The Gasper and Rahman multivariate ( − q ) ‐Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank q ‐Bannai–Ito algebra A n q . Lifting the action of the algebra to the connection coefficients, we find a realization of A n q by means of difference operators. This provides an algebraic interpretation for the

We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category O of representations of the quantum loop algebra introduced by Hernandez–Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type A , we prove that this

In the integral cohomology ring of the classifying space of the projective linear group P G L n (over C ), we find a collection of p ‐torsion classes y p , k of degree 2 ( p k + 1 + 1 ) for any odd prime divisor p of n , and k ⩾ 0 . If, in addition, p 2 ∤ n , there are p ‐torsion classes ρ p , k of degree p k + 1 + 1 in the Chow ring of the classifying stack of P G L n , such that the cycle class map

Let p be a rational prime and q a power of p . Let ℘ be a monic irreducible polynomial of degree d in F q [ t ] . In this paper, we define an analogue of the Hodge–Tate map which is suitable for the study of Drinfeld modules over F q [ t ] and, using it, develop a geometric theory of ℘ ‐adic Drinfeld modular forms similar to Katz's theory in the case of elliptic modular forms. In particular, we show

We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3‐manifolds and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds for the overcommutation length in SL 2 , related to the Steinberg relation.

We classify del Pezzo non‐commutative surfaces that are finite over their centres and have no worse than canonical singularities. Using the minimal model program, we introduce the minimal model of such surfaces. We first classify the minimal models and then give the classification of these surfaces in general. This presents a complementary result and method to the classification of del Pezzo orders

Let J be the set of inner functions whose derivative lies in the Nevanlinna class. In this paper, we discuss a natural topology on J where F n → F if the critical structures of F n converge to the critical structure of F . We show that this occurs precisely when the critical structures of the F n are uniformly concentrated on Korenblum stars. The proof uses Liouville's correspondence between holomorphic

We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly degenerate hyperbolic 3‐manifolds developed by Brock, Canary and Minsky during the course of their proof of the Ending Lamination Theorem. Thus we obtain uniformly

For spherical and parabolic averages of the Fourier transform of fractal measures, we obtain new upper bounds on rates of decay by an ‘intermediate dimension’ trick.

We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A 1 and A 2 , vanishes in any (co)homological degree p > 2 . Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to its (higher) homological calculus, by exchanging degrees p and 2 − p , and we prove a generalised version of the 2‐Calabi–Yau property. For the ADE Dynkin graphs, the preprojective

The classical completeness problem raised by Beurling and independently by Wintner asks for which ψ ∈ L 2 ( 0 , 1 ) , the dilation system { ψ ( k x ) : k = 1 , 2 , … } is complete in L 2 ( 0 , 1 ) , where ψ is identified with its extension to an odd 2‐periodic function on R . This difficult problem is nowadays commonly called as the periodic dilation completeness problem (PDCP). By Beurling's idea

Nielsen realization problem for the mapping class group Mod ( S g ) asks whether the natural projection p g : Homeo + ( S g ) → Mod ( S g ) has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion‐free subgroups of Mod ( S g ) . The main result of this paper is that the Torelli group

In this paper, we study the evolution problem u t ( x , t ) − λ j ( D 2 u ( x , t ) ) = 0 , in Ω × ( 0 , + ∞ ) , u ( x , t ) = g ( x , t ) , on ∂ Ω × ( 0 , + ∞ ) , u ( x , 0 ) = u 0 ( x ) , in Ω , where Ω is a bounded domain in R N (which verifies a suitable geometric condition on its boundary) and λ j ( D 2 u ) stands for the j th eigenvalue of the Hessian matrix D 2 u . We assume that u 0 and g are

We derive a fully explicit version of the Selberg trace formula for twist‐minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2‐dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the

We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle

Let O be an order in the imaginary quadratic field K . For positive integers M ∣ N , we determine the least degree of an O ‐CM point on the modular curve X ( M , N ) / K ( ζ M ) and also on the modular curve X ( M , N ) / Q ( ζ M ) : that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally

We initiate a study of distortion elements in the Polish groups Diff + k ( S 1 ) ( 1 ⩽ k < ∞ ), as well as Diff + 1 + A C ( S 1 ) , in terms of maximal metrics on these groups. We classify distortion in the k = 1 case: a C 1 circle diffeomorphism is C 1 ‐undistorted if and only if it has a hyperbolic periodic point. On the other hand, answering a question of Navas, we exhibit analytic circle diffeomorphisms

We show that for any subgroup H of Out ( F N ) , either H contains an atoroidal element or a finite index subgroup H ′ of H fixes a nontrivial conjugacy class in F N . This result is an analog of Ivanov's subgroup theorem for mapping class groups and Handel–Mosher's subgroup theorem for Out ( F N ) in the setting of irreducible elements.

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