We present a characterization of one‐component inner functions in terms of the location of their zeros and their associated singular measure. As consequence we answer several questions posed by Cima and Mortini. In particular, we prove that for any inner function Θ whose singular set has measure zero, one can find a Blaschke product B such that B Θ is one‐component. We also obtain a characterization

Building on the work of and answering a question by Michael Harrison, we show that any contact structure on R 3 induced by a line fibration of R 3 is diffeomorphic to the standard contact structure.

We improve some recent results of Sagiv and Steinerberger that quantify the following uncertainty principle: for a function f with mean zero, then either the size of the zero set of the function or the cost of transporting the mass of the positive part of f to its negative part must be big. We also provide a sharp upper estimate of the transport cost of the positive part of an eigenfunction of the

If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominators of very high degrees. When the base field is C , we give a completely explicit bound for the degrees of the monic right factors in terms of the degree and the order of the original operator, as well as of the largest modulus of the local exponents at

We discuss the relationship between two analogues in a 3‐manifold of the set of prime ideals in a number field. We prove that if ( K i ) i ∈ N > 0 is a sequence of knots obeying the Chebotarev law in the sense of Mazur and McMullen, then K = ∪ i K i is a stably generic link in the sense of Mihara. An example we investigate is the planetary link of a fibered hyperbolic finite link in S 3 . We also observe

Let F { d x } be a relatively stable (r.s.) probability distribution on the whole real line and S n the random walk started at the origin with step distribution F . We obtain an exact asymptotic form of the Green measure U { x + d y } = ∑ n = 0 ∞ P [ S n − x ∈ d y ] as x → ∞ when S n is transient and S n → ∞ in probability. If F is concentrated on [ 0 , ∞ ) , it is r.s. if and only if ℓ ( x ) : = ∫

Band surgery is an operation relating pairs of knots or links in the three‐sphere. We prove that if two quasi‐alternating knots K and K ′ of the same square‐free determinant are related by a band surgery, then the absolute value of the difference in their signatures is either 0 or 8. This obstruction follows from a more general theorem about the difference in the Heegaard Floer d ‐invariants for pairs

In this paper, we study the action of Homeo 0 ( M ) , the identity component of the group of homeomorphisms of an n ‐dimensional manifold M with an F p ‐free action, on another manifold N of dimension n + k < 2 n . We prove that if M is not an F p ‐homology sphere, then N ≅ M × K for a homology manifold K such that the action of Homeo 0 ( M ) on M is natural and on K is trivial. We also prove that

Given an induced representation of Langlands type ( π , V π ) of GL n ( F ) with F non‐Archimedean, we show that there exist explicit choices of matrix coefficient β and Schwartz–Bruhat function Φ for which the Godement–Jacquet zeta integral Z ( s , β , Φ ) attains the L ‐function L ( s , π ) .

Let χ Δ ( R n ) denote the minimum number of colors needed to color R n so that there will not be a monochromatic equilateral triangle with side length 1. Using the slice rank method, we reprove a result of Frankl and Rodl, and show that χ Δ ( R n ) grows exponentially with n . This technique substantially improves upon the best known quantitative lower bounds for χ Δ ( R n ) , and we obtain χ Δ R

Let Γ be the hyperbola { ( x , y ) ∈ R 2 : x y = 1 } and Λ β be the lattice‐cross defined by Λ β = ( Z × { 0 } ) ∪ ( { 0 } × β Z ) in R 2 , where β is a positive real. A result of Hedenmalm and Montes‐Rodríguez says that ( Γ , Λ β ) is a Heisenberg uniqueness pair if and only if β ⩽ 1 . In this paper, we show that for a rational perturbation of Λ β , namely Λ β θ = ( Z + { θ } ) × { 0 } ∪ { 0 } × β

In this paper, we consider some generalized holomorphic maps between pseudo‐Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz‐type results. As corollaries, the Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carathéodory pseudo‐distance on CR manifolds.

A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. Examples of finitely universal varieties of semigroups have been available since the early 1970s, but it is unknown if there exists a finitely universal variety of monoids. The main objective of the present article is to exhibit the first examples of finitely universal varieties of monoids

We study the failure of the integral Hasse principle and strong approximation for the Erdős–Straus conjecture using the Brauer–Manin obstruction.

