In this paper we compute the intersection number of two double ramification (DR) cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic DR integrals are the main ingredients for the computation of the DR hierarchy associated to the infinite‐dimensional partial cohomological field theory given by exp

We show that the imaginary multiplicative chaos exp ( i β Γ ) determines the gradient of the underlying field Γ for all log‐correlated Gaussian fields with covariance of the form − log | x − y | + g ( x , y ) with mild regularity conditions on g , for all d ⩾ 2 and for all β ∈ ( 0 , d ) . In particular, we show that the 2D continuum zero boundary Gaussian free field is measurable with respect to its

In this paper, we study conjugacy classes for pivotal fusion categories. In particular, we prove a Burnside type formula for the structure constants concerning the product of two conjugacy class sums of such a fusion category. For a braided weakly integral fusion category C , we show that these structure constants multiplied by dim ( C ) are non‐negative integers, extending some results obtained by

For an irreducible character χ of a finite group G , the codegree of χ is defined by | G : ker χ | / χ ( 1 ) . In this note, we show that if a finite solvable group G has an element of order m , then G admits an irreducible character of codegree divisible by m .

Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag

For every positive integer n , we construct a Hasse diagram with n vertices and independence number O ( n 3 / 4 ) . Such graphs have chromatic number Ω ( n 1 / 4 ) , which significantly improves the previously best‐known constructions of Hasse diagrams having chromatic number Θ ( log n ) . In addition, if we also require girth of at least k ⩾ 5 , we construct such Hasse diagrams with independence number

We construct new finite semigroups in β N , in particular, an m ‐element null semigroup for any m ⩾ 2 and a 7‐element semigroup generated by two idempotents. We also point out a connection of finite semigroups in β N with Ramsey theory. In particular, we deduce that for every m ⩾ 2 , there exists a partition { A 1 , … , A m } of N with the following property: for any finite partitions B i of A i ,

We prove a contractive Hardy–Littlewood type inequality for functions from H p ( T ) , 0 < p ⩽ 2 which is sharp in the first two Taylor coefficients and asymptotically at infinity.

We investigate a variant of the Beurling–Ahlfors extension of quasisymmetric homeomorphisms of the real line that is given by the convolution of the heat kernel, and prove that the complex dilatation of such a quasiconformal extension of a strongly symmetric homeomorphism (that is, its derivative is an A ∞ ‐weight whose logarithm is in VMO) induces a vanishing Carleson measure on the upper half‐plane

We show the existence of non‐Einstein cones which are critical for all quadratic curvature functionals.

We continue our study of ‘no‐dimension’ analogues of basic theorems in combinatorial and convex geometry in Banach spaces. We generalize some results of the paper (Adiprasito, Bárány and Mustafa, ‘Theorems of Carathéodory, Helly, and Tverberg without dimension’, Proceedings of the Thirtieth Annual ACM‐SIAM Symposium on Discrete Algorithms (Society for Industrial and Applied Mathematics, San Diego,

We determine the diffraction intensity of the k ‐free integers near the origin.

Let X be a compact Kähler manifold of dimension n . For 1 ⩽ j ⩽ m , let T j be a closed positive (1,1)‐current on X whose cohomology class is Kähler. Let T be a closed positive current of bi‐degree ( p , p ) with m + p ⩽ n . We prove that if T 1 , … , T m are of T ‐relative full mass intersection, then the T ‐relative non‐pluripolar product of T 1 , … , T m is equal to the Dinh–Sibony product of T

We prove that every n ‐vertex tournament G has an acyclic subgraph with chromatic number at least n 5 / 9 − o ( 1 ) , while there exists an n ‐vertex tournament G whose every acyclic subgraph has chromatic number at most n 3 / 4 + o ( 1 ) . This establishes in a strong form a conjecture of Nassar and Yuster and improves on another result of theirs. Our proof combines probabilistic and spectral techniques

In Theorems A and B of [1], we state that for 1 < p < ∞ the lattice of closed ideals of L ( ℓ p , c 0 ) , L ( ℓ p , ℓ ∞ ) and of L ( ℓ 1 , ℓ p ) are at least of cardinality c , the continuum. What our proof actually shows is that the cardinality of the lattice of closed ideals of L ( ℓ p , c 0 ) , L ( ℓ p , ℓ ∞ ) and of L ( ℓ 1 , ℓ p ) , is at least 2 c , and thus equal to it.

We prove that the chromatic number of a circle graph with clique number ω is at most 7 ω 2 .

