Authors:Pascale Roesch; Xiaoguang Wang; Yongcheng Yin Pages: 1 - 59 Abstract: Publication date: 15 December 2017 Source:Advances in Mathematics, Volume 322 Author(s): Pascale Roesch, Xiaoguang Wang, Yongcheng Yin In this article, we study the topology and bifurcations of the moduli space M 3 of cubic Newton maps. It's a subspace of the moduli space of cubic rational maps, carrying the Riemann orbifold structure ( C ˆ , ( 2 , 3 , ∞ ) ) . We prove two results: • The boundary of the unique unbounded hyperbolic component is a Jordan arc and the boundaries of all other hyperbolic components are Jordan curves; • The Head's angle map is surjective and monotone. The fibers of this map are characterized completely. The first result is a moduli space analogue of the first author's dynamical regularity theorem [37]. The second result confirms a conjecture of Tan Lei.

Authors:Nitu Kitchloo; Vitaly Lorman; W. Stephen Wilson Pages: 60 - 82 Abstract: Publication date: 15 December 2017 Source:Advances in Mathematics, Volume 322 Author(s): Nitu Kitchloo, Vitaly Lorman, W. Stephen Wilson We take advantage of the internal algebraic structure of the Bockstein spectral sequence converging to E R ( n ) ⁎ ( p t ) to prove that for spaces Z that are part of Landweber flat real pairs with respect to E ( n ) (see Definition 2.9), the cohomology ring E R ( n ) ⁎ ( Z ) can be obtained from E ( n ) ⁎ ( Z ) by base change. In particular, our results allow us to compute the Real Johnson–Wilson cohomology of the Eilenberg–MacLane spaces Z = K ( Z , 2 m + 1 ) , K ( Z / 2 q , 2 m ) , K ( Z / 2 , m ) for any natural numbers m and q, as well as connective covers of BO: BO , BSO , BSpin and BO 〈 8 〉 .

Authors:Chongying Dong; Li Ren; Feng Xu Pages: 1 - 30 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Chongying Dong, Li Ren, Feng Xu Let V be a simple vertex operator algebra and G a finite automorphism group of V such that V G is regular. It is proved that every irreducible V G -module occurs in an irreducible g-twisted V-module for some g ∈ G . Moreover, the quantum dimensions of irreducible V G -modules are determined and a global dimension formula for V in terms of twisted modules is obtained. In particular, the orbifold theory conjecture is completely solved if G is solvable.

Authors:Huyi Hu; Yongxia Hua; Weisheng Wu Pages: 31 - 68 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Huyi Hu, Yongxia Hua, Weisheng Wu We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphisms, which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier–Young, though we do not use increasing partitions. The unstable topological entropy is defined equivalently via separated sets, spanning sets and open covers along a piece of unstable leaf, and it coincides with the unstable volume growth along unstable foliation. We also obtain some properties for the unstable metric entropy such as affineness, upper semi-continuity and a version of Shannon–McMillan–Breiman theorem.

Authors:Mikhail Belolipetsky; Benjamin Linowitz Pages: 69 - 79 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Mikhail Belolipetsky, Benjamin Linowitz Given a simple Lie group H of real rank at least 2 we show that the maximum cardinality of a set of isospectral non-isometric H-locally symmetric spaces of volume at most x grows at least as fast as x c log x / ( log log x ) 2 where c = c ( H ) is a positive constant. In contrast with the real rank 1 case, this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of Belolipetsky and Lubotzky [2]. Our proof uses Sunada's method, results of [2], and some deep results from number theory. We also discuss an open number-theoretical problem which would imply an even faster growth estimate.

Authors:Joseph Chuang; Hyohe Miyachi; Kai Meng Tan Pages: 80 - 159 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Joseph Chuang, Hyohe Miyachi, Kai Meng Tan We provide closed formulas for a large subset of the canonical basis vectors of the Fock space representation of U q ( sl ˆ e ) . These formulas arise from parallelotopes which assemble to form polytopal complexes. The subgraphs of the Ext 1 -quivers of v-Schur algebras at complex e-th roots of unity generated by simple modules corresponding to these canonical basis vectors are given by the 1-skeletons of the polytopal complexes.

