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In the 1st paper of this series we studied the asymptotic behavior of Betti numbers, twisted torsion, and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo–Stuck–Zimmer theorem classifies all IRSs of G. Using the classification, one can deduce asymptotic statements about spectral invariants of lattices. When G has real rank one, the space of IRSs is more complicated. We construct here several uncountable families of IRSs in the groups SO(n , 1), n ≥ 2. We give dimension-specific constructions when n = 2, 3, and also describe a general gluing construction that works for every n. Part of the latter construction is inspired by Gromov and Piatetski-Shapiro's construction of non-arithmetic lattices in SO(n , 1). [ABSTRACT FROM AUTHOR]

Let S be a K3 surface with primitive curve class β⁠. We solve the relative Gromov–Witten theory of S × P1 in classes (β,1) and (β,2)⁠. The generating series are quasi-Jacobi forms and equal to a corresponding series of genus 0 Gromov–Witten invariants on the Hilbert scheme of points of S⁠. This proves a special case of a conjecture of Pandharipande and the author. The new geometric input of the paper is a genus bound for hyperelliptic curves on K3 surfaces proven by Ciliberto and Knutsen. By exploiting various formal properties we find that a key generating series is determined by the very first few coefficients. Let E be an elliptic curve. As collorary of our computations, we prove that Gromov–Witten invariants of S × E in classes (β,1) and (β,2) are coefficients of the reciprocal of the Igusa cusp form. We also calculate several linear Hodge integrals on the moduli space of stable maps to a K3 surface and the Gromov–Witten invariants of an abelian threefold in classes of type (1,1,d)⁠. [ABSTRACT FROM AUTHOR]

We compute the class of the classifying stack of the exceptional algebraic group G2 and of the spin groups Spin7 and Spin8 in the Grothendieck ring of stacks, and show that they are equal to the inverse of the class of the corresponding group. Furthermore, we show that the computation of the motivic classes of the stacks BSpinn can be reduced to the computation of the classes of B Δn⁠, where Δn ⊂ Pinn is the "extraspecial 2-group", the preimage of the diagonal matrices under the projection Pinn → On to the orthogonal group. [ABSTRACT FROM AUTHOR]

Given an elliptic curve |$E$| defined over |$\mathbb{Q}$|⁠, we are motivated by the 2-part of the Birch and Swinnerton-Dyer formula to study the relation between the 2-Selmer rank of |$E$| and the 2-Selmer rank of an abelian variety |$A$| obtained by Ribet's level raising theorem. For certain imaginary quadratic fields |$K$| satisfying the Heegner hypothesis, we prove that the 2-Selmer ranks of |$E$| and |$A$| over |$K$| have different parity, as predicted by the BSD conjecture. When the 2-Selmer rank of |$E$| is one, we further prove that the 2-Selmer rank of |$A$| can never be zero, revealing an obstruction to rank lowering which is unseen for |$p$| -Selmer groups for odd |$p$|⁠. [ABSTRACT FROM AUTHOR]

In this article, we study the Legendrian knots in the knot types of satellite knots. In particular, we classify Legendrian Whitehead patterns and learn a great deal about Legendrian braided patterns. We also show how the classification of Legendrian patterns can lead to a classification of the associated satellite knots if the companion knot is Legendrian simple and uniformly thick. This leads to new Legendrian and transverse classification results for knots in the 3-sphere with its standard contact structure as well as a more general perspective on some previous classification results. [ABSTRACT FROM AUTHOR]

We study whether there exist infinitely many surfaces with given discrete invariants for which the $$H^2$$ is of CM type. This is a surface analogue of a conjecture of Coleman about curves. We construct a large number of examples of families of surfaces with $$p_{\text{g}} = 1$$ that in the moduli space cut out a special subvariety; these provide a positive answer to our question. Most of these are families of K3 surfaces but we also obtain some families of surfaces of general type. As input for our construction we use the work of Deligne and Mostow. Finally we prove that a very general K3 surface cannot be dominated by a product of curves of small genus. [ABSTRACT FROM AUTHOR]

