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  • On the Discretized Sum-Product Problem
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-13
    Guth L, Katz N, Zahl J.

    We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset \mathbb{R}$ is a $(\delta ,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac{1}{68}}$.

    更新日期:2020-01-15
  • A Family of Riesz Distributions for Differential Forms on Euclidian Space
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-13
    Fischmann M, Ørsted B.

    In this paper, we introduce a new family of operator-valued distributions on Euclidian space acting by convolution on differential forms. It provides a natural generalization of the important Riesz distributions acting on functions, where the corresponding operators are $(-\Delta )^{-\alpha /2}$, and we develop basic analogous properties with respect to meromorphic continuation, residues, Fourier transforms, and relations to conformal geometry and representations of the conformal group.

    更新日期:2020-01-15
  • Genus Six Curves, K3 Surfaces, and Stable Pairs
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-15
    Goluboff J.

    A general smooth curve of genus six lies on a quintic del Pezzo surface. Artebani and Kondō [ 4] construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this paper, we construct a smooth Deligne–Mumford stack ${\mathfrak{P}}_0$ parametrizing certain stable surface-curve pairs, which essentially resolves this map. Moreover, we give an explicit description of pairs in ${\mathfrak{P}}_0$ containing special curves.

    更新日期:2020-01-15
  • The Jacobian, Reflection Arrangement and Discriminant for Reflection Hopf Algebras
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-15
    Kirkman E, Zhang J.

    We study finite-dimensional semisimple Hopf algebra actions on noetherian connected graded Artin–Schelter regular algebras and introduce definitions of the Jacobian, the reflection arrangement, and the discriminant in a noncommutative setting.

    更新日期:2020-01-15
  • Topology of the Nodal Set of Random Equivariant Spherical Harmonics on 𝕊3
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Jung J, Zelditch S.

    We show that real and imaginary parts of equivariant spherical harmonics on ${{\mathbb{S}}}^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left (\frac{N^2 - m^2}{2} + N\right ) $. Hence, if $\frac{m}{N}= c $ for fixed $0 < c < 1$, then the genus has order $N^3$.

    更新日期:2020-01-15
  • Mass Transference Principle: From Balls to Arbitrary Shapes
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Koivusalo H, Rams M.

    The mass transference principle, proved by Beresnevich and Velani in 2006, is a strong result that gives lower bounds for the Hausdorff dimension of limsup sets of balls. We present a version for limsup sets of open sets of arbitrary shape.

    更新日期:2020-01-13
  • On the Modular Completion of Certain Generating Functions
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Bringmann K, Ehlen S, Schwagenscheidt M.

    We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights $\frac 32$ and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight $\frac 32$, which prominently appear in work of Zagier [ 27] on traces of singular moduli.

    更新日期:2020-01-13
  • Toeplitz Operators on the Symmetrized Bidisc
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Bhattacharyya T, Das B, Sau H.

    The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results.(1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules.(2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators.(3) Theorem III gives several characterizations of an analytic Toeplitz operator.(4) Theorem IV characterizes asymptotic Toeplitz operators.(5) Theorem V is a commutant lifting theorem.(6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

    更新日期:2020-01-13
  • The Weil Algebra of a Double Lie Algebroid
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Meinrenken E, Pike J.

    Given a double vector bundle $D\to M$, we define a bigraded bundle of algebras $W(D)\to M$ called the “Weil algebra bundle”. The space ${\mathcal{W}}(D)$ of sections of this algebra bundle ”realizes” the algebra of functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebra bundles of $D$ and those of the double vector bundles $D^{\prime},\ D^{\prime\prime}$ obtained from $D$ by duality operations. We show that ${\mathcal{V}\mathcal{B}}$-algebroid structures on $D$ are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the 3rd. Furthermore, Mackenzie’s definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a ”classical” version of Voronov’s result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that ${\mathcal{W}}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad–Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multi-vector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras.

    更新日期:2020-01-13
  • Self-Contracted Curves in Spaces With Weak Lower Curvature Bound
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Lebedeva N, Ohta S, Zolotov V.

    We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author’s previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. Here we apply this strategy to spaces with lower curvature bound.

    更新日期:2020-01-13
  • Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Bringmann B.

    We study the derivative nonlinear wave equation $- \partial _{tt} u + \Delta u = |\nabla u|^2$ on $\mathbb{R}^{1 +3}$. The deterministic theory is determined by the Lorentz-critical regularity $s_L = 2$, and both local well-posedness above $s_L$ as well as ill-posedness below $s_L$ are known. In this paper, we show the local existence of solutions for randomized initial data at the super-critical regularities $s\geqslant 1.984$. In comparison to the previous literature in random dispersive equations, the main difficulty is the absence of a (probabilistic) nonlinear smoothing effect. To overcome this, we introduce an adaptive and iterative decomposition of approximate solutions into rough and smooth components. In addition, our argument relies on refined Strichartz estimates, a paraproduct decomposition, and the truncation method of de Bouard and Debussche.

