We fill in a gap in the proof of the main theorem in our earlier paper [the above-mentioned paper]. At the same time, we prove a slightly stronger version of the theorem needed for another paper.

In this survey we describe how the so-called Dold congruence arises in topology, and how it relates to periodic point counting in dynamical systems.

We formalize an equivariant version of Bestvina–Brady discrete Morse theory, and apply it to Vietoris–Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.

We show that, for every transitive permutation group G of degree n ⩾ 2 , the largest abelian quotient of G has cardinality at most 4 n / log 2 n . This gives a positive answer to a 1989 outstanding question of László Kovács and Cheryl Praeger.

In this survey article, we explain how false theta functions can be embedded into a modular framework and show some of the applications of this modularity.

We provide the second-known example of an extremally Ricci pinched closed G 2 -structure on a compact 7-manifold, by finding a lattice in the only unimodular solvable Lie group admitting a left-invariant G 2 -structure. Furthermore, the Laplacian coflow and its solitons are studied on a 6-parameter family of left-invariant coclosed G 2 -structures on this Lie group. In this way, we obtain a 4-parameter

We show that the space of anti-symplectic involutions of a monotone S 2 × S 2 whose fixed point set is a Lagrangian sphere is connected. This follows from a stronger result, namely that any two anti-symplectic involutions in that space are Hamiltonian isotopic.

Let X be a smooth, projective curve defined over a finite field F r , and ∞ a (closed) point of X . Let κ be the function field of X and A the elements of κ regular everywhere except possibly at ∞ . Let ϕ : A → K { τ } be a Drinfeld module defined over K , where ι : A → K is an injective map that identifies K as a finite extension of κ . Suppose that the rank of ϕ is n ⩾ 2 .

In this paper, we characterize the extremal points of the unit ball of the Benamou–Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points consist of pairs of measures concentrated on absolutely continuous curves which are characteristics of the continuity equation. Then, we apply this result to provide a

We prove that for a generic element in a nonhyperelliptic component of an abelian stratum H g ( μ ) in genus g , the underlying curve has maximal gonality. We extend this result to the case of quadratic strata when the partition μ has positive entries. As a consequence we deduce that all nonhyperelliptic components of H 9 ( μ ) are uniruled when μ is a positive partition of 16 and all nonhyperelliptic

For an extension of associative algebras B ⊂ A over a field and an A -bimodule X , we obtain a Jacobi–Zariski long nearly exact sequence relating the Hochschild homologies of A and B , and the relative Hochschild homology, all of them with coefficients in X . This long sequence is exact twice in three. There is a spectral sequence which converges to the gap of exactness.

We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case.

We display a symmetric monoidal equivalence between the stable ∞ -category of filtered spectra, and quasi-coherent sheaves on A 1 / G m , the quotient in the setting of spectral algebraic geometry of the flat affine line by the canonical action of the flat multiplicative group scheme. Via a Tannaka duality argument, we identify the underlying spectrum and associated graded functors with pull-backs

Building on recent work of Harper, and using various results of Chang and Iwaniec on the zero-free regions of L -functions L ( s , χ ) for characters χ with a smooth modulus q , we establish a conjecture of Soundararajan on the distribution of smooth numbers over reduced residue classes for such moduli q . A crucial ingredient in our argument is that, for such q , there is at most one ‘problem character’

Pascal's theorem gives a synthetic geometric condition for six points a , … , f in P 2 to lie on a conic. Namely, that the intersection points a b ¯ ∩ d e ¯ , a f ¯ ∩ d c ¯ , e f ¯ ∩ b c ¯ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d + 4 points in P d to lie on a degree d rational normal curve? In this paper we find many of these conditions

Given an orthogonal representation of a compact group, we show that any element of the connected component of the isometry group of the orbit space lifts to an equivariant isometry of the original Euclidean space. Corollaries include a simple formula for this connected component, which applies to ‘most’ representations.

