We study smooth bundles over surfaces with highly connected almost parallelizable fiber M of even dimension, providing necessary conditions for a manifold to be bordant to the total space of such a bundle and showing that, in most cases, these conditions are also sufficient. Using this, we determine the characteristic numbers realized by total spaces of bundles of this type, deduce divisibility constraints

We construct an integrable colored five‐vertex model whose partition function is a Lascoux atom based on the five‐vertex model of Motegi and Sakai and the colored five‐vertex model of Brubaker, the first author, Bump and Gustafsson. We then modify this model in two different ways to construct a Lascoux polynomial, yielding the first proven combinatorial interpretation of a Lascoux polynomial and atom

We calculate the Masur–Veech volume of the gothic locus G in the stratum H ( 2 3 ) of genus 4. Our method is based on the use of the formulae for the Euler characteristics of gothic Teichmüller curves to determine the number of lattice points of given area. We also use this method to recalculate the Masur–Veech volumes of the Prym loci P 3 ⊂ H ( 4 ) and P 4 ⊂ H ( 6 ) in genus 3 and 4.

We give a complete characterization of the forking independence relation over any set of parameters in the free groups of finite rank, in terms of the J S J decompositions relative to those parameters.

Let C be the category of finite‐dimensional modules over the quantum affine algebra U q ( g ̂ ) of a simple complex Lie algebra g . Let C − be the subcategory introduced by Hernandez and Leclerc. We prove the geometric q ‐character formula conjectured by Hernandez and Leclerc in types A and B for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial

We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C ∗ ‐algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved C ∗ ‐algebras.

This paper considers the propagation of TE‐modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model

In this paper, we show an equi‐distributed property in 2‐dimensional finite abelian groups Z p n × Z p m , where p is a prime number. By using this equi‐distributed property, we prove that Fuglede's spectral set conjecture holds on groups Z p 2 × Z p , namely, a set in Z p 2 × Z p is a spectral set if and only if it is a translational tile.

A group word w is said to be strongly concise in a class C of profinite groups if, for every group G in C such that w takes less than 2 ℵ 0 values in G , the verbal subgroup w ( G ) is finite. Detomi, Morigi and Shumyatsky established that multilinear commutator words — and the particular words x 2 and [ x 2 , y ] — have the property that the corresponding verbal subgroup is finite in a profinite group

In this paper, we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form x 2 + y 2 with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive positive definite binary quadratic form. In particular, for any positive definite binary quadratic form F and binary linear form G , there exist infinitely many ℓ , m ∈

We essentially determine the saturated 2‐fusion systems of J‐component type in which the centralizer of some fully centralized involution of maximal 2‐rank contains a component that is the 2‐fusion system of an alternating group A n for some n ⩾ 8 .

The extremal index (EI) is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic processes with an EI for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised

In this paper, we study power series having a fixed point of multiplier 1. First, we give a closed formula for the residue fixed point index, in terms of the first coefficients of the power series. Then, we use this formula to study wildly ramified power series in positive characteristic. Among power series having a multiple fixed point of small multiplicity, we characterize those having the smallest

We prove sharp ℓ p L p decoupling inequalities for two quadratic forms in four variables. We also recover several previous results (Bourgain and Demeter, J. Funct. Anal. 270 (2016) 1299–1318; Bourgain and Demeter, J. Anal. Math. 133 (2017) 79–311; Demeter, Guo and Shi, Rev. Mat. Iberoam 35 (2019) 423–460; Oh, Math. Z. 290 (2018) 389–419) in a unified way.

We consider the question of whether solutions of variants of Teichmüller harmonic map flow from surfaces M to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least 2, as well as the flow from cylinders, we prove that such a finite‐time degeneration must occur in situations where the image of thin collars is ‘stretching out’ at a rate of at least

Slow convergence of cyclic projections implies divergence of random projections and vice versa. Let L 1 , L 2 , ⋯ , L K be a family of K closed subspaces of a Hilbert space. It is well known that although the cyclic product of the orthogonal projections on these spaces always converges in norm, random products might diverge. Moreover, in the cyclic case there is a dichotomy: the convergence is fast

With respect to the analytic‐algebraic dichotomy, the theory of Siegel modular forms of half‐integral weight is lopsided; the analytic theory is strong, whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture — the p ‐adic L ‐function obtained by interpolating the complex L ‐function at special

For s = 3 , 4 , we prove the existence of arbitrarily long sequences of consecutive integers none of which is a sum of s nonnegative s th powers. More generally, we study the existence of gaps between the values ⩽ N of diagonal forms of degree s in s variables with positive integer coefficients. We find: (1) gaps of size ≫ log N ( log log N ) 2 when s = 3 ; (2) gaps of size ≫ log log log N log log

We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so‐called special Picard curves over Q with good reduction outside 2 and 3, and use this to determine the smallest possible conductor a special Picard curve may have. We also collect a database of Picard curves over Q of small conductor.