Consider the holomorphic bundle with connection on $${\mathbb P}^1-\{0,1,\infty\}$$ corresponding to the regular hypergeometric differential operator   ∏ j = 1 h (D − α j) − z ∏ j = 1 h (D − β j), D = z d d z. If the numbers $$\alpha_i$$ and $$\beta_j$$ are real and for all $$i$$ and $$j$$ the number $$\alpha_i-\beta_j$$ is not integer, then the bundle with connection is known to underlie a complex polarizable variation of Hodge structures. We calculate some Hodge invariants for this variation, in particular, the Hodge numbers. From this, we derive a conjecture of Alessio Corti and Vasily Golyshev. We also use non-abelian Hodge theory to interpret our theorem as a statement about parabolic Higgs bundles. [ABSTRACT FROM AUTHOR]

We study Pandharipande–Thomas’s stable pair theory on $$K3$$ fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai–Yoshioka’s formula for the Euler characteristics of moduli spaces of stable pairs on $$K3$$ surfaces and Noether–Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the $$K3$$ fibration. In the case that the $$K3$$ fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson–Thomas invariants of 2D Gieseker semistable sheaves supported on the fibers. [ABSTRACT FROM AUTHOR]

Donovan and Wemyss [8] introduced the contraction algebra of flopping curves in 3-folds. When the flopping curve is smooth and irreducible, we prove that the contraction algebra together with its A∞-structure recovers various invariants associated to the underlying singularity and its small resolution, including the derived category of singularities and the genus zero Gopakumar-Vafa invariants. [ABSTRACT FROM AUTHOR]

We compute global log canonical thresholds, or equivalently alpha invariants, of birationally rigid Fano three-folds embedded in weighted projective spaces as codimension two or three. As an important application, we prove that most of them are weakly exceptional, $$K$$ -stable and admit Kähler–Einstien metric. [ABSTRACT FROM AUTHOR]

We use Green's transference principle to show that any subset of the dth powers of primes with positive relative density contains nontrivial solutions to a translation-invariant linear equation in d² + 1 or more variables, with explicit quantitative bounds. [ABSTRACT FROM AUTHOR]

We introduce L²-Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II1 factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups #915;, we recover the ordinary (co)homology of #915;. For Cartan subalgebras, we recover Gaboriau's L²-Betti numbers for the associated equivalence relation. In this common framework, we prove that the L²-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property, and amenability. We compute the L²-Betti numbers for the standard invariants of the Temperley--Lieb--Jones subfactors and of the Fuss--Catalan subfactors, as well as for free products and tensor products. [ABSTRACT FROM AUTHOR]

Following an idea of Ishida, we develop polynomial equations for certain unramified double covers of surfaces with pg = q = 1 and K² = 2. Our first main result provides an explicit surface X with these invariants defined over ... that has Picard number ρ(X) = 2, which is the smallest possible for these surfaces. This is done by giving equations for the double cover ... of X, calculating the zeta function of the reduction of ... to 픽3, and extracting from this the zeta function of the reduction of X to 픽3; the basic idea used in this process may be of independent interest. Our second main result is a big monodromy theorem for a family that contains all surfaces with pg = q = 1, K² = 2, and K ample. It follows from this that a certain Hodge correspondence of Kuga and Satake, between such a surface and an abelian variety, is motivated (and hence absolute Hodge). This allows us to deduce our third main result, which is that the Tate Conjecture in characteristic zero holds for all surfaces with pg = q = 1, K² = 2, and K ample. [ABSTRACT FROM AUTHOR]

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent Laplacian and Schramm's orthogonal circle patterns. We show that the corresponding discrete minimal surfaces unify the earlier notions of discrete minimal surfaces: circular minimal surfaces via the integrable systems approach and conical minimal surfaces via the curvature approach. In fact they form conjugate pairs of minimal surfaces. Furthermore, discrete holomorphic quadratic differentials obtained from discrete integrable systems yield discrete minimal surfaces which are critical points of the total area. [ABSTRACT FROM AUTHOR]