Let q be a unimodular quadratic form over a field K . Pfister's famous local–global principle asserts that q represents a torsion class in the Witt group of K if and only if it has signature 0, and that in this case, the order of Witt class of q is a power of 2. We give two analogues of this result to systems of quadratic forms, the second of which applying only to nonsingular pairs. We also prove

We show that rounding to a δ ‐net in SO ( 3 ) is not close to a group operation, thus confirming a conjecture of Gowers and Long.

Let K be a field equipped with a valuation ν and let L = K ( x ) be a transcendental extension of K . Every valuation μ of L which extends ν is determined by its restriction to the polynomial ring K [ x ] . We know how to associate to this valuation μ a family of valuations A = ( μ i ) i ∈ I of K [ x ] , called the associated admitted family that converges in a certain sense towards the valuation μ

A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi–de Zeeuw, Makhul–Shaffaf, Shaffaf and Tao. In the process, we give a criterion for certain varieties with non‐canonical singularities

In 1909, Hardy gave an example of a transcendental entire function, f , with the property that the set of points where f achieves its maximum modulus, M ( f ) , has infinitely many discontinuities. This is one of only two known examples of such an entire function.

We introduce a new type of aperiodic hexagonal monotile; a prototile that admits infinitely many tilings of the plane, but any such tiling lacks any translational symmetry. Adding a copy of our monotile to a patch of tiles must satisfy two rules that apply only to adjacent tiles. The first is inspired by the Socolar–Taylor monotile, but can be realised by shape alone. The second is a dendrite rule;

Let ( H R , U t ) be any strongly continuous orthogonal representation of R on a real (separable) Hilbert space H R . For any q ∈ ( − 1 , 1 ) , we denote by Γ q ( H R , U t ) ′ ′ the q ‐deformed Araki–Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that Γ q ( H R , U t ) ′ ′ has trivial bicentralizer if it is a type III 1 factor. In particular, we obtain that Γ q ( H R ,

A lamination of a graph embedded on a surface is a collection of pairwise disjoint non‐contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice polytope, then we characterize the polytopes that arise as the lamination space of some

We construct a space P for which the canonical homomorphism π 1 ( P , p ) → π ̌ 1 ( P , p ) from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following properties: (1) P ∖ { p } is a 2‐manifold with connected non‐compact boundary; (2) P is connected and locally path connected; (3) P is strongly homotopically Hausdorff; (4) P is homotopically path

What kind of reduced monomial schemes can be obtained as a Gröbner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley–Reisner schemes associated to acyclic Cohen–Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over totally real

Suh (Math. Nachr . 290 (2017) 442–451) proved that there are no Hopf real hypersurfaces in the complex quadric that have parallel normal Jacobi operators. Motivated by this result, in this paper, we introduce the notions of C ‐parallel and Reeb parallel for normal Jacobi operators, which generalize the notion ‘parallel’. First we obtain a non‐existence theorem of Hopf real hypersurfaces with C ‐parallel

Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an irreducible immersion on a finite graph. We give a new proof of this and also show a partial converse holds when the immersion has connected Whitehead graphs with no cut vertices. The next result is a characterization of finitely generated subgroups of the free group that are invariant under an irreducible

This paper studies, by employing analytical tools, the small‐time behavior of the heat content for the Poisson kernel over the unit ball in R d , d ⩾ 2 by working with its related set covariance function. As a result, we obtain a representation for the third term in the expansion of the heat content over the unit ball and provide the explicit form of this term in the particular cases d = 2 and d =

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: (4.6) The missing content has been

Let g be a finite‐dimensional semisimple complex Lie algebra and θ an involutive automorphism of g . According to Letzter, Kolb and Balagović the fixed‐point subalgebra k = g θ has a quantum counterpart B , a coideal subalgebra of the Drinfeld–Jimbo quantum group U q ( g ) possessing a universal K ‐matrix K . The objects θ , k , B and K can all be described in terms of Satake diagrams. In the present

It is conjectured that the dual variety of a smooth nonlinear variety X ⊆ P N of dimension dim ( X ) > 2 N 3 is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of Hartshorne's complete intersection conjecture but nevertheless remains open for the case of subvarieties of codimension > 2 . A combinatorial approach to prove the conjecture in the

We derive an analytic class number formula valid for an order in a product of S ‐integers in global fields, or equivalently for reduced finite‐type affine schemes of pure dimension 1 over Z .

