We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.

Let V be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on V is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on V are obtained, and the distribution of visible points into different congruence classes is investigated.

We prove Manin's conjecture for four singular quartic del Pezzo surfaces over imaginary quadratic number fields, using the universal torsor method.

In this paper, we give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type Nn ×f Qq, where f ∈ C∞(N). This setting allows us to deal, among other things, with minimal submanifolds bounded by cylinders, cones, spheres and pseudo-hyperbolic spaces where most of these examples are not covered in the literature. Applications also include the study of the essential spectrum of hyperbolic graphs over compact regions of the boundary at infinity.

We prove that the regular hendecagon (11-gon) is constructible by marked ruler and compass.

In this paper we study the existence and the dynamics of a very special class of motions, which satisfy a strong global minimization property. More precisely, we call a free time minimizer a curve which satisfies the least action principle between any pair of its points without the constraint of time for the variations. An example of a free time minimizer defined on an unbounded interval is a parabolic homothetic motion by a minimal central configuration. The existence of a large amount of free time minimizers can be deduced from the weak KAM theorem. In particular, for any choice of x0, there should be at least one free time minimizer x(t) defined for all t ≥ 0 and satisfying x(0)=x0. We prove that such motions are completely parabolic. Using Marchal's theorem we deduce as a corollary that there are no entire free time minimizers, i.e. defined on $\mathbb{R}$. This means that the Mañé set of the Newtonian N-body problem is empty.

The coinvariant algebra of a Weyl group plays a fundamental role in several areas of mathematics. The fake degrees are the graded multiplicities of the irreducible modules of a Weyl group in its coinvariant algebra, and they were computed by Steinberg, Lusztig and Beynon–Lusztig. In this paper we formulate a notion of spin coinvariant algebra for every Weyl group. Then we compute all the spin fake degrees for each classical Weyl group, which are by definition the graded multiplicities of the simple modules of a spin Weyl group in the spin coinvariant algebra. The spin fake degrees for the exceptional Weyl groups are given in a sequel.

We give new Casson–Gordon style obstructions for a two–component link to be topologically concordant to the Hopf link.

In this paper, we extend the definition of property T and strong property T to general C*-algebras (not necessarily unital). We show that if an inclusion pair of locally compact groups (G,H) has property T, then (C*(G), C*(H)) has property T. As a partial converse, if T is abelian and C*(G) has property T, then T is compact. We also show that if Ω is a first countable locally compact Hausdorff space, then C0(Ω) has (strong) property T if and only if Ω is discrete. Furthermore, the non-unital C*-algebra $c_0(\mathbb{Z}^n)\rtimes SL_n(\mathbb{Z})$ has strong property T when n ≥ 3. We also give some equivalent forms of strong property T, which are new even in the unital case.

In this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi.

A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing projection and hence, every knot has a minimal quadruple crossing number c4(K). In this paper, we investigate quadruple crossing number, and in particular, use the span of the bracket polynomial to determine quadruple crossing number for a variety of knots and links.

Hua domains, generalized Hua domains and Hua constructions, named after the great Chinese mathematician Luogeng Hua (Loo-Keng Hua), are generalizations of Cartan–Hartogs domains introduced by Weiping Yin around the end of the 20th century. In this paper, we give a complete description of automorphism groups of generalized Hua domains. We also discuss the corresponding problem for Hua constructions.

We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the ‘twistor’ bundle of conformal inner products having holomorphic image. The second solution allows to use standard results in algebraic geometry to show that the Weyl connections on the two-sphere whose geodesics are the great circles are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane.

For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X,${\cal E}$,H) for the twisted odd signature operator valued in a flat hermitian vector bundle ${\cal E}$, where H = ∑ ij+1H2j+1 is an odd-degree closed differential form on X and H2j+1 is a real-valued differential form of degree 2j+1. We show that ρ(X,${\cal E}$,H) is independent of the choice of metrics on X and ${\cal E}$ and of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah–Patodi–Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary. The homotopy invariance of the rho invariant ρ(X,${\cal E}$,H) is more delicate to establish, and is settled under further hypotheses on the fundamental group of X.

Recently, Conlon, Fox and the author gave a new proof of a relative Szemerédi theorem, which was the main novel ingredient in the proof of the celebrated Green–Tao theorem that the primes contain arbitrarily long arithmetic progressions. Roughly speaking, a relative Szemerédi theorem says that if S is a set of integers satisfying certain conditions, and A is a subset of S with positive relative density, then A contains long arithmetic progressions, and our recent results show that S only needs to satisfy a so-called linear forms condition.

This paper contains an alternative proof of the new relative Szemerédi theorem, where we directly transfer Szemerédi's theorem, instead of going through the hypergraph removal lemma. This approach provides a somewhat more direct route to establishing the result, and it gives better quantitative bounds.

The proof has three main ingredients: (1) a transference principle/dense model theorem of Green–Tao and Tao–Ziegler (with simplified proofs given later by Gowers, and independently, Reingold–Trevisan–Tulsiani–Vadhan) applied with a discrepancy/cut-type norm (instead of a Gowers uniformity norm as it was applied in earlier works); (2) a counting lemma established by Conlon, Fox and the author; and (3) Szemerédi's theorem as a black box.

Two related issues are explored for bond percolation on ${\mathbb{Z}^d$ (with d ≥ 3) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point pfin satisfies pfin ≥ pc, and strict inequality is proved when either d is sufficiently large, or d ≥ 7 and the model is sufficiently spread out. It is not known whether d ≥ 3 suffices. Secondly, for what p does there exist an infinite dual surface of plaquettes? The associated critical point psurf satisfies psurf ≥ pfin.

Necessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

In a previous paper, Button and Thillaisundaram proved that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, we show that the aforementioned result of Button and Thillaisundaram implies a result of Lackenby.

For an odd prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z}_p$-extension of the rational number field. In this paper, we classify all S such that the Galois group is a metacyclic pro-p group.

The spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on ${\mathbb R}^1$, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on ${\mathbb Z}_n$, ${\mathbb Z}$ and ${\mathbb R}^1$ and also the seemingly stronger universal tiling (spectrum) conjectures on the respective groups. In this paper, we clarify the equivalences between these statements in dimension one. In addition, we show that if the Fuglede conjecture on ${\mathbb R}^1$ is true, then every spectral set with rational measure must have a rational spectrum. We then investigate the Coven–Meyerowitz property for finite sets of integers, introduced in [1], and we show that if the spectral sets and the tiles in ${\mathbb Z}$ satisfy the Coven–Meyerowitz property, then both sides of the Fuglede conjecture on ${\mathbb R}^1$ are true.