Authors:Sylvie Corteel; Olya Mandelshtam; Lauren Williams Pages: 160 - 204 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Sylvie Corteel, Olya Mandelshtam, Lauren Williams In previous work [12–14], the first and third authors introduced staircase tableaux, which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process (ASEP) and for the moments of the Askey–Wilson weight function. The fact that the ASEP and Askey–Wilson moments are related at all is unexpected, and is due to [45]. The ASEP is a model of particles hopping on a one-dimensional lattice of N sites with open boundaries; particles can enter and exit at both left and right borders. It was introduced around 1970 [34,43] and is cited as a model for both traffic flow and translation in protein synthesis. Meanwhile, the Askey–Wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials. So from this previous work, we have the relationship ASEP −− staircase tableaux −− Askey–Wilson moments. The Askey–Wilson polynomials can be viewed as the one-variable case of the multivariate Koornwinder polynomials, which are also known as the Macdonald polynomials attached to the non-reduced affine root system ( C n ∨ , C n ). It is natural then to ask whether one can generalize the relationships among the ASEP, Askey–Wilson moments, and staircase tableaux, in such a way that Koornwinder moments replace Askey–Wilson moments. In [15], we made a precise link between Koornwinder moments and the two-species ASEP, a generalization of the ASEP which has two species of particles with different “weights.” In this article we introduce rhombic staircase tableaux, and show that we have the relationship 2-species ASEP −− rhombic staircase tableaux −− Koornwinder moments. In particular, we give formulas for the stationary distribution of the two-species ASEP and for Koornwinder moments, in terms of rhombic staircase tableaux.

Authors:Keomkyo Seo Pages: 205 - 220 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Keomkyo Seo In this paper, we derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L 2 norm of the mean curvature vector in a Riemannian manifold.

Authors:Jesse Leo Kass; Nicola Pagani Pages: 221 - 268 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Jesse Leo Kass, Nicola Pagani The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth pointed curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable pointed curves using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with this parameter. We use this result to analyze divisors on the moduli space of smooth pointed curves that have recently been studied by Grushevsky–Zakharov, Hain and Müller. Finally, we compute the pullback of the theta divisor studied in Alexeev's work on stable semiabelic varieties and in Caporaso's work on theta divisors of compactified Jacobians.

Authors:John H. Johnson; Florian Karl Richter Pages: 269 - 286 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): John H. Johnson, Florian Karl Richter Answering a question posed by Bergelson and Leibman in [6], we establish a nilpotent version of the Polynomial Hales–Jewett Theorem that contains the main theorem in [6] as a special case. Important to the formulation and the proof of our main theorem is the notion of a relative syndetic set (relative with respect to a closed non-empty subsets of β G ) [25]. As a corollary of our main theorem we prove an extension of the restricted van der Waerden Theorem to nilpotent groups, which involves nilprogressions.

Authors:Jacob Fox; László Miklós Lovász Pages: 287 - 297 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Jacob Fox, László Miklós Lovász Let p be a fixed prime. A triangle in F p n is an ordered triple ( x , y , z ) of points satisfying x + y + z = 0 . Let N = p n = F p n . Green proved an arithmetic triangle removal lemma which says that for every ϵ > 0 and prime p, there is a δ > 0 such that if X , Y , Z ⊂ F p n and the number of triangles in X × Y × Z is at most δ N 2 , then we can delete ϵN elements from X, Y, and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1 / δ can be taken to be an exponential tower of twos of height logarithmic in 1 / ϵ . We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in F p n . We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is an explicit number C p such that we may take δ = ( ϵ / 3 ) C p , and we must have δ ≤ ϵ C p − o ( 1 ) . In particular, C 2 = 1 + 1 / ( 5 / 3 − log 2 3 ) ≈ 13.239 , and C 3 = 1 + 1 / c 3 with c 3 = 1 − log b log 3 , b = a − 2 / 3 + a 1 / 3 + a 4 / 3 , and a = 33 − 1 8 , which gives C 3 ≈ 13.901 . The proof uses the essentially sharp bound on mu... PubDate: 2017-10-08T14:37:17Z DOI: 10.1016/j.aim.2017.09.037 Issue No:Vol. 321 (2017)