An r -uniform hypergraph ( r -graph for short) is called linear if every pair of vertices belongs to at most one edge. A linear r -graph is complete if every pair of vertices is in exactly one edge. The famous Brown–Erdős–Sós conjecture states that for every fixed k and r , every linear r -graph with Ω ( n 2 ) edges contains k edges spanned by at most ( r − 2 ) k + 3 vertices. As an intermediate step

One of the main results in the paper mentioned in the title is the following. Theorem 1.Let Q 0 = [ 0 , 1 ] d be the unit cube in R d and let f : Q 0 → R be a continuous function with zero mean. Let Z ( f ) be the nodal set Z ( f ) = { x ∈ Q : f ( x ) = 0 } . Let H d − 1 ( Z ( f ) ) denote the ( d − 1 ) ‐dimensional Hausdorff measure of Z ( f ) . Then W 1 ( f + , f − ) H d − 1 ( Z ( f ) ) ∥ f ∥ ∞ ∥

We study the geometry of simply connected wandering domains for entire functions and we prove that every bounded connected regular open set, whose closure has a connected complement, is a wandering domain of some entire function. In particular such a domain can be realized as an escaping or an oscillating wandering domain. As a consequence, we obtain that every Jordan curve is the boundary of a wandering

Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank‐one projections). Suppose dim H ⩾ 3 . We prove the following Wigner‐type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti‐unitary operator. This statement will be obtained as a consequence of the following result: every

We show that a random group Γ in the triangular binomial model Γ ( n , p ) is asymptotically almost surely (a.a.s.) not left‐orderable for p ∈ ( c n − 2 , n − 3 / 2 − ε ) , where c , ε are any constants satisfying ε > 0 , c > ( 1 / 8 ) log 4 / 3 2 ≈ 0.3012 . We also prove that if p ⩾ ( 1 + ε ) ( log n ) n − 2 for any fixed ε > 0 , then a random Γ ∈ Γ ( n , p ) has a.a.s. no non‐trivial left‐orderable

Given two sets of natural numbers A and B of natural density 1, we prove that their product set A · B : = { a b : a ∈ A , b ∈ B } also has natural density 1. On the other hand, for any ε > 0 , we show there are sets A of density > 1 − ε for which the product set A · A has density < ε . This answers two questions of Hegyvári, Hennecart and Pach.

We classify abelian subgroups of two‐dimensional Artin groups.

Let Ω ( n ) denote the number of prime factors of n . We show that for any bounded f : N → C one has 1 N ∑ n = 1 N f ( Ω ( n ) + 1 ) = 1 N ∑ n = 1 N f ( Ω ( n ) ) + o N → ∞ ( 1 ) . This yields a new elementary proof of the Prime Number Theorem.

We present a direct proof of asymptotic consensus in the non‐linear Hegselmann–Krause model with transmission‐type delay, where the communication weights depend on the particle distance in phase space. Our approach is based on an explicit estimate of the shrinkage of the group diameter on finite time intervals and avoids the usage of Lyapunov‐type functionals or results from non‐negative matrix theory

In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it has no minor obtained canonically from either an ( ℵ 0 , ℵ 1 ) ‐regular bipartite graph or an order‐theoretic Aronszajn tree. In particular, this refuted an earlier conjecture of Halin's that only the first of these obstructions was needed to characterize

In the study of normal numbers using the base‐ b representation, b ⩾ 2 , we know that a number is normal if and only if it is simply normal in all the bases b n , n ⩾ 1 . We prove the analogue of this result for continued fraction normality. In particular, we show that two notions of continued fraction normality, one where overlapping occurrences of finite patterns are counted as distinct occurrences

Let G be a group definable in a weakly o‐minimal non‐valuational structure M . Then G / G 00 , equipped with the logic topology, is a compact Lie group, and if G has finitely satisfiable generics, then dim ( G / G 00 ) = dim ( G ) . Our main technical result is that G is a dense subgroup of a group definable in the canonical o‐minimal extension of M .

Given a Fano complete intersection defined by sections of a collection nef line bundles L 1 , … , L c on a Fano toric manifold Y , a construction of Givental/Hori–Vafa provides a mirror‐dual Landau–Ginzburg model. This construction depends on a choice of suitable nef partition; that is, a partition of the rays of the fan determined by Y . We show that Landau–Ginzburg models constructed from different

We prove a Cayley–Bacharach type theorem for points in projective space P n that lie on a complete intersection of n hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections.

