We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of q-deformed irreducible oscillator representations. We also give a combinatorial description of the embedding from the crystal of a q-oscillator representation to that of the negative half subalgebra given in terms of a PBW type basis. It is given as a composition of embeddings into the crystals of intermediate parabolic Verma modules, where the most nontrivial one is from an oscillator module to a maximally parabolic Verma module with respect to a quantum subsuperalgebra for $\mathfrak {gl}_{m|n}$. A new crystal theoretic realization of Burge correspondence of orthosymplectic type plays an important role in the description of this embedding.

Let G be a totally disconnected locally compact (tdlc) group. The contraction group $\mathrm {con}(g)$ of an element $g\in G$ is the set of all $h\in G$ such that $g^n h g^{-n} \to 1_G$ as $n \to \infty $. The nub of g can then be characterized as the intersection $\mathrm {nub}(g)$ of the closures of $\mathrm {con}(g)$ and $\mathrm {con}(g^{-1})$.

Contraction groups and nubs provide important tools in the study of the structure of tdlc groups, as already evidenced in the work of G. Willis. It is known that $\mathrm {nub}(g) = \{1\}$ if and only if $\mathrm {con}(g)$ is closed. In general, contraction groups are not closed and computing the nub is typically a challenging problem.

Maximal Kac–Moody groups over finite fields form a prominent family of nondiscrete compactly generated simple tdlc groups. In this paper we give a complete description of the nub of any element in these groups.

We develop the deformation theory of primitive Enriques varieties, which are defined as quasi-étale quotients of primitive symplectic varieties by nonsymplectic group actions. In particular, we establish a local Torelli theorem for primitive Enriques varieties. As applications thereof, we describe the behavior of certain primitive Enriques varieties under locally trivial deformations.

We establish the full asymptotic stability of solitary waves for the focusing cubic Schrödinger equation on the line under small even perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the symmetries of the problem, namely the invariance under scaling and phase shifts. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the nonselfadjoint linearized matrix Schrödinger operator around the solitary waves. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

We call a family of $s$ sets $\{F_1, \ldots , F_s\}$ a sunflower with $s$ petals if, for any distinct $i, j \in [s]$, one has $F_i \cap F_j = \cap _{u = 1}^s F_u$. The set $C = \cap _{u = 1}^s F_u$ is called the core of the sunflower. It is a classical result of Erdős and Rado that there is a function $\phi (s,k)$ such that any family of $k$-element sets contains a sunflower with $s$ petals. In 1977, Duke and Erdős asked for the size of the largest family $\mathcal {F}\subset {[n]\choose k}$ that contains no sunflower with $s$ petals and core of size $t-1$. In 1987, Frankl and Füredi asymptotically solved this problem for $k\geqslant 2t+1$ and $n>n_0(s,k)$. Their paper is one of the pinnacles of the so-called Delta-system method.

In this paper, we extend the result of Frankl and Füredi to a much broader range of parameters: $n>poly(s,t) k$, when additionally $k>poly(s,t)\log n$ or $n>n_0(s,t)$. We also extend this result to other domains, such as $[n]^k$ and ${n\choose k/w}^w$ and obtain even stronger and more general results for forbidden sunflowers with core at most $t-1$ (including results for families of permutations and subfamilies of the $k$-th layer in a simplicial complex).

The methods of the paper, among other things, combine the spread approximation approach, introduced by Zakharov and the first author, with the Delta-system approach of Frankl and Füredi and the hypercontractivity for global functions approach, developed by Keller, Lifshitz, and coauthors. Previous works in extremal set theory relied on at most one of these approaches. Creating such a unified view was one of the goals for the paper.

Let G be a product of unitary groups and let $(M,\omega )$ be a compact symplectic manifold with Hamiltonian G-action. We prove an equivariant formality result for any complex-oriented cohomology theory $\mathbb {E}^*$ (in particular, integral cohomology). This generalizes the celebrated result of Atiyah–Bott–Kirwan for rational cohomology from the 1980s. The proof does not use classical ideas but instead relies on a recent cohomological splitting result of Abouzaid–McLean–Smith [AMS21] for Hamiltonian fibrations over $\mathbb {CP}^1.$ Moreover, we establish analogues of the ‘localization’ and ‘injectivity to fixed points’ theorems for certain cohomology theories studied by Hopkins–Kuhn–Ravenel in [HKR00]. As an application of these results, we establish a Goresky–Kottwitz–MacPherson theorem with Morava K-theory coefficients for Hamiltonian T-manifolds.

Let ${\mathbb F}$ be an algebraically closed field and $G$ be an almost quasi-simple group. An important problem in representation theory is to classify the subgroups $H<G$ and ${\mathbb F} G$-modules $L$ such that the restriction $L{{\downarrow }}_H$ is irreducible. This problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where $G$ is the Schur’s double cover of an alternating or symmetric group.

The limited dependence between the additive and the multiplicative structure of fields is in the background of a number of explicit constructions of various types of pseudorandom objects. In this direction we study the size of the intersection of (the additive) translates of fibers of the (multiplicative) norm function over finite fields. Besides extending earlier upper bounds, our main focus here is on obtaining lower bounds.