We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary contact 3-manifold. In contrast, we observe that there are only finitely many possibilities for the homology groups of Stein fillings of a given contact 3-manifold with support genus zero. We also show that there are 4-strand braids which admit infinitely many distinct Hurwitz classes of quasipositive factorizations, yielding in particular an infinite family of knotted complex analytic annuli in the 4-ball bounded by the same transverse link up to transverse isotopy. These realize the smallest possible examples in terms of the number of boundary components a genus-1 mapping class and the number of strands a braid can have with infinitely many positive/quasipositive factorizations. [ABSTRACT FROM AUTHOR]

We study the orbits and polynomial invariants of certain affine action of the super Weyl groupoid of Lie superalgebra gl(n,m), depending on a parameter. We show that for generic values of the parameter all the orbits are finite and separated by certain explicitly given invariants. We also describe explicitly the special set of parameters, for which the algebra of invariants is not finitely generated and does not separate the orbits, some of which are infinite. [ABSTRACT FROM AUTHOR]

We show that the S¹-equivariant Yamabe invariant of the 3-sphere, endowed with the Hopf action, is equal to the (non-equivariant) Yamabe invariant of the 3-sphere. More generally, we establish a topological upper bound for the S¹-equivariant Yamabe invariant of any closed oriented 3-manifold endowed with an S¹-action. Furthermore, we prove a convergence result for the equivariant Yamabe constants of an accumulating sequence of subgroups of a compact Lie group acting on a closed manifold. [ABSTRACT FROM AUTHOR]

We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework for various multidimensional spinning tops, as well as Beauville systems, we prove that if the spectral curve is nodal, then all singularities on the corresponding fiber of the system are non-degenerate. We also show that the type of a non-degenerate singularity can be read off from the behavior of double points on the spectral curve under an appropriately defined antiholomorphic involution. Our analysis is based on linearization of integrable flows on generalized Jacobian varieties, as well as on the possibility to express the eigenvalues of an integrable vector field linearized at a singular point in terms of residues of certain meromorphic differentials. [ABSTRACT FROM AUTHOR]

We study the behavior of the signature of colored links [6, 9] under the splice operation. We extend the construction to colored links in integral homology spheres and show that the signature is almost additive, with a correction term independent of the links. We interpret this correction term as the signature of a generalized Hopf link and give a simple closed formula to compute it. [ABSTRACT FROM AUTHOR]

In this paper, we prove that, over an algebraically closed field of odd characteristic, a weakly tame automorphism of a K3 surface of finite height can be lifted over the ring of Witt vectors of the base field. Also we prove that a non-symplectic tame automorphism of a supersingular K3 surface or a symplectic tame automorphism of a supersingular K3 surface of Artin-invariant at least 2 can be lifted to the ring of Witt vectors. Using these results, we prove, for a weakly tame K3 surface of finite height, there is a lifting over the ring of Witt vectors to which the entire automorphism group extends. Also we prove a K3 surface equipped with a purely non-symplectic automorphism of a certain high order is unique up to isomorphism. The following is the outline of the paper. In Section 1, we review some background of the problem and state the main results. In Section 2, we recall the deformation theory of K3 surfaces defined over a field of positive characteristic and prove that formal liftings of a K3 surface over the Witt vectors are classified by the Hodge structure on the second crystalline cohomology. In Section 3, we prove the main results based on the classification of liftings and some properties of the crystalline cohomology of K3 surfaces. In Section 4, we prove, using the main result, the uniqueness of a tame purely non-symplectic automorphism of some high order of a K3 surface over a field of positive characteristic. [ABSTRACT FROM AUTHOR]