Authors:Josep Àlvarez Montaner; Craig Huneke; Luis Núñez-Betancourt Pages: 298 - 325 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Josep Àlvarez Montaner, Craig Huneke, Luis Núñez-Betancourt We study the structure of D-modules over a ring R which is a direct summand of a polynomial or a power series ring S with coefficients over a field. We relate properties of D-modules over R to D-modules over S. We show that the localization R f and the local cohomology module H I i ( R ) have finite length as D-modules over R. Furthermore, we show the existence of the Bernstein–Sato polynomial for elements in R. In positive characteristic, we use this relation between D-modules over R and S to show that the set of F-jumping numbers of an ideal I ⊆ R is contained in the set of F-jumping numbers of its extension in S. As a consequence, the F-jumping numbers of I in R form a discrete set of rational numbers. We also relate the Bernstein–Sato polynomial in R with the F-thresholds and the F-jumping numbers in R.

Authors:Gus Schrader; Alexander Shapiro Pages: 431 - 474 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Gus Schrader, Alexander Shapiro We construct an algebra embedding of the quantum group U q ( g ) into a central extension of the quantum coordinate ring O q [ G w 0 , w 0 / H ] of the reduced big double Bruhat cell in G. This embedding factors through the Heisenberg double H q of the quantum Borel subalgebra U ≥ 0 , which we relate to O q [ G ] via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck–Springer resolution studied in [10], and the quantum Beilinson–Bernstein theorem investigated in [2] and [36].

Authors:Ashutosh Kumar; Saharon Shelah Pages: 475 - 485 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Ashutosh Kumar, Saharon Shelah We show that for every partition of a set of reals into countable sets there is a transversal of the same outer measure.

Authors:Leandro Arosio; Filippo Bracci Pages: 486 - 512 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Leandro Arosio, Filippo Bracci We prove that a finite family of commuting holomorphic self-maps of the unit ball B q ⊂ C q admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball, and that such conjugacy satisfies a universal property. As an application we describe when a hyperbolic and a parabolic holomorphic self-map of B q can commute.

Authors:Xuhua He; Sian Nie Pages: 513 - 528 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Xuhua He, Sian Nie In this paper, we study the μ-ordinary locus of a Shimura variety with parahoric level structure. Under the axioms in [12], we show that μ-ordinary locus is a union of certain maximal Ekedahl–Kottwitz–Oort–Rapoport strata introduced in [12] and we give criteria on the density of the μ-ordinary locus.

Authors:Ariel Rapaport Pages: 529 - 546 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Ariel Rapaport We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than 1, such that there exist lines for which dimension conservation does not hold and the projection of the measure is singular. In fact, the set of such directions is residual and the typical slices of the measure, perpendicular to these directions, are discrete.

Authors:Luis Barreira; Davor Dragičević; Claudia Valls Pages: 547 - 591 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Luis Barreira, Davor Dragičević, Claudia Valls For a sequence of bounded linear operators acting on a Banach space, we consider the notion of nonuniform spectrum. This is defined in terms of the existence of nonuniform exponential dichotomies with an arbitrarily small nonuniform part and can be seen as a nonuniform version of the spectrum introduced by Sacker and Sell in the case of a single trajectory, although now in the infinite-dimensional setting. We give a complete characterization of all possible forms of the nonuniform spectrum for sequences of compact linear operators and, more generally, for sequences of bounded linear operators satisfying a certain asymptotic compactness. Moreover, we provide explicit examples of sequences of compact linear operators acting on the l 2 space of sequences of real numbers for all the possible forms of the nonuniform spectrum. As nontrivial applications, we show that the nonuniform spectrum of a Lyapunov regular sequence is the set of Lyapunov exponents and that the asymptotic behavior persists under sufficiently small nonlinear perturbations, in the sense that the lower and upper Lyapunov exponents of the perturbed dynamics belong to a connected component of the nonuniform spectrum. Finally, we obtain appropriate versions of the results for nonuniformly hyperbolic cocycles.

Authors:Marco Schlichting Pages: 1 - 81 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Marco Schlichting We improve homology stability ranges for elementary and special linear groups over rings with many units. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative local rings with infinite residue fields, we show that the obstruction to further stability is given by Milnor–Witt K-theory. As an application we construct Euler classes of projective modules with values in the cohomology of the Milnor–Witt K-theory sheaf. For d-dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the Euler class is necessary and sufficient for an oriented projective module P of rank d to split off a rank 1 free direct summand. Along the way we obtain a new presentation of Milnor–Witt K-theory and of symplectic K 2 simplifying the classical Matsumoto–Moore presentation.