Every closed geodesic γ on a surface has a canonically associated knot γ ̂ in the projective unit tangent bundle. We study, for γ filling, the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We provide a lower bound for the volume relative to the number of homotopy classes of γ ‐arcs in each pair of pants of a pants decomposition of the surface.

The purpose of this article is to define and study new invariants of topological spaces: the p ‐adic Betti numbers and the p ‐adic torsion. These invariants take values in the p ‐adic numbers and are constructed from a virtual pro‐ p completion of the fundamental group. The key result of the article is an approximation theorem which shows that the p ‐adic invariants are limits of their classical analogues

In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p ‐local finite group ( S , F , L ) in terms of representation rings of subgroups of S . We also prove a stable elements formula for generalized cohomological invariants of p ‐local finite groups, which is used to show the existence of unitary embeddings of p ‐local finite groups

This paper focuses on maximum and comparison principles related to the Lane–Emden problem − L 1 u = λ ρ ( x ) | v | α − 1 v in Ω , − L 2 v = μ τ ( x ) | u | β − 1 u in Ω , u = v = 0 on ∂ Ω , where α , β > 0 , α β = 1 , Ω is a smooth bounded open subset of R n with n ⩾ 1 , ρ and τ are positive functions on Ω and L 1 and L 2 are second‐order uniformly elliptic linear operators in Ω . We characterize

We show that a strong well‐based cylindrical algebraic decomposition P of a bounded semi‐algebraic set S is a regular cell decomposition, in any dimension and independently of the method by which P is constructed. Being well‐based is a global condition on P that holds for the output of many widely used algorithms. We also show the same for S of dimension at most 3 and P a strong cylindrical algebraic

The spectral factorization mapping F → F + puts a positive definite integrable matrix function F having an integrable logarithm of the determinant in correspondence with an outer analytic matrix function F + such that F = F + ( F + ) ∗ almost everywhere. The main question addressed here is to what extent ∥ F + − G + ∥ H 2 is controlled by ∥ F − G ∥ L 1 and ∥ log det F − log det G ∥ L 1 .

Consider a sequence of finite regular graphs converging, in the sense of Benjamini–Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non‐zero coupling constant α ) and a symmetric potential U on the edges. We show that in the spectral regions where the infinite quantum tree has absolutely continuous spectrum

Given a shifted order ideal U , we associate to it a family of simplicial complexes ( Δ t ( U ) ) t ⩾ 0 that we call squeezed complexes. In a special case, our construction gives squeezed balls that were defined and used by Kalai to show that there are many more simplicial spheres than boundaries of simplicial polytopes. We study combinatorial and algebraic properties of squeezed complexes. In particular

We extend a classical theorem of Courrège to Lie groups in a global setting, thus characterising all linear operators on the space of smooth functions of compact support that satisfy the positive maximum principle. We show that these are Lévy type operators (with variable characteristics), and pseudo‐differential operators when the group is compact. If the characteristics are constant, then the operator

A subset of a group is said to be product free if it does not contain three elements satisfying the equation x y = z . We give a negative answer to a question of Babai and Sós on the existence of large product‐free sets in finite groups by model theoretic means. This question was originally answered by Gowers. Furthermore, we give a natural and sufficient model theoretic condition for a group to have

The extension space conjecture of oriented matroid theory states that the space of all one‐element, non‐loop, non‐coloop extensions of a realizable oriented matroid of rank d has the homotopy type of a sphere of dimension d − 1 . We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space. This also resolves

We consider the higher order asymptotics for the modified Korteweg‐de Vries equation in the Painlevé sector. We first show that the solution admits a uniform expansion to all orders in powers of t − 1 / 3 with coefficients that are smooth functions of x ( 3 t ) − 1 / 3 . We then consider the special case when the reflection coefficient vanishes at the origin. In this case, the leading coefficient which

We study the distance in the Zygmund class Λ * to the subspace I ( BMO ) of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of I ( BMO ) in the Zygmund seminorm. We also generalise this result to Zygmund measures on R d . Finally, we apply the techniques developed in the article to characterise the closure of the subspace of functions in

In 1960, Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases.