We prove the Myers–Steenrod theorem for local topological groups of isometries acting on pointed C k , α ‐Riemannian manifolds, with k + α > 0 . As an application, we infer a new regularity result for a certain class of locally homogeneous Riemannian metrics.

In this work, we discuss completeness for the lattice orders of first and second order stochastic dominance. The main results state that both first‐ and second‐order stochastic dominance induce Dedekind super complete lattices, that is, lattices in which every bounded nonempty subset has a countable subset with identical least upper bound and greatest lower bound. Moreover, we show that, if a suitably

We investigate the relationship between the maximum of the zeta function on the 1‐line and the maximal order of S ( t ) , the error term in the number of zeros up to height t . We show that the conjectured upper bounds on S ( t ) along with the Riemann hypothesis imply a conjecture of Littlewood that max t ∈ [ 1 , T ] | ζ ( 1 + i t ) | ∼ e γ log log T . The relationship in the region 1 / 2 < σ < 1

We give two proofs of the following theorem and a partial generalization: if a finite‐dimensional algebra A is derived equivalent to a smooth projective scheme, then any derived equivalence between A and another algebra B is standard, that is, isomorphic to the derived tensor functor by a two‐sided tilting complex. The main ingredients of the proofs are as follows: (1) between the derived categories

We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which

We generalize the results of Skorobogatov and Zarhin considering the commutativity of Brauer groups (and Brauer–Manin sets) with taking product of two varieties, by relaxing the condition that varieties are projective.

We show that for arithmetic weights with a fixed finite‐order character, the slopes of U p for p = 2 (which is inert) acting on overconvergent Hilbert modular forms of level U 0 ( 4 ) are independent of the (algebraic part of the) weight and can be obtained by a simple recipe from the classical slopes in parallel weight 3.

We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by‐product of our methods, we also prove a generalization of a result of Flavell, which itself generalizes Wielandt's Zipper Lemma and provides a characterization of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems

In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there are only finitely many torsion points whose coordinates are multiplicatively dependent. Moreover, we produce an effective result when the elliptic curve is defined

We study the disproportionate version of the classical cake‐cutting problem: how efficiently can we divide a cake, here [0,1], among n ⩾ 2 agents with different demands α 1 , α 2 , ⋯ , α n summing to 1? When all the agents have equal demands of α 1 = α 2 = ⋯ = α n = 1 / n , it is well known that there exists a fair division with n − 1 cuts, and this is optimal. For arbitrary demands on the other hand

In this paper, we investigate the weak‐type regularity of the Bergman projection. The two domains we focus on are the polydisc and the Hartogs triangle. For the polydisc, we provide a proof that the weak‐type behavior is of ‘ L log L ’ type. This result is likely known to the experts, but does not appear to be in the literature. For the Hartogs triangle, we show that the operator is of weak‐type (4

The Nottingham group is the unique Sylow pro‐ p subgroup of the automorphism group of the formal power series algebra over a finite field. This group has been intensively studied in the last decades for its interesting properties, just infiniteness being one of the most remarkable. Shalev and Leedham‐Green introduced the generalized Nottingham groups and confirmed that they are just infinite, however

Every finite simple group can be generated by two elements, and Guralnick and Kantor proved that, moreover, every nontrivial element is contained in a generating pair. Groups with this property are said to be 3 2 ‐generated. Thompson's group V was the first finitely presented infinite simple group to be discovered. The Higman–Thompson groups V n and the Brin–Thompson groups m V are two families of

We classify the primitive idempotents of the p ‐local complex representation ring of a finite group G in terms of the cyclic subgroups of order prime to p and show that they all come from idempotents of the Burnside ring. Our results hold without adjoining roots of unity or inverting the order of G , thus extending classical structure theorems. We then derive explicit group‐theoretic obstructions for

We show that the generating series of traces of reciprocal singular moduli is a mixed mock modular form of weight 3 / 2 whose shadow is given by a linear combination of products of unary and binary theta functions. To prove these results, we extend the Kudla–Millson theta lift of Bruinier and Funke to meromorphic modular functions.

Given a vector bundle V of rank r over a curve X , we define and study a surjective rational map Hilb d ( P V ) ⤍ Quot 0 , d ( V ∗ ) generalising the natural map Sym d X → Quot 0 , d ( O X ) . We then give a generalisation of the geometric Riemann–Roch theorem to vector bundles of higher rank over X . We use this to give a geometric description of the tangent cone to the higher rank Brill–Noether locus

We work on ℓ p uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of Farah and the authors to mutual embeddings of uniform Roe algebras of operators on ℓ p spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe C ∗ ‐algebras which depend only on their Banach algebra structure.