    更新日期:2020-01-13
  • On Moduli Spaces of Positive Scalar Curvature Metrics on Highly Connected Manifolds
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Wiemeler M.

    Let $M$ be a simply connected spin manifold of dimension at least six, which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar curvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds that includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.

    更新日期:2020-01-13
  • Spectral Properties of the Neumann–Poincaré Operator in 3D Elasticity
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Miyanishi Y, Rozenblum G.

    We consider the adjoint double layer potential (Neumann–Poincaré (NP)) operator appearing in 3-dimensional elasticity. We show that the recent result about the polynomial compactness of this operator for the case of a homogeneous media follows without additional calculations from previous considerations by Agranovich et al., based upon pseudodifferential operators. Further on, we define the NP operator for the case of a nonhomogeneous isotropic media and show that its properties depend crucially on the character of nonhomogeneity. If the Lamé parameters are constant along the boundary, the NP operator is still polynomially compact. On the other hand, if these parameters are not constant, two or more intervals of continuous spectrum may appear, so the NP operator ceases to be polynomially compact. However, after a certain modification, it becomes polynomially compact again. Finally, we evaluate the rate of convergence of discrete eigenvalues of the NP operator to the tips of the essential spectrum.

    更新日期:2020-01-13
  • On Perfect Matchings in k-Complexes
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-11
    Han J.

    Keevash and Mycroft [ 19] developed a geometric theory for hypergraph matchings and characterized the dense simplicial complexes that contain a perfect matching. Their proof uses the hypergraph regularity method and the hypergraph blow-up lemma recently developed by Keevash. In this note we give a new proof of their results, which avoids these complex tools. In particular, our proof uses the lattice-based absorbing method developed by the author and a recent probabilistic argument of Kohayakawa, Person, and the author.

    更新日期:2020-01-13
  • The Cone Structure Theorem
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-13
    Batista E, Costa J, Nuño-Ballesteros J.

    We consider the topological classification of finitely determined map germs $f:(\mathbb{R}^n,0)\to (\mathbb{R}^p,0)$ with $f^{-1}(0)\neq \{0\}$. Associated with $f$ we have a link diagram, which is well defined up to topological equivalence. We prove that $f$ is topologically $\mathcal{A}$-equivalent to the generalized cone of its link diagram.

    更新日期:2020-01-13
  • Zeros of Holomorphic One-Forms and Topology of Kähler Manifolds
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-08
    Schreieder S.

    A conjecture of Kotschick predicts that a compact Kähler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao [ 10], we use our approach to prove Kotschick’s conjecture for smooth projective three-folds.

    更新日期:2020-01-09
  • Uniqueness of Real Lagrangians up to Cobordism
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-08
    Kim J.

    We prove that a real Lagrangian submanifold in a closed symplectic manifold is unique up to cobordism. We then discuss the classification of real Lagrangians in ${\mathbb{C}} P^2$ and $S^2\times S^2$. In particular, we show that a real Lagrangian in ${\mathbb{C}} P^2$ is unique up to Hamiltonian isotopy and that a real Lagrangian in $S^2\times S^2$ is either Hamiltonian isotopic to the antidiagonal sphere or Lagrangian isotopic to the Clifford torus.

    更新日期:2020-01-09
  • Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-08
    Oeh D.

    Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak{g}}} = {{\mathfrak{h}}} \oplus{{\mathfrak{q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal{H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak{q}}},{{\mathfrak{q}}}] \oplus i{{\mathfrak{q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak{h}}} \oplus i{{\mathfrak{q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

    更新日期:2020-01-09
  • Homotopy Invariance of Convolution Products
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-08
    Sagave S, Schwede S.

    The purpose of this paper is to show that various convolution products are fully homotopical, meaning that they preserve weak equivalences in both variables without any cofibrancy hypothesis. We establish this property for diagrams of simplicial sets indexed by the category of finite sets and injections and for tame $M$-simplicial sets, with $M$ the monoid of injective self-maps of the positive natural numbers. We also show that a certain convolution product studied by Nikolaus and the 1st author is fully homotopical. This implies that every presentably symmetric monoidal $\infty $-category can be represented by a symmetric monoidal model category with a fully homotopical monoidal product.

    更新日期:2020-01-09
  • Eigenvector Convergence for Minors of Unitarily Invariant Infinite Random Matrices
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-09
    Najnudel J.