For a bounded Hankel matrix Γ , we describe the structure of the Schmidt subspaces of Γ , namely the eigenspaces of Γ ∗ Γ corresponding to non‐zero eigenvalues. We prove that these subspaces are in correspondence with weighted model spaces in the Hardy space on the unit circle. Here we use the term ‘weighted model space’ to describe the range of an isometric multiplier acting on a model space. Further

The elliptic curve E k : y 2 = x 3 + k admits a natural 3‐isogeny ϕ k : E k → E − 27 k . We compute the average size of the ϕ k ‐Selmer group as k varies over the integers. Unlike previous results of Bhargava and Shankar on n ‐Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on k ; this sensitivity can be precisely controlled by the Tamagawa

The Powell conjecture offers a finite generating set for the genus g Goeritz group, the group of automorphisms of S 3 that preserve a genus g Heegaard surface Σ g , generalizing a classical result of Goeritz in the case g = 2 . We study the relationship between the Powell conjecture and the reducing sphere complex R ( Σ g ) , the subcomplex of the curve complex C ( Σ g ) spanned by the reducing curves

We develop a theory of Čech‐Bott‐Chern cohomology and in this context we naturally come up with the relative Bott‐Chern cohomology. In fact, Bott‐Chern cohomology has two relatives and they all arise from a single complex. Thus, we study these three cohomologies in a unified way and obtain a long exact sequence involving the three. We then study the localization problem of characteristic classes in

We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are n balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than n − n , then the total number of collisions is bounded by n ( 5 / 2 + ε ) n , for any fixed

We consider sets of trace‐normalized non‐negative operators in Hilbert–Schmidt balls that maximize their mutual Hilbert–Schmidt distance; these are optimal arrangements in the sets of purity‐limited classical or quantum states on a finite‐dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also

In this article, we construct a new family of semifields, containing and extending two well‐known families, namely Albert's generalised twisted fields and Petit's cyclic semifields (also known as Johnson–Jha semifields). The construction also gives examples of semifields with parameters for which no examples were previously known. In the case of semifields two dimensions over a nucleus and four‐dimensional

We revisit Sapozhenko's classic proof on the asymptotics of the number of independent sets in the discrete hypercube { 0 , 1 } d and Galvin's follow‐up work on weighted independent sets. We combine Sapozhenko's graph container methods with the cluster expansion and abstract polymer models, two tools from statistical physics, to obtain considerably sharper asymptotics and detailed probabilistic information

Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl–Tillmann: we show that the marked polytopes they constructed for two‐generator one‐relator groups with nice presentations are independent of the presentations used. We also show that, when the groups are additionally torsion‐free, the agrarian polytope encodes the splitting complexity of the group

We give explicit criteria that imply the resurgence of a self‐radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. One criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning

An explicit form of genus character L ‐functions of quadratic orders is presented in full generality. As an application, we generalize a formula due to Hirzebruch and Zagier on the class number of imaginary quadratic fields expressed in term of the continued fraction expansion.

In this paper, we establish a weak Serrin‐type blowup criterion for the Cauchy problem of the three‐dimensional (3D) compressible barotropic Navier–Stokes equations in the whole space. It shows that the strong or smooth solution exists globally if the velocity satisfies the weak Serrin's condition and the L ∞ ( 0 , T ; L α 0 ) ‐norm of the density is bounded, where α 0 is positive constant. Therefore

We study property A for metric spaces X with bounded geometry introduced by Guoliang Yu. Property A is an amenability‐type condition, which is less restrictive than amenability for groups. The property has a connection with finite‐dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space X with bounded geometry is equivalent

We study the Zariski closure of the monodromy group Mon of Lauricella's hypergeometric function F C . If the identity component Mon 0 acts irreducibly, then Mon ¯ ∩ SL 2 n ( C ) must be one of classical groups SL 2 n ( C ) , SO 2 n ( C ) and Sp 2 n ( C ) . We also study Calabi–Yau varieties arising from integral representations of F C .