Authors:Ivan Izmestiev; Steven Klee; Isabella Novik Pages: 82 - 114 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Ivan Izmestiev, Steven Klee, Isabella Novik We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly ( d + 1 ) -colored) triangulation of a combinatorial d-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial d-manifold can be connected by a sequence of cross-flips. Along the way we prove that for every m ≥ d + 2 and any closed combinatorial d-manifold M, two m-colored triangulations of M can be connected by a sequence of bistellar flips that preserve the vertex colorings.

Authors:Thomas Hudson; Takeshi Ikeda; Tomoo Matsumura; Hiroshi Naruse Pages: 115 - 156 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Thomas Hudson, Takeshi Ikeda, Tomoo Matsumura, Hiroshi Naruse We prove a determinantal formula that describes the K-theoretic degeneracy loci classes for Grassmann bundles. We further prove Pfaffian formulas for symplectic and odd orthogonal Grassmann bundles. The former generalizes Damon–Kempf–Laksov's determinantal formula, and the latter generalize Pragacz–Kazarian's formulas for the Chow ring.

Authors:Tomoyuki Arakawa; Anne Moreau Pages: 157 - 209 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Tomoyuki Arakawa, Anne Moreau We show that sheet closures appear as associated varieties of affine vertex algebras. Further, we give new examples of non-admissible affine vertex algebras whose associated variety is contained in the nilpotent cone. We also prove some conjectures from our previous paper and give new examples of lisse affine W-algebras.

Authors:Kenneth Ascher; Dori Bejleri Pages: 210 - 243 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Kenneth Ascher, Dori Bejleri We classify the log canonical models of elliptic surface pairs ( f : X → C , S + F A ) where f : X → C is an elliptic fibration, S is a section, and F A is a weighted sum of marked fibers. In particular, we show how the log canonical models depend on the choice of the weights. We describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes. In addition, we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers. This is the first step in constructing compactifications of moduli spaces of elliptic surfaces using the minimal model program.

Authors:Yann Pequignot Pages: 244 - 249 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Yann Pequignot The shift graph G S is defined on the space of infinite subsets of natural numbers by letting two sets be adjacent if one can be obtained from the other by removing its least element. We show that this graph is not a minimum among the graphs of the form G f defined on some Polish space X, where two distinct points are adjacent if one can be obtained from the other by a given Borel function f : X → X . This answers the primary outstanding question from [8].

Authors:David Glickenstein; Joseph Thomas Pages: 250 - 278 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): David Glickenstein, Joseph Thomas A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating the simplex has an isometric embedding into a constant curvature background geometry (Euclidean, hyperbolic, or spherical) with the specified edge lengths. Additional geometric structure leads to a notion of discrete conformal structure, generalizing circle packings and their generalizations as studied by Thurston and others. We analyze discrete conformal variations of piecewise constant curvature 2-manifolds, giving particular attention to the variation of angles. Formulas are derived for the derivatives of angles in each background geometry, which yield formulas for the derivatives of curvatures and to curvature functionals. Finally, we provide a complete classification of possible definitions of discrete conformal structures in each of the background geometries.

Authors:Liran Shaul Pages: 279 - 328 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Liran Shaul Let K be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings A over K , such that H 0 ( A ) is essentially of finite type over K , and A has finite flat dimension over K . We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for f ! in this setting. As application, we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings. Our results generalize and solve some recent conjectures of Yekutieli.

Authors:Roland Lötscher; Mark MacDonald Pages: 329 - 360 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Roland Lötscher, Mark MacDonald The notion of a ( G , N ) -slice of a G-variety was introduced by P.I. Katsylo in the early 80's for an algebraically closed base field of characteristic 0. Slices (also known under the name of relative sections) have ever since provided a fundamental tool in invariant theory, allowing reduction of rational or regular invariants of an algebraic group G to invariants of a “simpler” group. We refine this notion for a G-scheme over an arbitrary field, and use it to get reduction of structure group results for G-torsors. Namely we show that any ( G , N ) -slice of a versal G-scheme gives surjective maps H 1 ( L , N ) → H 1 ( L , G ) in fppf-cohomology for infinite fields L containing F. We show that every stabilizer in general position H for a geometrically irreducible G-variety V gives rise to a ( G , N G ( H ) ) -slice in our sense. The combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type E 7 .