Extending methods first used by Casson, we show how to verify a hyperbolic structure on a finite triangulation of a closed 3‐manifold using interval arithmetic methods. A key ingredient is a new theoretical result (akin to a theorem by Neumann–Zagier and Moser for ideal triangulations on which HIKMOT is based) showing that there is a redundancy among the edge equations if the edges avoid ‘gimbal lock’

This paper surveys results concerning peak sets and boundary interpolation sets for the unit disc. It includes hitherto unpublished results proved by Ullrich on peak sets for the Zygmund class.

We study a notion of ‘width’ for Jordan curves in CP 1 , paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three‐space. A similar invariant in the setting of anti‐de Sitter geometry was used by Bonsante–Schlenker to characterize quasicircles among a larger class of Jordan curves in the boundary of

The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties). The generalization for singular varieties is due to Brasselet–Schürmann–Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed

Any group of type F 4 is obtained as the automorphism group of an Albert algebra. We prove that such a group is R ‐trivial whenever the Albert algebra is obtained from the first Tits construction. Our proof uses cohomological techniques and the corresponding result on the structure group of such Albert algebras.

In an algebraic family of rational maps of P 1 , we show that, for almost every parameter for the trace of the bifurcation current of a marked critical value, the critical value is Collet–Eckmann. This extends previous results of Graczyk and Świa̧tek in the unicritical family, using Makarov theorem. Our methods are based instead on ideas of laminar currents theory.

Recently, Van‐Erp and Yuncken and independently Choi and Ponge defined an inhomogeneous deformation groupoid. As shown by Van‐Erp and Yuncken, this deformation groupoid allows to fully recovers the general inhomogeneous pseudo‐differential calculus.

In this article, we consider the flat bundle U and the kernel K of the Higgs field naturally associated to any (polarized) variation of Hodge structures of weight 1. We study how strict the inclusion U ⊆ K can be in the geometric case. More precisely, for any smooth projective curve C of genus g ⩾ 2 and any r = 0 , … , g − 1 , we construct non‐isotrivial deformations of C over a quasi‐projective base

Let X and Y be Banach spaces, and T : X ∗ → Y be an operator. We prove that if X is Asplund and Y has the approximation property, then for each Radon probability μ on ( B X ∗ , w ∗ ) there is a sequence of w ∗ ‐to‐norm continuous operators T n : X ∗ → Y such that ∥ T n ( x ∗ ) − T ( x ∗ ) ∥ → 0 for μ ‐a.e. x ∗ ∈ B X ∗ ; if Y has the λ ‐bounded approximation property for some λ ⩾ 1 , then the sequence

We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a graph, there exists a mesoscopic scale on which the graph looks like R in the sense that the rescaled ball is close to a line segment in the Gromov–Hausdorff metric

We find a substantial class of pairs of ∗ ‐homomorphisms between graph C*‐algebras of the form C ∗ ( E ) ↪ C ∗ ( G ) ↞ C ∗ ( F ) whose pullback C*‐algebra is an AF graph C*‐algebra. Our result can be interpreted as a recipe for determining the quantum space obtained by shrinking a quantum subspace. There are numerous examples from noncommutative topology, such as quantum complex projective spaces (including

Let A ⊂ N , α ∈ ( 0 , 1 ) , and e ( x ) : = e 2 π i x for x ∈ R . We set S A ( α , N ) : = ∑ n ∈ A n ⩽ N e ( n α ) . Recently, A'Campo posed the following question: Is there an infinite non‐cofinite set A ⊂ N such that for all α ∈ ( 0 , 1 ) the sum S A ( α , N ) has bounded modulus as N → + ∞ ? In this note, we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on

We discuss the problem of lifting projective bundles to vector bundles, giving necessary and sufficient conditions for a lift to exist both in the smooth and in the holomorphic categories. These criteria are formulated and proved in the language of topology and complex differential geometry, respectively.