We give a sufficient condition for birational superrigidity of del Pezzo fibrations of degree 1 with only 1 2 ( 1 , 1 , 1 ) singular points, generalizing the so‐called K 2 ‐condition. As an application, we also prove that a del Pezzo fibration of degree 1 with only 1 2 ( 1 , 1 , 1 ) singular points embedded in a toric P ( 1 , 1 , 2 , 3 ) ‐bundle over P 1 is birationally superrigid if and only if it

We provide several new examples in dynamics on the 2‐sphere, with the emphasis on better understanding the induced boundary dynamics of invariant domains in parametrised families. First, motivated by a topological version of the Poincaré–Bendixson theorem obtained recently by Koropecki and Passeggi, we show the existence of homeomorphisms of S 2 with Lakes of Wada rotational attractors, with an arbitrarily

In this paper, we establish space like strong unique continuation property for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established in (Ruland, J. Differential Equations 265 (2018) 6009–6035) for elliptic sublinear equations with analogous structure

We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if p n denotes the n th prime and L is the set of limit points of the sequence { ( p n + 1 − p n ) / log p n } n = 1 ∞ , then for all T ⩾ 0 the Lebesque measure of L ∩ [ 0 , T ] is at least T / 3 . This improves the result of Pintz from 2015 that the Lebesque measure of L ∩ [

A closed set K of a Hilbert space H is said to be invariant under the evolution equation X ′ ( t ) = A X ( t ) + f t , X ( t ) ( t > 0 ) , whenever all solutions starting from a point of K , at any time t 0 ⩾ 0 , remain in K as long as they exist.

Let G = SL ( 2 , R ) ⋉ ( R 2 ) ⊕ k and let Γ be a congruence subgroup of SL ( 2 , Z ) ⋉ ( Z 2 ) ⊕ k . We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in Γ ∖ G which project to pieces of closed horocycles in SL ( 2 , Z ) ∖ SL ( 2 , R ) . As an application, we prove an effective quantitative Oppenheim‐type result for the quadratic form ( m 1

Let E / F be a finite Galois extension of totally real number fields and let p be a prime. The ‘ p ‐adic Stark conjecture at s = 1 ’ relates the leading terms at s = 1 of p ‐adic Artin L ‐functions to those of the complex Artin L ‐functions attached to E / F . We prove this conjecture unconditionally when E / Q is abelian. We also show that for certain non‐abelian extensions E / F the p ‐adic Stark

Based on reverse isoperimetric inequalities on Grassmann manifolds, a Grassmanian Loomis–Whitney inequality and its dual are established, which provides a lower bound for the volume of an origin‐symmetric convex body in terms of its lower dimensional sections.

Let G be a group, and let M = ( μ n ) n = 1 ∞ be a sequence of finitely supported probability measures on G . Consider the probability that two elements chosen independently according to μ n commute. Antolín, Martino and Ventura define the degree of commutativity dc M ( G ) of G with respect to this sequence to be the lim sup of this probability. The main results of the present paper give quantitative

Building off of the work of Kervaire–Milnor and Hill–Hopkins–Ravenel, Wang and Xu showed that the only odd dimensions n for which S n has a unique differentiable structure are 1, 3, 5, and 61. We show that the only even dimensions below 140 for which S n has a unique differentiable structure are 2, 6, 12, 56, and perhaps 4.

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with nonnegative Ricci curvature and maximal volume growth, further assume the dimension of the manifold is not less than 3, we prove that

We study the Zariski cancellation problem for Poisson algebras asking whether A [ t ] ≅ B [ t ] implies A ≅ B when A and B are Poisson algebras. We resolve this affirmatively in the cases when A and B are both connected graded Poisson algebras finitely generated in degree 1 without degree 1 Poisson central elements and when A is a Poisson integral domain of Krull dimension two with nontrivial Poisson

McDuff has previously shown that one four‐dimensional symplectic ellipsoid can be symplectically embedded into another if and only if a certain combinatorial criteria holds. We reinterpret this combinatorial criteria using the theory of Ehrhart quasipolynomials, and we use this to give purely combinatorial proofs of theorems of McDuff–Schlenk and Frenkel–Müller, concerning the existence of ‘infinite