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)

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 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

We show that for a quasicompact quasiseparated scheme X , the following assertions are equivalent: (1) the category QCoh ( X ) of all quasicoherent sheaves on X has a flat generator; (2) for every injective object E of QCoh ( X ) , the internal hom functor into E is exact; (3) the scheme X is semiseparated.

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 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.

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

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.

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

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

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.

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

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;

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 =

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

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

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 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 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