We describe a probabilistic model involving iterated Brownian motion for constructing a random chainable continuum. We show that this random continuum is indecomposable. We use our probabilistic model to define a Wiener‐type measure on the space of all chainable continua.

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a. topological full groups). We determine that any piecewise full group that is locally compact in the compact‐open topology on the group of self‐homeomorphisms of the Cantor set must be uniformly discrete, in a precise sense that we introduce here. Uniformly discrete groups of self‐homeomorphisms

We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a hyperbolic torus with a hole. To our knowledge, this is the first example of a manifold with a cylindrical end whose number of eigenvalues is known to be finite and nonzero. The construction can be varied to give

In the present work we have presented proof of Ball's integral inequality using symmetric B ‐spline functions. The proof is different from the before known.

We give two sufficient and necessary conditions for the Hochschild extension of a finite‐dimensional algebra by its dual bimodule and a Hochschild 2‐cocycle to be a symmetric algebra.

Let Ω be a C 4 ‐smooth bounded pseudoconvex domain in C 2 . We show that if the ∂ ¯ ‐Neumann operator N 1 is compact on L ( 0 , 1 ) 2 ( Ω ) , then the embedding operator J : D o m ( ∂ ¯ ) ∩ D o m ( ∂ ¯ ∗ ) → L ( 0 , 1 ) 2 ( Ω ) is L p ‐regular for all 2 ⩽ p < ∞ .

We prove that there exists at most one minimal diffeomorphism in a given homotopy class between any two closed Riemannian surfaces. This results was previously known only under the assumption that the Riemannian metrics have constant Gaussian curvature. Along the way, we prove the New Main Inequality which substantially strengthens the classical Reich–Strebel inequality for quasiconformal maps.

Let n ⩾ 6 be an integer. We prove that the number of number fields with Galois group A n and absolute discriminant at most X is asymptotically at least X 1 / 8 + O ( 1 / n ) . For n ⩾ 8 , this improves upon the previously best‐known lower bound of X ( 1 − 2 n ! ) / ( 4 n − 4 ) − ε , due to Pierce, Turnage‐Butterbaugh, and Wood.

A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the stable equivalence class. This is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and

We show a Wolff–Denjoy type theorem in complete geodesic spaces in the spirit of Beardon's framework that unifies several results in this area. In particular, it applies to strictly convex bounded domains in R n or C n with respect to a large class of metrics including Hilbert's and Kobayashi's metrics. The results are generalized to 1‐Lipschitz compact mappings in infinite‐dimensional Banach spaces

In the article ‘Pseudo‐loop conditions’ by Pierre Gillibert, Julius Jonušas and Michael Pinsker [1], the authors would like to amend an incorrect grant number referenced in the Acknowledgement section. All authors have received funding from the Austrian Science Fund (FWF) through project No P27600. Michael Pinsker has received funding from the Czech Science Foundation grant No 18‐20123S.

The paper contains an alternative proof of the celebrated L p estimates for differentially subordinate martingales established by Burkholder and Wang in the eighties and nineties. The approach links the validity of the estimate to the existence of a lower solution to a novel boundary value problem.

Rickard proved that for certain self‐injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Broué. In this paper we study singular equivalences of finite‐dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex

For a prime p , we determine a Sylow p ‐subgroup D of a finite group G such that the principal p ‐block B of G has four irreducible ordinary characters. It has been determined already for the cases where the number is up to three by work by R. Brauer, J. Brandt, and V.A. Belonogov 30 years ago. Our proof relies on the classification of finite simple groups.

We prove some necessary conditions for a link to be either concordant to a quasi‐positive link, quasi‐positive, positive or the closure of a positive braid. The main applications of our results are a characterisation of positive links with unlinking numbers 1 and 2, and a combinatorial criterion to test if a positive link is the closure of a positive braid. Finally, we compile a table of all positive

We give a simple construction of new, complete, finite volume manifolds M of bounded, nonpositive curvature. These manifolds have ends that look like a mixture of locally symmetric ends of different ranks and their fundamental groups are not duality groups.