Authors:Jungkai Chen; Zhi Jiang; Zhiyu Tian Pages: 361 - 390 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Jungkai Chen, Zhi Jiang, Zhiyu Tian We study the Albanese image of a compact Kähler manifold whose geometric genus is one. In particular, we prove that if the Albanese map is not surjective, then the manifold maps surjectively onto an ample divisor in some abelian variety, and in many cases the ample divisor is a theta divisor. With a further natural assumption on the topology of the manifold, we prove that the manifold is an algebraic fiber space over a genus two curve. Finally we apply these results to study the geometry of a compact Kähler manifold which has the same Hodge numbers as those of an abelian variety of the same dimension.

Authors:Jean-Marie Lescure; Stéphane Vassout Pages: 391 - 450 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Jean-Marie Lescure, Stéphane Vassout As announced in [36], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study Lagrangian conic submanifolds of the symplectic groupoid T ⁎ G . This includes their product, transposition and inversion. We also study the relationship between these Lagrangian submanifolds and the equivariant families of Lagrangian submanifolds of T ⁎ G x × T ⁎ G x parametrized by the units x ∈ G ( 0 ) of G. This allows us to select a subclass of Lagrangian distributions on any Lie groupoid G that deserve the name of Fourier integral G-operators (G-FIOs). By construction, the class of G-FIOs contains the class of equivariant families of ordinary Fourier integral operators on the manifolds G x , x ∈ G ( 0 ) . We then develop for G-FIOs the first stages of the calculus in the spirit of Hormander's work. Finally, we illustrate this calculus in the case of manifolds with boundary.

Authors:Qingchun Ji; Ke Zhu Pages: 451 - 474 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Qingchun Ji, Ke Zhu This paper develops a weighted L 2 -method for the (half) Dirac equation. For Dirac bundles over closed Riemann surfaces, we give a sufficient condition for the solvability of the (half) Dirac equation in terms of a curvature integral. Applying this to the Dolbeault–Dirac operator, we establish an automatic transversality criteria for holomorphic curves in Kähler manifolds. On compact Riemannian manifolds, we give a new perspective on some well-known results about the first eigenvalue of the Dirac operator, and improve the estimates when the Dirac bundle has a Z 2 -grading. On Riemannian manifolds with cylindrical ends, we obtain solvability in the L 2 -spaces with suitable exponential weights while allowing mild negativity of the curvature.

Authors:Motoo Tange Pages: 475 - 499 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Motoo Tange We consider a homology sphere M n ( K 1 , K 2 ) presented by two knots K 1 , K 2 with linking number 1 and framing ( 0 , n ) . We call the manifold Matsumoto's manifold. We show that M n ( T 2 , 3 , K 2 ) never bounds any contractible 4-manifold if n < 2 τ ( K 2 ) holds. We also give a formula of Ozsváth–Szabó's τ-invariant as the total sum of the Euler numbers of the reduced filtration. We compute the δ-invariants of the twisted Whitehead doubles of torus knots and correction terms of the branched covers of the Whitehead doubles. By using Owens and Strle's obstruction we show that the 12-twisted Whitehead double of the ( 2 , 7 ) -torus knot and the 20-twisted Whitehead double of the ( 3 , 7 ) -torus knot are not slice but the double branched covers bound rational homology 4-balls. These are new examples having a gap between what a knot is slice and what a double branched cover bounds a rational homology 4-ball.

Authors:Scott Mullane Pages: 500 - 519 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Scott Mullane For every g ≥ 2 and n ≥ g + 1 we exhibit infinitely many extremal effective divisors in M ‾ g , n coming from the strata of abelian differentials.