    In [10], Pickrell fully characterizes the unitarily invariant probability measures on infinite Hermitian matrices. An alternative proof of this classification is given by Olshanski and Vershik in [9], and in [3] Borodin and Olshanski deduce from this proof that under any of these invariant measures, the extreme eigenvalues of the minors, divided by the dimension, converge almost surely. In this paper, we prove that one also has a weak convergence for the eigenvectors, in a sense that is made precise. After mapping Hermitian to unitary matrices via the Cayley transform, our result extends a convergence proven in our paper with Maples and Nikeghbali [6], for which a coupling of the circular unitary ensemble of all dimensions is considered.

    更新日期:2020-01-09
  • Exponential-Square Integrability, Weighted Inequalities for the Square Functions Associated to Operators, and Applications
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2020-01-08
    Chen P, Duong X, Wu L, et al.

    Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and Hölder continuity in $x$, but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\mathbb R^n}$. We then apply this result to obtain: (1) estimates of the norm on $L^p$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (2) weighted norm inequalities for the square functions; and (3) eigenvalue estimates for Schrödinger operators on ${\mathbb R}^n$ or Lipschitz domains of ${\mathbb R}^n$.

    更新日期:2020-01-08
  • On the Number of Products Which Form Perfect Powers and Discriminants of Multiquadratic Extensions
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2019-12-30
    de la Bretèche R, Kurlberg P, Shparlinski I.

    We study some counting questions concerning products of positive integers $u_1, \ldots , u_n$, which form a nonzero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields that are multiquadratic extensions of ${\mathbb{Q}}$ and improve and generalize a result of N. Rome (2017).

    更新日期:2020-01-04
  • The Complex Orthogonal Gelfand–Zeitlin System
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2019-12-30
    Colarusso M, Evens S.

    In this paper, we use the theory of algebraic groups to prove a number of new and fundamental results about the orthogonal Gelfand–Zeitlin system. We show that the moment map (orthogonal Kostant–Wallach map) is surjective and simplify criteria of Kostant and Wallach for an element to be strongly regular. We further prove the integrability of the orthogonal Gelfand–Zeitlin system on regular adjoint orbits and describe the generic flows of the integrable system. We also study the nilfibre of the moment map and show that in contrast to the general linear case it contains no strongly regular elements. This extends results of Kostant, Wallach, and Colarusso from the general linear case to the orthogonal case.

    更新日期:2020-01-04
  • Ulrich Bundles on Intersections of Two 4-Dimensional Quadrics
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2019-12-31
    Cho Y, Kim Y, Lee K.

    In this paper, we investigate the moduli space of Ulrich bundles on a smooth complete intersection of two $4$-dimensional quadrics in $\mathbb P^5$. The main ingredient is the semiorthogonal decomposition by Bondal–Orlov, combined with the categorical methods pioneered by Kuznetsov and Lahoz–Macrì–Stellari. Using these methods, we prove that any smooth intersection of two 4-dimensional quadrics in $\mathbb P^5$ carries an Ulrich bundle of rank $r$ for every $r \ge 2$. Moreover, we provide a description of the moduli space of stable Ulrich bundles.

    更新日期:2020-01-04
  • On the Second Boundary Value Problem for a Class of Fully Nonlinear Flows I
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2017-11-13
    Huang R, Ye Y.

    In this article, a class of fully nonlinear flows with nonlinear Neumann type boundary condition is considered. This problem was solved partly by the first author under the assumption that the flow is the parabolic type special Lagrangian equation in $\mathbb{R}^{2n}$. We show that the convexity is preserved for solutions of the fully nonlinear parabolic equations and prove the long time existence and convergence of the flow. In particular, we can prescribe the second boundary value problems for a family of special Lagrangian graphs in Euclidean and pseudo-Euclidean space.

    更新日期:2020-01-04
  • Tilting and Cluster Tilting for Preprojective Algebras and Coxeter Groups
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2017-11-15
    Kimura Y.

    We study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\boldsymbol{w})$ of this category associated with each reduced expression $\boldsymbol{w}$ of $w$ and give a sufficient condition on $\boldsymbol{w}$ such that $M(\boldsymbol{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\boldsymbol{w})$. Moreover, we compare it with a triangle equivalence given by Amiot–Reiten–Todorov for a cluster category.

    更新日期:2020-01-04
  • Infinitely Many Solutions for Centro-affine Minkowski Problem
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2017-11-15
    Li Q.

    We study the multiplicity result for the centro-affine Minkowski problem. It is well-known that all ellipsoids with constant volume have the same centro-affine curvature. In this article, we construct a positive, Hölder continuous function $f\in C^\alpha (\mathbb S^n)$ such that there are infinitely many $C^{2,\alpha}$ hypersurfaces which are not affine-equivalent, but have the same centro-affine curvature $1/f$.

    更新日期:2020-01-04
  • A Relative Version of Gieseker’s Problem on Stratifications in Characteristic $\boldsymbol{p>0}$
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2017-11-18
    Esnault H, Srinivas V.