A well‐known result of Stanley from 1980 implies that the weak order on a maximal parabolic quotient of the symmetric group S n has the Sperner property; this same property was recently established for the weak order on all of S n by Gaetz and Gao, resolving a long‐open problem. In this paper, we interpolate between these results by showing that the weak order on any parabolic quotient of S n (and

We characterize the holomorphic self‐maps of the unit disk for which the corresponding composition operator is an isometry of BMOA equipped with the Möbius invariant H p norm, answering a question posed by Laitila, for p ∈ [ 1 , 2 ) . Also, we show that the image of the symbol of an isometric composition operator for BMOA, with respect to the Möbius invariant H p norm, p ∈ [ 1 , + ∞ ) , covers the

Let R be a real closed field. We prove that if R is uncountable, then any separately Nash (respectively, arc‐Nash) function defined over R is semialgebraic (respectively, continuous semialgebraic). To complete the picture, we provide an example showing that the assumption on R to be uncountable cannot be dropped. Moreover, even if R is uncountable but non‐Archimedean, then the shape of the domain of

Much has been written about Grothendieck duality. This survey will make the point that most of this literature is now obsolete: there is a brilliant 1968 article by Verdier with the right idea on how to approach the subject. Verdier's article was largely forgotten for two decades until Lipman resurrected it, reworked it and developed the ideas to obtain the right statements for what had before been

We prove a decomposition formula for the dimensional reduction of an extended topological field theory that arises as an orbifold of an equivariant topological field theory. Our decomposition formula can be expressed in terms of a categorification of the integral with respect to groupoid cardinality. The application of our result to topological field theories of Dijkgraaf–Witten type proves a recent

The expected signature is an analogue of the Laplace transform for probability measures on rough paths. A key question in the area has been to identify a general condition to ensure that the expected signature uniquely determines the measures. A sufficient condition has recently been given by Chevyrev and Lyons and requires a strong upper bound on the expected signature. While the upper bound was verified

We study Gabor orthonormal windows in L 2 ( Z p d ) for translation and modulation sets, where p is prime and d ⩾ 2 . We prove that for a set E ⊂ Z p d , the indicator function 1 E is a Gabor window if and only if E tiles and is spectral. Moreover, we prove that for any function g : Z p d → C with support E , if the cardinality of E coincides with the cardinality of the modulation set or if g is positive

The theory of F ‐manifolds, and more generally, manifolds endowed with commutative and associative multiplication of their tangent fields, was discovered and formalised in various models of quantum field theory involving algebraic and analytic geometry, at least since the 1990s.

We show that Picard–Fuchs equations of periods of certain families of abelian surfaces with quaternionic multiplication have fractional powers of algebraic modular forms as solutions. We give several applications of this modularity property, including a computation of exceptional sets of certain transcendental functions.

In this paper, we prove the irreducibility of the monodromy action on the anti‐invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surface S of degree ⩾ 7 in P 3 branched at a very general

The Markov group conjecture, a long‐standing open problem in the theory of Markov processes with countable state space, asserts that a strongly continuous Markov semigroup T = ( T t ) t ∈ [ 0 , ∞ ) on ℓ 1 has bounded generator if the operator T 1 is bijective. Attempts to disprove the conjecture have often aimed at glueing together finite‐dimensional matrix semigroups of growing dimension — that is

We show that the Smoluchowski coagulation equation with the solvable kernels K ( x , y ) equal to 2, x + y or x y is contractive in suitable Laplace norms. In particular, this proves exponential convergence to a self‐similar profile in these norms. These results are parallel to similar properties of Maxwell models for Boltzmann‐type equations, and extend already existing results on exponential convergence

Assuming the existence of certain large cardinal numbers, we prove that for every projective filter F over the set of natural numbers, F ‐bases in Banach spaces have continuous coordinate functionals. In particular, this applies to the filter of statistical convergence, thereby we solve a problem by Kadets (at least under the presence of certain large cardinals). In this setting, we recover also a

We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right‐angled Artin group. The algorithm works in the relative setting; in particular, it also applies to untwisted automorphism groups and basis‐conjugating automorphism groups. The main new tool is the construction of free abelian subgroups of certain Fouxe–Rabinovitch groups of rank equal to their

We establish a hidden extension in the Adams spectral sequence converging to the stable homotopy groups of spheres at the prime 2 in the 54‐stem. This extension is exceptional in that the only proof we know proceeds via Pstragowski's category of synthetic spectra. This was the final unresolved hidden 2‐extension in the Adams spectral sequence through dimension 80. We hope this provides a concise demonstration

In this paper, we prove a conjecture of Aharoni and Howard on the existence of rainbow (transversal) matchings in sufficiently large families F 1 , … , F s of tuples in { 1 , … , n } k , provided s ⩾ 470 .