Authors:Zakhar Kabluchko; Vladislav Vysotsky; Dmitry Zaporozhets Pages: 595 - 629 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Zakhar Kabluchko, Vladislav Vysotsky, Dmitry Zaporozhets Consider a sequence of partial sums S i = ξ 1 + … + ξ i , 1 ≤ i ≤ n , starting at S 0 = 0 , whose increments ξ 1 , … , ξ n are random vectors in R d , d ≤ n . We are interested in the properties of the convex hull C n : = Conv ( S 0 , S 1 , … , S n ) . Assuming that the tuple ( ξ 1 , … , ξ n ) is exchangeable and a certain general position condition holds, we prove that the expected number of k-dimensional faces of C n is given by the formula E [ f k ( C n ) ] = 2 ⋅ k ! n ! ∑ l = 0 ∞ [ n + 1 d − 2 l ] { d − 2 l k + 1 } , for all 0 ≤ k ≤ d − 1 , where [ n m ] and { n m } are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0 ≤ i 1 < … < i k + 1 ≤ n , the points S i 1 , … , S i k + 1 form a k-dimensional face of Conv ( S 0 , S 1 , … , S n ) . This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments PubDate: 2017-09-17T11:43:03Z DOI: 10.1016/j.aim.2017.09.002 Issue No:Vol. 320 (2017)

Authors:Konstantin M. Dyakonov Pages: 630 - 651 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Konstantin M. Dyakonov Let θ be an inner function on the unit disk, and let K θ p : = H p ∩ θ H 0 p ‾ be the associated star-invariant subspace of the Hardy space H p , with p ≥ 1 . While a nontrivial function f ∈ K θ p is never divisible by θ, it may have a factor h which is ‘‘not too different” from θ in the sense that the ratio h / θ (or just the anti-analytic part thereof) is smooth on the circle. In this case, f is shown to have additional integrability and/or smoothness properties, much in the spirit of the Hardy–Littlewood–Sobolev embedding theorem. The appropriate norm estimates are established, and their sharpness is discussed.

Authors:Juan Dávila; Manuel del Pino; Xuan Hien Nguyen Pages: 674 - 729 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Juan Dávila, Manuel del Pino, Xuan Hien Nguyen Finite topology self-translating surfaces for the mean curvature flow constitute a key element in the analysis of Type II singularities from a compact surface because they arise as limits after suitable blow-up scalings around the singularity. We prove the existence of such a surface M ⊂ R 3 that is orientable, embedded, complete, and with three ends asymptotically paraboloidal. The fact that M is self-translating means that the moving surface S ( t ) = M + t e z evolves by mean curvature flow, or equivalently, that M satisfies the equation H M = ν ⋅ e z where H M denotes mean curvature, ν is a choice of unit normal to M, and e z is a unit vector along the z-axis. This surface M is in correspondence with the classical three-end Costa–Hoffman–Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded complete minimal surfaces with finite total curvature.

Authors:Graeme Wilkin Pages: 730 - 794 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Graeme Wilkin In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical point. Therefore we can classify all of the isomorphism classes which contain an initial condition that flows up to a given critical point. As an application, we then show that Nakajima's Hecke correspondence for quivers has a Morse-theoretic interpretation as pairs of critical points connected by flow lines for the norm-square of a moment map. The results are valid in the general setting of finite quivers with relations.

Authors:J.P. Pridham Pages: 795 - 826 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): J.P. Pridham We show that real Deligne cohomology of a complex manifold X arises locally as a topological vector space completion of the analytic Lie groupoid of holomorphic vector bundles. Thus Beilinson's regulator arises naturally as a comparison map between K-theory groups of different types.

Authors:Alexander Koldobsky; Alexander S. Merkurjev; Vladyslav Yaskin Pages: 876 - 886 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Alexander Koldobsky, Alexander S. Merkurjev, Vladyslav Yaskin An infinitely smooth convex body in R n is called polynomially integrable of degree N if its parallel section functions are polynomials of degree N. We prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids, if N ≥ n − 1 . This is in contrast with the case of even dimensions and the case of odd dimensions with N < n − 1 , where such bodies do not exist, as it was recently shown by Agranovsky.

Authors:Gunter Malle Pages: 887 - 903 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Gunter Malle We investigate the action of outer automorphisms of finite groups of Lie type on their irreducible characters. We obtain a definite result for cuspidal characters. As an application we verify the inductive McKay condition for some further infinite families of simple groups at certain primes.