    We prove that the vanishing of the functoriality morphism for the étale fundamental group between smooth projective varieties over an algebraically closed field of characteristic $p>0$ forces the same property for the fundamental groups of stratifications.

    更新日期:2020-01-04
  • Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2017-11-23
    Chung H, Kim D, Kim M, et al.

    Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of $n$-th power residue symbols. This formalism leads to a precise arithmetic analogue of a “path-integral formula” for linking numbers.

    更新日期:2020-01-04
  • Non-Umbilical Quaternionic Contact Hypersurfaces in Hyper-Kähler Manifolds
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2017-11-23
    Ivanov S, Minchev I, Vassilev D.

    It is shown that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. In the non-compact case, it is proved that a seven-dimensional everywhere non-umbilical qc-hypersurface embedded in a hyper-Kähler manifold is qc-conformal to a qc-Einstein structure which is locally qc-equivalent to the 3-Sasakian sphere, the quaternionic Heisenberg group or the hyperboloid.

    更新日期:2020-01-04
  • Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2017-12-08
    Humphries P.

    We show that if a positive integer $q$ has $s(q)$ odd prime divisors $p$ for which $p^2$ divides $q$, then a positive proportion of the Laplacian eigenvalues of Maaß newforms of weight $0$, level $q$, and principal character occur with multiplicity at least $2^{s(q)}$. Consequently, the new part of the cuspidal spectrum of the Laplacian on $\Gamma_0(q) \backslash \mathbb{H}$ cannot be simple for any odd non-squarefree integer $q$. This generalises work of Strömberg who proved this for $q = 9$ by different methods.

    更新日期:2020-01-04
  • Perverse Schobers and Wall Crossing
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2017-12-09
    Donovan W.

    For a balanced wall crossing in geometric invariant theory (GIT), there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterize when this is an intersection cohomology complex of a local system on the punctured disk.

    更新日期:2020-01-04
  • Composite Knots Obtained by Twisting Torus Knots
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2017-12-09
    Lee S.

    A twisted torus knot is obtained from a torus knot by performing full twists on some adjacent strands of the torus knot. Morimoto constructed infinitely many twisted torus knots which are composite knots and he conjectured that his knots are all of composite twisted torus knots. We show that his conjecture is almost true by giving a complete list of such twisted torus knots.

    更新日期:2020-01-04
  • Modular finite W-algebras
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2018-01-16
    Goodwin S, Topley L.

    Let ${\mathbb{k}}$ be an algebraically closed field of characteristic p > 0 and let G be a connected reductive algebraic group over ${\mathbb{k}}$. Under some standard hypothesis on G, we give a direct approach to the finite W-algebra $U(\mathfrak{g},e)$ associated to a nilpotent element $e \in \mathfrak{g} = \textrm{Lie}\ G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the p-centre of $U(\mathfrak{g},e)$, which allows us to define reduced finite W-algebras $U_{\eta}(\mathfrak{g},e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabin’s equivalence of categories, generalizing recent work of the second author.

    更新日期:2020-01-04
  • Erratum: “Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers”
    Int. Math. Res. Notices (IF 1.452) Pub Date : 2019-05-31
    Chung H, Kim D, Kim M, et al.

    We wish to point out errors in the paper “Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers”, International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1–29. The main error concerns the symmetry of the “ramified case” of the height pairing, which relies on the vanishing of the Bockstein map in Proposition 3.5. The surjectivity claimed in the 1st line of the proof of Proposition 3.5 is incorrect. The specific results that are affected are Proposition 3.5; Lemmas 3.6, 3.7, 3.8, and 3.9; and Corollary 3.11. The definition of the $(S,n)$-height pairing following Lemma 3.9 is also invalid, since the symmetry of the pairing was required for it to be well defined. The results of Section 3 before Proposition 3.5 as well as those of the other Sections are unaffected.Proposition 3.10 is correct, but the proof is unclear and has some sign errors. So we include here a correction. As in the paper, let $I$ be an ideal such that $I^n$ is principal in ${\mathcal{O}}_{F,S}$. Write $I^n=(f^{-1})$. Then the Kummer cocycles $k_n(f)$ will be in $Z^1(U, {{\mathbb{Z}}/{n}{\mathbb{Z}}})$. For any $a\in F$, denote by $a_S$ its image in $\prod _{v\in S} F_v$. Thus, we get an element $$\begin{equation*}[f]_{S,n}:=[(k_n(f), k_{n^2}(f_S), 0)] \in Z^1(U, {{{\mathbb{Z}}}/{n}{{\mathbb{Z}}}} \times_S{\mathbb{Z}}/n^2{\mathbb{Z}}),\end{equation*}$$which is well defined in cohomology independently of the choice of roots used to define the Kummer cocycles. (We have also trivialized both $\mu _{n^2}$ and $\mu _n$.)

    更新日期:2020-01-04
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