We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.

We study the following rainbow version of subgraph containment problems in a family of (hyper)graphs, which generalizes the classical subgraph containment problems in a single host graph. For a collection \mathit {\mathbf {G}}=\{G_1, G_2,\ldots , G_{m}\}$\mathit {\mathbf {G}}=\{G_1, G_2,\ldots , G_{m}\}$ of not necessarily distinct k-graphs on the same vertex set [n]$[n]$, a (sub)graph H on [n]$[n]$ is rainbow if there exists an injection \varphi : E(H)\rightarrow [m]$\varphi : E(H)\rightarrow [m]$, such that e\in E(G_{\varphi (e)})$e\in E(G_{\varphi (e)})$ for each e\in E(H)$e\in E(H)$. Note that if |E(H)|=m$|E(H)|=m$, then \varphi $\varphi$ is a bijection, and thus H contains exactly one edge from each G_i$G_i$.

Our main results focus on rainbow clique-factors in (hyper)graph systems with minimum d-degree conditions. Specifically, we establish the following:

1. (1) A rainbow analogue of an asymptotical version of the Hajnal–Szemerédi theorem, namely, if t\mid n$t\mid n$ and \delta (G_i)\geq (1-\frac {1}{t}+\varepsilon )n$\delta (G_i)\geq (1-\frac {1}{t}+\varepsilon )n$ for each i\in [\frac {n}{t}\binom {t}{2}]$i\in [\frac {n}{t}\binom {t}{2}]$, then \mathit {\mathbf {G}}$\mathit {\mathbf {G}}$ contains a rainbow K_t$K_t$-factor;

2. (2) Essentially, a minimum d-degree condition forcing a perfect matching in a k-graph also forces rainbow perfect matchings in k-graph systems for d\in [k-1]$d\in [k-1]$.

The degree assumptions in both results are asymptotically best possible (although the minimum d-degree condition forcing a perfect matching in a k-graph is in general unknown). For (1), we also discuss two directed versions and a multipartite version. Finally, to establish these results, we in fact provide a general framework to attack this type of problem, which reduces it to subproblems with finitely many colors.

In this paper, together with the preceding Part I [10], we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to [10] the treatment of the mixed characteristic case. Hensel minimality is inspired by o-minimality and its role in real geometry and diophantine applications. We develop geometric results and applications for Hensel minimal structures that were previously known only under stronger or less axiomatic assumptions, and which often have counterparts in o-minimal structures. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under Hensel minimality – more precisely, 1$1$-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.

We consider the Vlasov equation in any spatial dimension, which has long been known [ZI76, Mor80, Gib81, MW82] to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie–Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. One may view this work as a classical counterpart to [MNP+20], which provided a rigorous derivation of the Hamiltonian structure of the cubic nonlinear Schrödinger equation from the many-body problem for interacting bosons in a certain infinite particle number limit, the first result of its kind. In particular, our work settles a question of Marsden, Morrison and Weinstein [MMW84] on providing a ‘statistical basis’ for the bracket structure of the Vlasov equation.

We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.

For the kernel B_{\kappa ,a}(x,y)$B_{\kappa ,a}(x,y)$ of the (\kappa ,a)$(\kappa ,a)$-generalized Fourier transform \mathcal {F}_{\kappa ,a}$\mathcal {F}_{\kappa ,a}$, acting in L^{2}(\mathbb {R}^{d})$L^{2}(\mathbb {R}^{d})$ with the weight |x|^{a-2}v_{\kappa }(x)$|x|^{a-2}v_{\kappa }(x)$, where v_{\kappa }$v_{\kappa }$ is the Dunkl weight, we study the important question of when \|B_{\kappa ,a}\|_{\infty }=B_{\kappa ,a}(0,0)=1$\|B_{\kappa ,a}\|_{\infty }=B_{\kappa ,a}(0,0)=1$. The positive answer was known for d\ge 2$d\ge 2$ and \frac {2}{a}\in \mathbb {N}$\frac {2}{a}\in \mathbb {N}$. We investigate the case d=1$d=1$ and \frac {2}{a}\in \mathbb {N}$\frac {2}{a}\in \mathbb {N}$. Moreover, we give sufficient conditions on parameters for \|B_{\kappa ,a}\|_{\infty }>1$\|B_{\kappa ,a}\|_{\infty }>1$ to hold with d\ge 1$d\ge 1$ and any a.

We also study the image of the Schwartz space under the \mathcal {F}_{\kappa ,a}$\mathcal {F}_{\kappa ,a}$ transform. In particular, we obtain that \mathcal {F}_{\kappa ,a}(\mathcal {S}(\mathbb {R}^d))=\mathcal {S}(\mathbb {R}^d)$\mathcal {F}_{\kappa ,a}(\mathcal {S}(\mathbb {R}^d))=\mathcal {S}(\mathbb {R}^d)$ only if a=2$a=2$. Finally, extending the Dunkl transform, we introduce nondeformed transforms generated by \mathcal {F}_{\kappa ,a}$\mathcal {F}_{\kappa ,a}$ and study their main properties.

