Let $p(k)$ denote the partition function of $k$. For each $k\geqslant 2$, we describe a list of $p(k)-1$ quasirandom properties that a $k$-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness by Kohayakawa, Rödl, and Skokan, and by Conlon, Hàn, Person, and Schacht, and the spectral approach of Friedman and Wigderson. For each of the quasirandom properties that is described, we define the largest and the second largest eigenvalues. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon, Hàn, Person, and Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung, Graham, and Wilson for graphs.

We introduce a novel solution concept, denoted ${\it\alpha}$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa–Holm system on the line with vanishing asymptotics. All the ${\it\alpha}$-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.

We investigate the interplay between the local and asymptotic geometry of a set $A\subseteq \mathbb{R}^{n}$ and the geometry of model sets ${\mathcal{S}}\subset {\mathcal{P}}(\mathbb{R}^{n})$, which approximate $A$ locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an $(n-1)$-dimensional asymptotically optimally doubling measure in $\mathbb{R}^{n}$ ($n\geqslant 4$) has upper Minkowski dimension at most $n-4$.

Arithmetic duality theorems over a local field $k$ are delicate to prove if $\text{char}\,k>0$. In this case, the proofs often exploit topologies carried by the cohomology groups $H^{n}(k,G)$ for commutative finite type $k$-group schemes $G$. These ‘Čech topologies’, defined using Čech cohomology, are impractical due to the lack of proofs of their basic properties, such as continuity of connecting maps in long exact sequences. We propose another way to topologize $H^{n}(k,G)$: in the key case when $n=1$, identify $H^{1}(k,G)$ with the set of isomorphism classes of objects of the groupoid of $k$-points of the classifying stack $\mathbf{B}G$ and invoke Moret-Bailly’s general method of topologizing $k$-points of locally of finite type $k$-algebraic stacks. Geometric arguments prove that these ‘classifying stack topologies’ enjoy the properties expected from the Čech topologies. With this as the key input, we prove that the Čech and the classifying stack topologies actually agree. The expected properties of the Čech topologies follow, and these properties streamline a number of arithmetic duality proofs given elsewhere.

We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub and Smale for computations over $\mathbb{R}$ (which in turn followed those of the classical, discrete, complexity theory as laid down by Cook, Karp, and Levin, among others). In particular, we focus on complexity classes of decision problems and, paramount among them, on appropriate versions of the classes $\mathsf{P}$, $\mathsf{NP}$, and $\mathsf{EXP}$ of polynomial, nondeterministic polynomial, and exponential time, respectively. We prove some basic relationships between these complexity classes, and provide natural NP-complete problems.

Recently, settling a question of Erdős, Balogh, and Petříčková showed that there are at most $2^{n^{2}/8+o(n^{2})}$$n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here, we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set.

Our proof uses the Ruzsa–Szemerédi removal lemma, the Erdős–Simonovits stability theorem, and recent results of Balogh, Morris, and Samotij and Saxton and Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_{3}$s, which is of independent interest.

We prove that the coherent cohomology of a proper morphism of noetherian schemes can be made arbitrarily $p$-divisible by passage to proper covers (for a fixed prime $p$). Under some extra conditions, we also show that $p$-torsion can be killed by passage to proper covers. These results are motivated by the desire to understand rational singularities in mixed characteristic, and have applications in $p$-adic Hodge theory.

Let $\text{H}$ be a subgroup of some locally compact group $\text{G}$. Assume that $\text{H}$ is approximable by discrete subgroups and that $\text{G}$ admits neighborhood bases which are almost invariant under conjugation by finite subsets of $\text{H}$. Let $m:\text{G}\rightarrow \mathbb{C}$ be a bounded continuous symbol giving rise to an $L_{p}$-bounded Fourier multiplier (not necessarily completely bounded) on the group von Neumann algebra of $\text{G}$ for some $1\leqslant p\leqslant \infty$. Then, $m_{\mid _{\text{H}}}$ yields an $L_{p}$-bounded Fourier multiplier on the group von Neumann algebra of $\text{H}$ provided that the modular function ${\rm\Delta}_{\text{G}}$ is equal to 1 over $\text{H}$. This is a noncommutative form of de Leeuw’s restriction theorem for a large class of pairs $(\text{G},\text{H})$. Our assumptions on $\text{H}$ are quite natural, and they recover the classical result. The main difference with de Leeuw’s original proof is that we replace dilations of Gaussians by other approximations of the identity for which certain new estimates on almost-multiplicative maps are crucial. Compactification via lattice approximation and periodization theorems are also investigated.

We incorporate the new theory of equivariant moving frames for Lie pseudogroups into Vessiot’s method of group foliation of differential equations. The automorphic system is replaced by a set of reconstruction equations on the pseudogroup jets. The result is a completely algorithmic and symbolic procedure for finding both invariant and noninvariant solutions of differential equations admitting a symmetry group.