Authors:Raphaël Danchin; Piotr Bogusław Mucha Pages: 904 - 925 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Raphaël Danchin, Piotr Bogusław Mucha Here we prove the existence of global in time regular solutions to the two-dimensional compressible Navier–Stokes equations supplemented with arbitrary large initial velocity v 0 and almost constant density ϱ 0 , for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible Navier–Stokes equations supplemented with the divergence-free projection of v 0 , is global. The systems are examined in R d with d ≥ 2 , in the critical B ˙ 2 , 1 s Besov spaces framework.

Authors:Changfeng Gui; Yong Liu; Juncheng Wei Pages: 926 - 992 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Changfeng Gui, Yong Liu, Juncheng Wei In this paper, we study axially symmetric solutions of Allen–Cahn equation in the three dimensional Euclidean space. Using a sophisticated continuation method, we show the existence of a complete family of axially symmetric solutions with a range of logarithmic growth rates, which may be regarded as the analogue of the family of catenoids and hence called two-end solutions. Nonexistence of two-end solution with a small growth rate is also shown, which differs from the theory of minimal surfaces.

Abstract: Publication date: 15 December 2017 Source:Advances in Mathematics, Volume 322 Author(s): Alice Rizzardo Given a Fourier–Mukai functor Φ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to Φ, and also give explicit formulas for them. These formulas are simple and natural, and recover the usual formulas when the Fourier–Mukai kernel is a perfect complex. This extends previous work of [1,12,13] and has applications to the twist autoequivalences of [9].

Abstract: Publication date: 15 December 2017 Source:Advances in Mathematics, Volume 322 Author(s): Alex Massarenti Let X [ n ] be the Fulton–MacPherson compactification of the configuration space of n ordered points on a smooth projective variety X. We prove that if either n ≠ 2 or dim ( X ) ≥ 2 , then the connected component of the identity of Aut ( X [ n ] ) is isomorphic to the connected component of the identity of Aut ( X ) . When X = C is a curve of genus g ( C ) ≠ 1 we classify the dominant morphisms C [ n ] → C [ r ] , and thanks to this we manage to compute the whole automorphism group of C [ n ] , namely Aut ( C [ n ] ) ≅ S n × Aut ( C ) for any n ≠ 2 , while Aut ( C [ 2 ] ) ≅ S 2 ⋉ ( Aut ( C ) × Aut ( C ) ) . Furthermore, we extend these results on the automorphisms to the case where X = C 1 × . . . × C r is a product of curves of genus g ( C i ) ≥ 2 . Finally, using the techniques developed to deal with Fulton–MacPherson spaces, we study the automorphism groups of some Kontsevich moduli spaces M ‾ 0 , n ( P N , d ) .

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Jean Bourgain, Ciprian Demeter, Shaoming Guo We prove a sharp decoupling for a certain two dimensional surface in R 9 . As an application, we obtain the full range of expected estimates for the cubic Parsell–Vinogradov system in two dimensions.

Authors:Geoffroy Horel Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Geoffroy Horel In this paper, we prove that the group of homotopy automorphisms of the profinite completion of the operad of little 2-disks is isomorphic to the profinite Grothendieck–Teichmüller group. In particular, the absolute Galois group of Q acts faithfully on the profinite completion of E 2 in the homotopy category of profinite weak operads.

Authors:Fan Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Fan Ge Assume the Riemann Hypothesis. We establish a local structure theorem for zeros of the Riemann zeta-function ζ ( s ) and its derivative ζ ′ ( s ) . As an application, we prove a stronger form of half of a conjecture of Radziwiłł [18] concerning the global statistics of these zeros. Roughly speaking, we show that on the Riemann Hypothesis, if there occurs a small gap between consecutive zeta zeros, then there is exactly one zero of ζ ′ ( s ) lying not only very close to the critical line but also between that pair of zeta zeros. This refines a result of Zhang [22]. Some related results are also shown. For example, we prove a weak form of a conjecture of Soundararajan, and suggest a repulsion phenomena for zeros of ζ ′ ( s ) .

Authors:Marcin Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Marcin Chałupnik We introduce the notion of an affine strict polynomial functor. We show how this concept helps to understand homological behavior of the operation of Frobenius twist in the category of strict polynomial functors over a field of positive characteristic. We also point out for an analogy between our category and the category of representations of the group of algebraic loops on G L n .