A generating set for a finite group G is minimal if no proper subset generates G, and m(G)$m(G)$ denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist a,b> 0$a,b> 0$ such that any finite group G satisfies m(G) \leqslant a \cdot \delta (G)^b$m(G) \leqslant a \cdot \delta (G)^b$, for \delta (G) = \sum _{p \text { prime}} m(G_p)$\delta (G) = \sum _{p \text { prime}} m(G_p)$, where G_p$G_p$ is a Sylow p-subgroup of G. To do this, we first bound m(G)$m(G)$ for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank 1$1$ or 2$2$). In particular, we prove that there exist a,b> 0$a,b> 0$ such that any finite simple group G of Lie type of rank r over the field \mathbb {F}_{p^f}$\mathbb {F}_{p^f}$ satisfies r + \omega (f) \leqslant m(G) \leqslant a(r + \omega (f))^b$r + \omega (f) \leqslant m(G) \leqslant a(r + \omega (f))^b$, where \omega (f)$\omega (f)$ denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist a,b> 0$a,b> 0$ such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over \mathbb {F}_{p^f}$\mathbb {F}_{p^f}$ has size at most ar^b + \omega (f)$ar^b + \omega (f)$.

Remixed Eulerian numbers are a polynomial q-deformation of Postnikov’s mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as q-binomial coefficients and Garsia–Remmel’s q-hit numbers. We study their combinatorics in more depth. As polynomials in q, they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the permutahedron into certain combinatorial cubes, we obtain a second combinatorial interpretation. At q=1$q=1$, the former recovers Postnikov’s interpretation whereas the latter recovers Liu’s interpretation, both of which were obtained via methods different from ours.

We prove that the class of all the rings \mathbb {Z}/m\mathbb {Z}$\mathbb {Z}/m\mathbb {Z}$ for all m>1$m>1$ is decidable. This gives a positive solution to a problem of Ax asked in his celebrated 1968 paper on the elementary theory of finite fields [1, Problem 5, p. 270]. In our proof, we reduce the problem to the decidability of the ring of adeles \mathbb {A}_{\mathbb {Q}}$\mathbb {A}_{\mathbb {Q}}$ of \mathbb {Q}$\mathbb {Q}$.

Let G be a real Lie group, \Lambda <G$\Lambda <G$ a lattice and H\leqslant G$H\leqslant G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures \mu $\mu$ on H and, applying recent work of Eskin–Lindenstrauss, prove that \mu $\mu$-stationary probability measures on G/\Lambda $G/\Lambda$ are homogeneous. Transferring a construction by Benoist–Quint and drawing on ideas of Eskin–Mirzakhani–Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on G/\Lambda $G/\Lambda$ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in G/\Lambda $G/\Lambda$ for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons–Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a nonconformal and weighted setting.

We give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves Hermitian lattices over number fields.

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack \mathcal {X}$\mathcal {X}$ , which specializes to the Batyrev–Manin conjecture when \mathcal {X}$\mathcal {X}$ is a scheme and to Malle’s conjecture when \mathcal {X}$\mathcal {X}$ is the classifying stack of a finite group.

For a smooth rigid space X over a perfectoid field extension K of \mathbb {Q}_p$\mathbb {Q}_p$ , we investigate how the v-Picard group of the associated diamond X^{\diamondsuit }$X^{\diamondsuit }$ differs from the analytic Picard group of X. To this end, we construct a left-exact ‘Hodge–Tate logarithm’ sequence

\begin{align*}0\to \operatorname{Pic}_{\mathrm{an}}(X)\to \operatorname{Pic}_v(X^{\diamondsuit})\to H^0(X,\Omega_X^1)\{-1\}. \end{align*} $$\begin{align*}0\to \operatorname{Pic}_{\mathrm{an}}(X)\to \operatorname{Pic}_v(X^{\diamondsuit})\to H^0(X,\Omega_X^1)\{-1\}. \end{align*}$$

We deduce some analyticity criteria which have applications to p-adic modular forms. For algebraically closed K, we show that the sequence is also right-exact if X is proper or one-dimensional. In contrast, we show that, for the affine space \mathbb {A}^n$\mathbb {A}^n$ , the image of the Hodge–Tate logarithm consists precisely of the closed differentials. It follows that, up to a splitting, v-line bundles may be interpreted as Higgs bundles. For proper X, we use this to construct the p-adic Simpson correspondence of rank one.

We prove a relative Lefschetz–Verdier theorem for locally acyclic objects over a Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal 2$2$ -category of cohomological correspondences. We show that local acyclicity is equivalent to dualisability and deduce that duality preserves local acyclicity. As another application of the category of cohomological correspondences, we show that the nearby cycle functor over a Henselian valuation ring preserves duals, generalising a theorem of Gabber.

This paper presents a framework for compressed sensing that bridges a gap between existing theory and the current use of compressed sensing in many real-world applications. In doing so, it also introduces a new sampling method that yields substantially improved recovery over existing techniques. In many applications of compressed sensing, including medical imaging, the standard principles of incoherence and sparsity are lacking. Whilst compressed sensing is often used successfully in such applications, it is done largely without mathematical explanation. The framework introduced in this paper provides such a justification. It does so by replacing these standard principles with three more general concepts: asymptotic sparsity, asymptotic incoherence and multilevel random subsampling. Moreover, not only does this work provide such a theoretical justification, it explains several key phenomena witnessed in practice. In particular, and unlike the standard theory, this work demonstrates the dependence of optimal sampling strategies on both the incoherence structure of the sampling operator and on the structure of the signal to be recovered. Another key consequence of this framework is the introduction of a new structured sampling method that exploits these phenomena to achieve significant improvements over current state-of-the-art techniques.