Suppose that g is a complex reductive Lie algebra. Let σ be an involution of g, which defines a Z/2Z-grading g = g0⊕g1, where g0 := gσ is a symmetric subalgebra of g. The corresponding Z2-contraction of g is the Lie algebra g˜ = g0⋉g1, where g1 is a commutative ideal. The goal of this article is to find out whether the algebra of symmetric invariants S(g˜) g˜ is a polynomial ring in rk g variables. To this end we develop a more general theory of symmetric invariants related to semi-direct products. [ABSTRACT FROM AUTHOR]

Let C be a finite tensor category with simple unit object, let Z(C) denote its monoidal center, and let L and R be a left adjoint and a right adjoint of the forgetful functor U : Z(C) → C. We show that the following conditions are equivalent: (1) C is unimodular, (2) U is a Frobenius functor, (3) L preserves the duality, (4) R preserves the duality, (5) L(1) is self-dual, and (6) R(1) is self-dual, where 1 ∈ C is the unit object. We also give some other equivalent conditions. As an application, we give a categorical understanding of some topological invariants arising from finite-dimensional unimodular Hopf algebras. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree ƍ ≥ 2 is said to be disintegrated if it is not linearly conjugate to xƍ or ±Tƍ(x), where Tƍ(x) is the Chebyshev polynomial of degree ƍ. Let d and n be integers >1, we prove that there exists an effectively computable constant c(d,n) depending only on d and n such that the following holds. Let f1,..., fn ∈ K[x] be disintegrated polynomials of degree at most d and let φ = f1×...×fn be the induced coordinate-wise self-map of AnK. Then the period of every irreducible φ-periodic subvariety of AnK with non-constant projection to each factor A¹K is at most c(d,n). As an immediate application, we prove an instance of the dynamical Mordell-Lang problem following recent work of Xie. The main technical ingredients are the Medvedev-Scanlon classification of invariant subvarieties together with classical and more recent results in Ritt's theory of polynomial decomposition. [ABSTRACT FROM AUTHOR]

In this paper, we conjecture an extension of the Hilbert basis theorem and the finite generation of invariants to commutative algebras in symmetric finite tensor categories over fields of positive characteristic. We prove the conjecture in the case of semisimple categories and more generally in the case of categories with fiber functors to the characteristic p>0 Verlinde category of SL2. We also construct a symmetric finite tensor category sVec2 over fields of characteristic 2 and show that it is a candidate for the category of supervector spaces in this characteristic. We further show that sVec2 does not fiber over the characteristic 2 Verlinde category of SL2 and then prove the conjecture for any category fibered over sVec2. [ABSTRACT FROM AUTHOR]

In this paper, we study bilipschitz equivalences of germs of holomorphic foliations in (C2,0). We prove that the algebraic multiplicity of a singularity is invariant by such equivalences. Moreover, for a large class of singularities, we show that the projective holonomy representation is also a bilipschitz invariant. [ABSTRACT FROM AUTHOR]

We study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order k at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierarchy of higher order analogs to the PIII equation, which reduces to the PIII equation itself when the pole is simple. [ABSTRACT FROM AUTHOR]

For a Riemannian manifold with dimension at least 6, we prove that the existence of a conformal metric with positive scalar and Q curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator. [ABSTRACT FROM AUTHOR]

Let p be a prime number, and F be a nonarchimedean local field of residual characteristic p. We explore the interaction between the pro-p-Iwahori-Hecke algebras of the group GLn(F) and its derived subgroup SLn(F). Using the interplay between these two algebras, we deduce two main results. The first is an equivalence of categories between Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SL2(Qp) and smooth mod-p representations of SL2(Qp) generated by their pro-p-Iwahori-invariants. The second is a "numerical correspondence" between packets of supersingular Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SLn(F), and irreducible, n-dimensional projective Galois representations. [ABSTRACT FROM AUTHOR]

We determine the 6D solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion, generalized Gauduchon, balanced and strongly Gauduchon metrics is studied. As an application, we construct a holomorphic family (M, Ja) of compact complex manifolds such that (M, Ja) satisfies the ∂ ∂-lemma and admits a balanced metric for any a≠ 0, but the central limit neither satisfies the ∂ ∂-lemma nor admits balanced metrics. [ABSTRACT FROM AUTHOR]