From our results we conclude several consequences in extremal combinatorics. Our motivation is the projective norm graph $\text {NG}(q,t)$ and its small subgraph statistics. $\text {NG}(q,t)$ provides a tight construction for the Turán number of complete bipartite graphs $K_{t,s}$ with $s>(t-1)!$; in particular, it does not contain $K_{t, (t-1)!+1}$. Yet, for $t\geq 4$ it is not even known whether $\text {NG}(q,t)$ contains $K_{t,t}$. The determination of the largest integer $s(t)$, such that $\text {NG}(q,t)$ contains $K_{t,s(t)}$ for all large enough prime powers q is an important open question with far-reaching consequences, and the best known bounds, $t-1\leq s(t) \leq (t-1)!$, are very far apart. In this paper we settle the first open case and establish $s(4)=6$. Along the way we also count subgraphs of $\text {NG}(q,t)$ isomorphic to H, for any fixed $3$-degenerate graph H, and find that projective norm graphs are quasirandom with respect to these parameters. These results go beyond the consequences of the Expander Mixing Lemma and also imply extensions of the work of Alon and Shikhelman on generalized Turán numbers.

Finally, we also give an elementary proof of the $K_{4,s}$-freeness of $\text {NG}(q,t)$ for $s= 6(\sum _{i=0}^{t-4} q^i) +1$. This was known before for $t=4$ only, via a less direct algebro-geometric argument.

Let $P$ be a finite partially ordered set. In a recent series of works, Proudfoot introduced the notion of $Z$-polynomials associated with $P$-kernels, providing a unified framework for various intersection cohomology Poincaré polynomials arising in diverse areas of mathematics. One of the problems posed by Proudfoot was to interpret the $Z$-polynomial in a fundamental setting – namely, when $P$ is the lattice of faces of a convex polytope (or, more generally, an Eulerian poset). We resolve this problem by proving that the $Z$-polynomial of any Eulerian poset coincides with the toric $h$-polynomial of the poset of all (possibly empty) closed intervals of $P$, ordered by reverse inclusion. Under suitable polyhedral conditions, this result identifies the $Z$-polynomial of a polytope with the Poincaré polynomial of the intersection cohomology of an associated auxiliary polytope. We prove some results about the Chow polynomials of the poset of intervals of an Eulerian poset and relate them to the Veronese transforms of polynomials.

In this paper, we introduce the notion of post-Hopf algebroids, generalizing the pre-Hopf algebroids introduced in [8] in the study of exotic aromatic S-series. We construct action post-Hopf algebroids through actions of post-Hopf algebras. We show that the universal enveloping algebra of a post-Lie-Rinehart algebra (post-Lie algebroid) is naturally a post-Hopf algebroid. As a byproduct, we construct the free post-Lie-Rinehart algebra using a magma algebra with a linear map to the derivation Lie algebra of a commutative associative algebra. Applications in geometric numerical integration on manifolds are given.

In this paper we study generalizations of classical results on intersection patterns of set systems in $\mathbb {R}^d$, such as the fractional Helly theorem or the $(p,q)$-theorem, in the setting of arbitrary triangulable spaces with a forbidden homological minor.

Given a simplicial complex K and an integer b, we say that a family $\mathcal {F}$ of subcomplexes of some simplicial complex X is a (K,b)-free cover if (i) K is a forbidden homological minor of X, and (ii) the jth reduced Betti number $\tilde {\beta }_j(\bigcap _{S\in {\mathcal {G}}}S,\mathbb {Z}_2)$ is strictly less than b for all $0\leq j < \dim K$ and all nonempty subfamilies $\mathcal {G}\subseteq \mathcal {F}$.

We show that for every K and b, the fractional Helly number of a $(K,b)$-free cover is at most $\mu (K)+1$, where $\mu (K)$ is the maximum sum of the dimensions of two disjoint faces in K. This implies that the assertion of the $(p,q)$-theorem holds for every $p \ge q> \mu (K)$ and every $(K,b)$-free cover $\mathcal {F}$. For $b=1$ and a suitable K, this recovers the original $(p,q)$-theorem and its generalization to good covers. Interestingly, our results show that the range of parameters $(p,q)$ for which the $(p,q)$-theorem holds is independent of b.

Our proofs use Ramsey-type arguments combined with the notion of stair convexity of Bukh et al. to construct (forbidden) homological minors in certain cubical complexes.

For a given hypergraph H and a vertex $v\in V(H)$, consider a random matching M chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if H is a d-regular linear k-uniform hypergraph, the probability that M does not cover v is $(1 + o_d(1))d^{-1/k}$ for all vertices $v\in V(H).$ This conjecture was proved for $k = 2$ by Kahn and Kim in $1998.$

In this paper, we disprove this conjecture for all $k \geq 3.$ For infinitely many values of $d,$ we construct d-regular linear k-uniform hypergraph H containing two vertices $v_1$ and $v_2$ such that $\mathcal {P}(v_1 \notin M) = 1 - \frac {(1 + o_d(1))}{d^{k-2}}$ and $\mathcal {P}(v_2 \notin M) = \frac {(1 + o_d(1))}{d+1}.$ The gap between $\mathcal {P}(v_1 \notin M)$ and $\mathcal {P}(v_2 \notin M)$ in this H is best possible. In the course of proving this, we also prove a hypergraph analog of Godsil’s result on matching polynomials and paths in graphs, which is of independent interest.