Let $w_{1}$ and $w_{2}$ be nontrivial words in free groups $F_{n_{1}}$ and $F_{n_{2}}$, respectively. We prove that, for all sufficiently large finite nonabelian simple groups $G$, there exist subsets $C_{1}\subseteq w_{1}(G)$ and $C_{2}\subseteq w_{2}(G)$ such that $|C_{i}|=O(|G|^{1/2}\log ^{1/2}|G|)$ and $C_{1}C_{2}=G$. In particular, if $w$ is any nontrivial word and $G$ is a sufficiently large finite nonabelian simple group, then $w(G)$ contains a thin base of order $2$. This is a nonabelian analog of a result of Van Vu [‘On a refinement of Waring’s problem’, Duke Math. J. 105(1) (2000), 107–134.] for the classical Waring problem. Further results concerning thin bases of $G$ of order $2$ are established for any finite group and for any compact Lie group $G$.

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

Let $R$ be a commutative noetherian local ring. As an analog of the notion of the dimension of a triangulated category defined by Rouquier, the notion of the dimension of a subcategory of finitely generated $R$-modules is introduced in this paper. We found evidence that certain categories over nice singularities have small dimensions. When $R$ is Cohen–Macaulay, under a mild assumption it is proved that finiteness of the dimension of the full subcategory consisting of maximal Cohen–Macaulay modules which are locally free on the punctured spectrum is equivalent to saying that $R$ is an isolated singularity. As an application, the celebrated theorem of Auslander, Huneke, Leuschke, and Wiegand is not only recovered but also improved. The dimensions of stable categories of maximal Cohen–Macaulay modules as triangulated categories are also investigated in the case where $R$ is Gorenstein, and special cases of the recent results of Aihara and Takahashi, and Oppermann and Št́ovíček are recovered and improved. Our key technique involves a careful study of annihilators and supports of $\mathsf{Tor}$, $\mathsf{Ext}$, and $\underline{\mathsf{Hom}}$ between two subcategories.

Let $p\geqslant 5$ be a prime, and let ${\mathcal{O}}$ be the ring of integers of a finite extension $K$ of $\mathbb{Q}_{p}$ with uniformizer ${\it\pi}$. Let ${\it\rho}_{n}:G_{\mathbb{Q}}\rightarrow \mathit{GL}_{2}\left({\mathcal{O}}/({\it\pi}^{n})\right)$ have modular mod-${\it\pi}$ reduction $\bar{{\it\rho}}$, be ordinary at $p$, and satisfy some mild technical conditions. We show that ${\it\rho}_{n}$ can be lifted to an ${\mathcal{O}}$-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when $K$ is a ramified extension of $\mathbb{Q}_{p}$. We also show that a prescribed ramified complete discrete valuation ring ${\mathcal{O}}$ is the weight-$2$ deformation ring for $\bar{{\it\rho}}$ for a suitable choice of auxiliary level. This implies that the field of Fourier coefficients of newforms of weight 2, square-free level, and trivial nebentype that give rise to semistable $\bar{{\it\rho}}$ of weight 2 can have arbitrarily large ramification index at $p$.

Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain local hypotheses. As a consequence, we prove that for a Hilbert modular form of parallel weight, trivial character, and good ordinary reduction at all primes dividing $p$, if the central critical $L$-value is zero then the $p$-adic Selmer group of it has rank at least one. We also prove that one of the local assumptions in the main result of Skinner and Urban can be removed by a base-change trick.

We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise.

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.

For each prime $p$ we construct a family $\{G_{i}\}$ of finite $p$-groups such that $|\text{Aut}(G_{i})|/|G_{i}|$ tends to zero as $i$ tends to infinity. This disproves a well-known conjecture that $|G|$ divides $|\text{Aut}(G)|$ for every nonabelian finite $p$-group $G$.

We prove global well-posedness of the time-dependent degenerate thermistor problem by establishing a uniform-in-time bounded mean ocsillation (BMO) estimate of inhomogeneous parabolic equations. Applying this estimate to the temperature equation, we derive a BMO bound of the temperature uniform with respect to time, which implies that the electric conductivity is an $A_{2}$ weight. The Hölder continuity of the electric potential is then proved by applying the De Giorgi–Nash–Moser estimate for degenerate elliptic equations with an $A_{2}$ coefficient. The uniqueness of the solution is proved based on the established regularity of the weak solution. Our results also imply the existence of a global classical solution when the initial and boundary data are smooth.

We introduce and study a multi-marginal optimal partial transport problem. Under a natural and sharp condition on the dominating marginals, we establish uniqueness of the optimal plan. Our strategy of proof establishes and exploits a connection with another novel problem, which we call the Monge–Kantorovich partial barycenter problem (with quadratic cost). This latter problem has a natural interpretation as a variant of the factories-and-mines description of optimal transport. We then turn our attention to various analytic properties of these two problems. Of particular interest, we show that monotonicity of the active marginals with respect to the amount $m$ of mass to be transported can fail, a surprising difference from the two-marginal case.

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.