Here we state a conjecture concerning a local version of Brunella's alternative: any codimension one foliation in (C³, 0) without germ of invariant surface has a neighborhood of the origin formed by leaves containing a germ of analytic curve at the origin. We prove the conjecture for the class of codimension one foliations whose reduction of singularities is obtained by blowing-up points and curves of equireduction and such that the final singularities are free of saddle-nodes. The concept of "partial separatrix" for a given reduction of singularities has a central role in our argumentations, as well as the quantitative control of the generic Camacho-Sad index in dimension three. The "nodal components" are the only possible obstructions to get such germs of analytic curves. We use the partial separatrices to push the leaves near a nodal component towards compact dicritical divisors, finding in this way the desired analytic curves. [ABSTRACT FROM AUTHOR]

We show that the equation 1n1²+...+sns²=0 admits nontrivial solutions in any subset of [N] of density (log N)-cs, provided that s=7 and the coefficients iZ\{0} sum to zero and satisfy certain sign conditions. This improves upon previous known density bounds of the form (log log N)-c. [ABSTRACT FROM AUTHOR]

Fix O a special nilpotent co-adjoint orbit and write A(O) for the group of components of the centralizer of an element f in O. Let 1 denotes the trivial representation of A(O). The Springer correspondence attaches to the pair (O, 1) an irreducible representation of the Weyl group W which we denote by Sp(O). Sp(O) admits two natural bases consisting of homogeneous polynomials on h*. One of the bases is parametrized by a set Primp (O), consisting of certain primitive ideals of the universal enveloping algebra. The second basis consists of character polynomials, parametrized by a set of geometric objects. The relation between these bases encodes significant representation theoretical information. We study the change-of-basis matrix and derive consequences relevant to the computation of invariants of Harish-Chandra modules. [ABSTRACT FROM AUTHOR]

For local Calabi-Yau (CY) manifolds which are total spaces of vector bundle over algebraic Goresky-Kottwitz-MacPherson (GKM) manifolds, we propose a formal definition of reduced genus 1 Gromov-Witten (GW) invariants, by assigning contributions from the refined decorated rooted trees. We show that this definition satisfies a localized version of the standard versus reduced (LSvR) formula, whose global version in the compact cases is due to Zinger. As an application, we prove the conjecture in a previous article on the genus 1 GW invariants of local CY manifolds which are total spaces of concave splitting vector bundles over projective spaces. In particular, we prove the mirror formulae for genus 1 GW invariants of KP2 and KP3 , conjectured by Klemm, Zaslow, and Pandharipande. In the appendix, we derive the modularity of genus 1 GW invariants for the local P2 as a consequence of the results on Ramanujan's cubic transformation. Inspired by the LSvR formula, we show that the ordinary genus 1 GW invariants of CY hypersurfaces in projective spaces can be computed by virtual localization and quantum hyperplane property after the contribution of a genus 1 vertex is replaced by a modified one. [ABSTRACT FROM AUTHOR]

We introduce a notion of refinements in the context of patching, in order to obtain new results about local-global principles and field invariants in the context of quadratic forms and central simple algebras. The fields we consider are finite extensions of the fraction fields of two-dimensional complete domains that need not be local. Our results in particular give the u-invariant and period-index bound for these fields, as consequences of more general abstract results. [ABSTRACT FROM AUTHOR]

Given a compact Lie group G and a closed subgroup H, the dimension datum of H is the map DH : Ĝ → ℤ which assigns to each irreducible complex linear representation W of G the dimension of WH , the subspace of H-invariant vectors. We prove that as a subspace of ZĜ the set consisting of dimension data of closed subgroups of G is compact. We obtain new proofs of some previously known finiteness and rigidity results and we give an application to spectral geometry. [ABSTRACT FROM AUTHOR]