We investigate when finite-order Hamiltonian diffeomorphisms extend to Hamiltonian circle actions, probing the transition from discrete to continuous symmetry in symplectic topology. Focusing on irrational ruled symplectic $4$-manifolds, we show that homologically trivial symplectic cyclic actions of order $k>2$ always extend to Hamiltonian $S^1$-actions, possibly after modifying the symplectic form. In contrast, we construct explicit symplectic involutions that cannot be so extended, even on minimal irrational ruled surfaces. These examples reveal geometric obstructions to extending discrete symmetries and highlight new exotic symplectic actions not equivalent to holomorphic ones. Our results also apply to higher-dimensional and noncyclic group actions, and we establish several structural results on the isomorphism types of finite groups that can act on irrational ruled symplectic $4$-manifolds.

Cohen proved that the infinite variable polynomial ring $R=k[x_1,x_2,\ldots ]$ is noetherian with respect to the action of the infinite symmetric group $\mathfrak {S}$. The first two authors began a program to understand the $\mathfrak {S}$-equivariant algebra of R in detail. In previous work, they classified the $\mathfrak {S}$-prime ideals of R. An important example of an $\mathfrak {S}$-prime is the ideal $\mathfrak {h}_s$ generated by $(s+1)$st powers of the variables. In this paper, we study the category of $R/\mathfrak {h}_s$-modules. We obtain a number of results, and mention just three here: (a) we determine the Grothendieck group of the category; (b) we show that the Krull–Gabriel dimension is s; and (c) we obtain generators for the derived category. This paper will play a key role in subsequent work where we study general modules.

We give a quadratic-time algorithm to compute the stretch factor and the invariant measured foliations for a pseudo-Anosov element of the mapping class group. As input, the algorithm accepts a word (in any given finite generating set for the mapping class group) representing a pseudo-Anosov mapping class, and the length of the word is our measure of complexity for the input. The output is a train track and an integer matrix where the stretch factor is the largest real eigenvalue and the unstable foliation is given by the corresponding eigenvector. This is the first algorithm to compute stretch factors and measured foliations that is known to terminate in subexponential time.

We prove that a smooth projective surface of degree d in $\mathbb P^3$ contains at most $d^2(d^2-3d+3)$ lines. We characterize the surfaces containing exactly $d^2(d^2-3d+3)$ lines: these occur only in prime characteristic p and, up to choice of projective coordinates, are cut out by equations of the form $x^{p^{e}+1}+y^{p^{e}+1}+z^{p^{e}+1}+ w^{p^{e}+1} = 0$.

We use a function field version of the circle method to prove that a positive proportion of elements in $\mathbb {F}_q[t]$ are representable as a sum of three cubes of minimal degree from $\mathbb {F}_q[t]$, assuming a suitable form of the Ratios Conjecture and that $\operatorname {\mathrm {char}}(\mathbb {F}_q)>3$. The analogue of this conjecture for quadratic Dirichlet L-functions is known for large fixed q, via recent developments in homological stability.

Let G be a transitive permutation group acting on $\Omega $. In this paper, we introduce and study the parameter $\mathrm {sep}(G)$, which denotes the size of the smallest set of points A such that, for every permutation $g\in G$, $A \cap A^g$ is nonempty. In particular, we focus on deriving general bounds for arbitrary transitive groups, and on the asymptotic behaviour of certain families of primitive groups. We also provide a classification of transitive groups with $\mathrm {sep}(G)$ largest possible, namely with $\mathrm {sep}(G)=\lceil (|\Omega |+1) / 2 \rceil $.

We consider polynomials which take integer values on the integers (IVPs), and satisfy an additional growth condition on the natural numbers. Elkies and Speyer, answering a question by Dimitrov, showed that there is a critical exponential growth threshold, such that there are infinitely many IVPs with growth above the threshold and finitely many IVPs below that threshold (of arbitrary degree). In this paper, we give more refined estimates for the number of IVPs satisfying such exponential growth constraints. In addition, we consider a similar problem on the integers, where not necessarily symmetric growth conditions are imposed. Notably, the critical threshold is determined by the logarithmic capacity of an explicit domain.

In this article, we describe completely the singularities appearing in Calogero–Moser varieties associated (at any parameter) to the wreath product symplectic reflection groups. We do so by parameterizing the symplectic leaves in the variety, describing combinatorially the resulting closure relation, and computing a transverse slice to each leaf. We also show that the normalization of the closure of each symplectic leaf is isomorphic to a Calogero–Moser variety for an associated (explicit) subquotient of the symplectic reflection group. This confirms a conjecture of Bonnafé for these groups.

We use the fact that the Calogero–Moser varieties associated to wreath products can be identified with certain Nakajima quiver varieties. In particular, our result identifying the normalization of the closure of each symplectic leaf with another quiver variety holds for arbitrary quiver varieties.