Let M be a semi‐finite von Neumann algebra with normal state space S ( M ) . For any ϕ ∈ S ( M ) , let M ϕ : = { x ∈ M : x ϕ = ϕ x } be the centralizer of ϕ with center Z ( M ϕ ) . We show that for ϕ , ψ ∈ S ( M ) , the following are equivalent. ϕ = ψ . Z ( M ψ ) ⊆ Z ( M ϕ ) and ϕ | Z ( M ϕ ) = ψ | Z ( M ϕ ) . ϕ , ψ have the same distances to all the closed faces of S ( M ) .

A theorem of Dorronsoro from the 1980s quantifies the fact that real‐valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last variable. As

We prove that if G is a finite simple group of Lie type and S 1 , ⋯ , S k are subsets of G satisfying ∏ i = 1 k | S i | ⩾ | G | c for some c depending only on the rank of G , then there exist elements g 1 , ⋯ , g k such that G = ( S 1 ) g 1 ⋯ ( S k ) g k . This theorem generalizes an earlier theorem of the authors and Short.

We present a simple uniqueness argument for a collection of McKean–Vlasov problems that have seen recent interest. Our first result shows that, in the weak feedback regime, there is global uniqueness for a very general class of random drivers. By weak feedback we mean the case where the contagion parameters are small enough to prevent blow‐ups in solutions. Next, we specialise to a Brownian driver

For a collection G = { G 1 , ⋯ , G s } of not necessarily distinct graphs on the same vertex set V , a graph H with vertices in V is a G ‐transversal if there exists a bijection ϕ : E ( H ) → [ s ] such that e ∈ E ( G ϕ ( e ) ) for all e ∈ E ( H ) . We prove that for | V | = s ⩾ 3 and δ ( G i ) ⩾ s / 2 for each i ∈ [ s ] , there exists a G ‐transversal that is a Hamilton cycle. This confirms a conjecture

It is a classical fact that for any ε > 0 , a random permutation of length n = ( 1 + ε ) k 2 / 4 typically contains an increasing subsequence of length k . As a far‐reaching generalization, Alon conjectured that a random permutation of this same length n is typically k ‐universal , meaning that it simultaneously contains every pattern of length k . He also made the simple observation that for n = O

We consider a double phase Robin problem with a Carathéodory nonlinearity. When the reaction is superlinear but without satisfying the Ambrosetti–Rabinowitz condition, we prove an existence theorem. When the reaction is resonant, we prove a multiplicity theorem. Our approach is Morse theoretic, using the notion of homological local linking.

A virtual Schottky group is a Kleinian group K containing a Schottky group G as a finite index normal subgroup. These groups correspond to those groups of automorphisms of closed Riemann surfaces which can be realized at the level of their lowest uniformizations. In this paper, we provide a geometrical structural decomposition of K . When K / G is an abelian group, an explicit free product decomposition

We study the existence, nonexistence and multiplicity of positive solutions for a general class of degenerate nonlocal problems. The nonexistence results are provided in two different situations: under monotonicity assumptions involving the nonlinearity and without monotonicity conditions. A multiplicity result of positive solutions with ordered L p ‐norms complement some recent results obtained in

We prove that the signature bound for the topological 4‐genus of 3‐strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4‐strand and 6‐strand torus knots, and improve the upper bound on the asymptotic ratio between the topological 4‐genus and the Seifert genus of torus knots from 2/3 to 14/27.

We revisit the geometry of involutions in groups of finite Morley rank. The focus is on specific configurations where, as in PGL 2 ( K ) , the group has a subgroup whose conjugates generically cover the group and intersect trivially. Our main result is the subtle yet strong statement that in such configurations the conjugates of the subgroup may not cover all strongly real elements. As an application

We study existence of linear isometric embedding of ℓ p m into S ∞ , for 1 ⩽ p < ∞ , and unique operator space structure on two‐dimensional Banach spaces. For p ∈ ( 2 , ∞ ) ∪ { 1 } , we show that indeed ℓ p 2 does not embed isometrically into S ∞ . This verifies a guess of Pisier and broadly generalizes the main result of Gupta and Reza (Houston J. Math . 44 (2018) 1205–1212). We also show that S 1