Let d1 and d2 be discriminants of distinct quadratic imaginary orders Od1 and Od2 and let J(d1 ,d2 ) denote the product of differences of CM j-invariants with discriminants d1 and d2 . In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer J(d1 ,d2 ) in the case that d1 and d2 are relatively prime and discriminants of maximal orders. To compute this formula, they first reduce the problem to counting the number of simultaneous embeddings of Od1 and Od2 into endomorphism rings of supersingular curves, and then solve this counting problem. Interestingly, this counting problem also appears when computing class polynomials for invariants of genus 2 curves. However, in this application, one must consider orders Od1 and Od2 that are non-maximal. Motivated by the application to genus 2 curves, we generalize the methods of Gross and Zagier and give a computable formula for vℓ (J(d1 ,d2 )) for any pair of discriminants d1 ≠ d2 and any prime ℓ >2. In the case that d1 is squarefree and d2 is the discriminant of any quadratic imaginary order, our formula can be stated in a simple closed form. We also give a conjectural closed formula when the conductors of d1 and d2 are relatively prime. [ABSTRACT FROM AUTHOR]

We construct some examples of origamis (square-tiled surfaces) such that the Hodge bundles over the corresponding SL(2,R)-orbits on the moduli space admit nontrivial isotropic SL(2,R)-invariant subbundles. This answers a question posed to the authors by Eskin and Forni. [ABSTRACT FROM AUTHOR]

A compact Lie group G and a faithful complex representation V determine the Sato- Tate measure μG,V on C, defined as the direct image of Haar measure with respect to the character map g → tr(g ∣ V). We give a necessary and sufficient condition for a Sato- Tate measure to be an isolated point in the set of all Sato-Tate measures, endowed with the topology defined by moments. In particular, we prove that if G is connected and semisimple and V is irreducible, then μG,V is an isolated point. [ABSTRACT FROM AUTHOR]

We present a new form of the logarithmic Hardy inequality on the unit ball in RN with its optimal constant. One of the novelties of our inequality is the scale invariance under the natural scaling. We also discuss the nonexistence of a minimizer for an associated minimizing problem via a scaling argument. [ABSTRACT FROM AUTHOR]

We prove vanishing of distribution on p-adic spherical spaces that are equivariant with respect to a generic character of the nilradical of a Borel subgroup and satisfy a certain condition on the wave-front set. We deduce from this nonvanishing of spherical Bessel functions for Galois symmetric pairs. [ABSTRACT FROM AUTHOR]

Given a knot K ⊂ S³, Seidel and Smith described a graded cohomology group Khsymp,inv(K), a variant of their symplectic Khovanov cohomology group [18]. They also constructed a spectral sequence converging to the Heegaard Floer homology group HF(Σ(K)#(S² × S¹)) with E¹-page isomorphic to a summand of Khsymp,inv(K). A previous paper [19] showed that the higher pages of this spectral sequence are knot invariants. Here we discuss a reduced version of the spectral sequence which directly computes HF(Σ(K)). Under some degeneration conditions, one obtains a new absolute Maslov grading on the group HF(Σ(K)). This occurs when K is a two-bridge knot, and we compute the grading in this case. We also extract some ℚ-valued knot invariants from this construction. [ABSTRACT FROM AUTHOR]

It is well known that a positive proportion of all points in a d-dimensional lattice is visible from the origin, and that these visible lattice points have constant density in ℝd. In the present paper, we prove an analogous result for a large class of quasicrystals, including the vertex set of a Penrose tiling. We furthermore establish that the statistical properties of the directions of visible points are described by certain SL(d,ℝ)-invariant point processes. Our results imply in particular existence and continuity of the gap distribution for directions in certain 2D cut-and-project sets. This answers some of the questions raised by Baake et al. [1]. [ABSTRACT FROM AUTHOR]

Manin's conjecture predicts an asymptotic formula for the number of rational points of bounded height on a smooth projective variety X in terms of global geometric invariants of X. The strongest form of the conjecture implies certain inequalities among geometric invariants of X and of its subvarieties. We provide a general geometric framework explaining these phenomena, via the notion of balanced line bundles, and prove the required inequalities for a large class of equivariant compactifications of homogeneous spaces. [ABSTRACT FROM AUTHOR]