We show that the set of all possible constant diagonals of a bounded Hilbert space operator is always convex. This, in particular, answers an open question of Bourin (2003). Moreover, we show that the joint numerical range of a commuting operator tuple is, in general, not convex, which fills a gap in the literature. We also prove that the Asplund–Ptak numerical range (which is convex for pairs of operators)

Let σ = ( σ 1 , σ 2 , ⋯ , σ n ) ∈ S n − 1 and d σ denote the normalized Lebesgue measure on S n − 1 , n ⩾ 2 . For functions f 1 , f 2 , ⋯ , f n defined on R , consider the multilinear operator given by T ( f 1 , f 2 , ⋯ , f n ) ( x ) = ∫ S n − 1 ∏ j = 1 n f j ( x − σ j ) d σ , x ∈ R . In this paper, we obtain necessary and sufficient conditions on exponents p 1 , p 2 , ⋯ , p n and r for which the operator

We prove that the kernel of the evaluation morphism of global sections — namely the syzygy bundle — of a sufficiently ample line bundle on an abelian variety is stable. This settles a conjecture of Ein–Lazarsfeld–Mustopa, in the case of abelian varieties.

Inspired by work of Lipshitz–Sarkar, we show that the module structure on link Floer homology detects split links. Using results of Ni, Alishahi–Lipshitz, and Lipshitz–Sarkar, we establish an analogous detection result for sutured Floer homology.

We provide a new and very short proof of the fact that a spherical functor between certain triangulated categories induces an auto-equivalence.

A classical result due to Eagleson states (in particular) that if appropriately normalized Birkhoff sums generated by a measurable function and an ergodic probability preserving transformation converge in distribution, then they also converge in distribution with respect to any probability measure which is absolutely continuous with respect to the invariant one. In this note, we prove several quantitative

We show that every bounded pseudo‐convex domain with Hölder boundary in C n is hyperconvex.

For a Lie algebra L , let S ( L ) denote the symmetric Poisson algebra and s ( L ) the truncated symmetric Poisson algebra of L . In characteristic p ≠ 2 , the conditions under which these Poisson algebras are solvable were established by Monteiro Alves and Petrogradsky in (J. Algebra 488 (2017) 244–281). The characterization in the harder case p = 2 was left as an open problem and a related conjecture

A matrix A is totally positive (or non‐negative) of order k , denoted T P k (or T N k ), if all minors of size ⩽ k are positive (or non‐negative). It is well known that such matrices are characterized by the variation diminishing property together with the sign non‐reversal property. We do away with the former, and show that A is T P k if and only if every submatrix formed from at most k consecutive

Let K be a complete, discretely valued field with finite residue field and G K its absolute Galois group. The subject of this note is the study of the set of positive integers d for which there exists an absolutely irreducible ℓ -adic representation of G K of dimension d with rational traces on inertia. Our main result is that non-Sophie Germain primes are not in this set when the residue characteristic

We show that the Fraïssé limit of a category of unital separable C ∗ -algebras which is sufficiently closed under tensor products of its objects and morphisms is strongly self-absorbing, given that it has approximately inner half-flip. We use this connection between Fraïssé limits and strongly self-absorbing C ∗ -algebras to give a rather elementary proof for the well-known fact that the Jiang–Su algebra

For all 1 ⩽ m ⩽ n − 1 , we investigate the interaction of locally finite measures in R n with the family of m -dimensional Lipschitz graphs. For instance, we characterize Radon measures μ , which are carried by Lipschitz graphs in the sense that there exist graphs Γ 1 , Γ 2 , ⋯ such that μ ( R n ∖ ⋃ 1 ∞ Γ i ) = 0 , using only countably many evaluations of the measure. This problem in geometric measure

Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order complexes of posets.

Let D be the unit disk. Kutzschebauch and Studer (Bull. Lond. Math. Soc. 51 (2019) 995–1004) recently proved that, for each continuous map A : D ¯ → SL ( 2 , C ) , which is holomorphic in D , there exist continuous maps E , F : D ¯ → sl ( 2 , C ) , which are holomorphic in D , such that A = e E e F . Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this