We investigate extremal properties of shape functionals which are products of Newtonian capacity cap ( Ω ¯ ) , and powers of the torsional rigidity T ( Ω ) , for an open set Ω ⊂ R d with compact closure Ω ¯ , and prescribed Lebesgue measure. It is shown that if Ω is convex, then cap ( Ω ¯ ) T q ( Ω ) is (i) bounded from above if and only if q ⩾ 1 , and (ii) bounded from below and away from 0 if and

Let F be a Siegel cusp form of weight k and degree n > 1 with Fourier‐Jacobi coefficients { ϕ m } m ∈ N . In this article, we investigate the Ramanujan–Petersson conjecture (formulated by Kohnen) for the Petersson norm of ϕ m . In particular, we show that this conjecture is true when F is a Hecke eigenform and a Duke–Imamoğlu–Ikeda lift. This generalizes a result of Kohnen and Sengupta. Further, we

For x ∈ R , let M ( P , δ ) be the measure m { x : | P ′ ( x ) / ( n P ( x ) ) | ⩾ δ } ( δ > 0 ), where P is an arbitrary polynomial of positive degree n . We prove that sup Q M ( Q , δ ) = π / δ in the class of all real polynomials Q . As a corollary, we obtain the estimate of the derivative of a rational function, improving the known results of Gonchar and Dolzhenko, and prove that the inequality

We study Borel homomorphisms θ : G → H for arbitrary locally compact second countable groups G and H for which the measure θ ∗ ( μ ) ( α ) = μ ( θ − 1 ( α ) ) for α ⊆ H a Borel set is absolutely continuous with respect to ν , where μ (respectively, ν ) is a Haar measure for G , (respectively, H ). We define a natural mapping G from the class of maximal abelian selfadjoint algebra bimodules (masa bimodules)

We investigate local–global principles for Galois cohomology, in the context of function fields of curves over semi‐global fields. This extends work of Kato's on the case of function fields of curves over global fields.

Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right (respectively, left) form a monoidal category.

In this note, we provide several lower bounds for the volume of a geodesic ball within the injectivity radius in a 3‐dimensional Riemannian manifold assuming only upper bounds for the Ricci curvature.

We revisit the classical problem by Weyl, as well as its generalisations, concerning the isometric immersions of S 2 into simply‐connected 3‐dimensional Riemannian manifolds with non‐negative Gauss curvature. A sufficient condition is exhibited for the existence of global C 1 , 1 ‐isometric immersions. Our developments are based on the framework à la Labourie (J. Differential Geom. 30 (1989) 395–424)

Let Σ be a surface with either boundary or marked points, equipped with an arbitrary framing. In this note we determine the action of the associated ‘framed mapping class group’ on the homology of Σ relative to its boundary (respectively, marked points), describing the image as the kernel of a certain crossed homomorphism related to classical spin structures. Applying recent work of the authors, we

For a multigraph F , the k ‐subdivision of F is the graph obtained by replacing the edges of F with pairwise internally vertex‐disjoint paths of length k + 1 . Conlon and Lee conjectured that if k is even, then the ( k − 1 ) ‐subdivision of any multigraph has extremal number O ( n 1 + 1 k ) , and moreover, that for any simple graph F there exists ε > 0 such that the ( k − 1 ) ‐subdivision of F has

Let L / K be a finite Galois extension of fields with Galois group G . It is known that L / K admits exactly two Hopf–Galois structures when G is non‐abelian simple. In this paper, we extend this result to the case when G is quasisimple.

We explain why every non‐trivial exact tensor functor on the triangulated category of mixed motives over a field F has zero kernel, if one assumes ‘all’ motivic conjectures. In other words, every non‐zero motive generates the whole category up to the tensor triangulated structure. Under the same assumptions, we also give a complete classification of triangulated étale motives over F with integral coefficients

AF‐embeddability, quasidiagonality and stable finiteness of a C ∗ ‐algebra have been studied by many authors and shown to be equivalent for certain classes of C ∗ ‐algebras. The crossed products C ( X ) ⋊ σ Z (by Pimsner) and A F ⋊ α Z (by Brown) are such classes, and recently Schfhauser proves the equivalence for C ∗ ‐algebras of compact topological graphs. Clark, an Huef and Sims prove similar results

Due to a typesetting error, one equation and two pieces of line art were omitted from this article 1 when it was originally published. In the introduction, the following image of a regular unit octagon inscribed in a unit circle was missing at the end of page 901: Furthermore, the missing equation 4.6 on page 913 of the article should correctly appear as follows: The missing content has been restored

Let M be a compact oriented even‐dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f : S → M of strictly positive degree. The construction relies on two deep results: the first is a theorem of Ontaneda that gives a Riemannian manifold N of tightly pinched negative curvature which admits a map to M of degree equal to 1; the second is a result