It is the forth paper in a series devoted to the investigation of the ADE subalgebras of the triplet vertex algebra W(p). In this paper, we prove C2-cofiniteness of the invariant subalgebras W(p)Γ and study their representations when Γ is the tetrahedral group T (of E6-type), or the octahedral group O (of E7-type). We give a list of 21p irreducible W(p)T-modules and a list of 28p irreducible W(p)O-modules. The completeness of these two lists can be reduced to some conjectures about constant term identities which have been verified in the case p=1. For the proof of C2-cofiniteness of W(p)Γ , and the calculations of the Zhu's algebras A(W(p)Γ), we develop several quite universal formulas (Theorem 4.3 and its corollaries). In Appendix, we further apply them to derive a useful tool of structure coefficients for the triplet vertex algebra W(p) and its irreducible modules, which unifies almost all the calculations in this paper; as an unexpected byproduct, we give a new proof of sl2-action on W(p). [ABSTRACT FROM AUTHOR]

The lower central series invariants Mk of an associative algebra A are the two-sided ideals generated by k-fold iterated commutators; the Mk provide a filtration of A. We study the relationship between the geometry of X =Spec Aab and the associated graded components Nk of this filtration. We show that the Nk form coherent sheaves on a certain nilpotent thickening of X, and that Zariski localization on X coincides with noncommutative localization of A. Under certain freeness assumptions on A, we give an alternative construction of Nk purely in terms of the geometry of X (and in particular, independent of A). Applying a construction of Kapranov, we exhibit the Nk as natural vector bundles on the category of smooth schemes. [ABSTRACT FROM AUTHOR]

In this paper, we study the knot Floer homology of a subfamily of twisted (p, q) torus knots where q=±1 (mod p). Specifically, we classify the knots in this subfamily that admit L-space surgeries. To do calculations, we use the fact that these knots are (1, 1) knots and, therefore, admit a genus one Heegaard diagram. [ABSTRACT FROM AUTHOR]

In this paper, we will show how to distinguish contact structures using Rabinowitz- Floer homology--a homology for strong exactly fillable contact manifolds, which was recently introduced by Cieliebak and Frauenfelder [3]. Unlike contact homology, this invariant is not S¹-equivariant. As an application we give a new proof for the following result due to Ilya Ustilovsky: There are infinitely many nondiffeomorphic contact structures in every formal homotopy class of S4m+1. Furthermore, we construct nondiffeomorphic contact structures lying in the same formal homotopy class on S4m-1. [ABSTRACT FROM AUTHOR]

We investigate the Mordell constant of certain families of lattices, in particular, of lattices arising from totally real fields. We define the almost sure value κμ of the Mordell constant with respect to certain homogeneous measures on the space of lattices, and establish a strict inequality κμ1 < κμ2 when the μi are finite and supp(μ1)⊈supp(μ2). In combination with known results regarding the dynamics of the diagonal group we obtain isolation results as well as information regarding accumulation points of the Mordell-Gruber spectrum, extending previous work of Gruber and Ramharter. One of the main tools we develop is the associated algebra, an algebraic invariant attached to the orbit of a lattice under a block group, which can be used to characterize closed and finite volume orbits. [ABSTRACT FROM AUTHOR]

We prove that symplectic 4-manifolds with b+ > 1 have nonvanishing Donaldson invariants, and that the canonical class is always a basic class when b1 =0. We also characterize in many situations the basic classes of a Lefschetz fibration over the sphere which evaluate maximally on a generic fiber. [ABSTRACT FROM AUTHOR]

The zero set of the hyperbolic Gaussian analytic function is a random point process in the unit disk whose distribution is invariant under automorphisms of the disk. We study the variance of the number of points in a disk of increasing radius. Somewhat surprisingly, we find a change of behavior at a certain value of the "intensity" of the process, which appears to be novel. [ABSTRACT FROM AUTHOR]