X}_{g,n}$ and $\gamma \in \Gamma _{g}$ , we let $\mathsf {fix}_{\gamma }(\phi )$ be the number of fixed points of the permutation $\phi (\gamma )$ . The function $\mathsf {fix}_{\gamma }$ is a special case of a natural family of functions on $\mathbb {X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $n\to \infty $ , for the expectation of $\mathsf {fix}_{\gamma }$ with respect to the uniform probability measure on $\mathbb {X}_{g,n}$ , which is denoted by $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ . We prove that if $\gamma \in \Gamma _{g}$ is not the identity and q is maximal such that $\gamma $ is a q th power in $\Gamma _{g}$ , then

$$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=d(q)+O(n^{-1}) \end{align*}$$

as $n\to \infty $ , where $d\left (q\right )$ is the number of divisors of q. Even the weaker corollary that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]=o(n)$ as $n\to \infty $ is a new result of this paper. We also prove that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$ .

For a finite extension F of ${\mathbf Q}_p$ , Drinfeld defined a tower of coverings of (the Drinfeld half-plane). For $F = {\mathbf Q}_p$ , we describe a decomposition of the p-adic geometric étale cohomology of this tower analogous to Emerton’s decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finiteness theorem for the arithmetic étale cohomology modulo p whose proof uses Scholze’s functor, global ingredients, and a computation of nearby cycles which makes it possible to prove that this cohomology has finite presentation. This last result holds for all F; for $F\neq {\mathbf Q}_p$ , it implies that the representations of $\mathrm{GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case $F = {\mathbf Q}_p$ .

Let $M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$ denote the matrix multiplication tensor (and write $M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$ ), and let $\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$ denote the determinant polynomial considered as a tensor. For a tensor T, let $\underline {\mathbf {R}}(T)$ denote its border rank. We (i) give the first hand-checkable algebraic proof that $\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$ , (ii) prove $\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$ and $\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$ , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was $M_{\langle 2\rangle }$ , (iii) prove $\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$ , (iv) prove $\underline {\mathbf {R}}(\operatorname {det}_3)=17$ , improving the previous lower bound of $12$ , (v) prove $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$ for all $\mathbf {n}\geq 25$ , where previously only $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$ was known, as well as lower bounds for $4\leq \mathbf {n}\leq 25$ , and (vi) prove $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$ for all $\mathbf {n} \ge 18$ , where previously only $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$ was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors.

The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, called border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensor T and an integer r, in a finite number of steps, either outputs that there is no border rank r decomposition for T or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable when T has a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.

A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \sum _{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets A. In 1986, he asked if this bound is attained for the set of prime numbers. In this article, we answer in the affirmative.

As further applications of the method, we make progress towards a question of Erdős, Sárközy and Szemerédi from 1968. We also refine the classical Davenport–Erdős theorem on infinite divisibility chains, and extend a result of Erdős, Sárközy and Szemerédi from 1966.

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.

We use the theta correspondence to study the equivalence between Godement–Jacquet and Jacquet–Langlands L-functions for ${\mathrm {GL}}(2)$. We show that the resulting comparison is in fact an expression of an exotic symmetric monoidal structure on the category of ${\mathrm {GL}}(2)$-modules. Moreover, this enables us to construct an abelian category of abstractly automorphic representations, whose irreducible objects are the usual automorphic representations. We speculate that this category is a natural setting for the study of automorphic phenomena for ${\mathrm {GL}}(2)$, and demonstrate its basic properties.

This paper is a part of the author’s thesis [4].