Authors:Anton Alekseev; Nariya Kawazumi; Yusuke Kuno; Florian Naef Pages: 1 - 53 Abstract: Publication date: 21 February 2018 Source:Advances in Mathematics, Volume 326 Author(s): Anton Alekseev, Nariya Kawazumi, Yusuke Kuno, Florian Naef In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket { − , − } and Turaev cobracket δ on the K -span of homotopy classes of free loops on Σ. Applying an expansion θ : K π → K 〈 x 1 , … , x n 〉 yields an algebraic description of the operations { − , − } and δ in terms of non-commutative variables x 1 , … , x n . If Σ is a surface of genus g = 0 the lowest degree parts { − , − } − 1 and δ − 1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31]. It was conjectured by the second and the third authors that one can define an expansion θ such that { − , − } = { − , − } − 1 and δ = δ − 1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24], Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2]).

Authors:Xianzhe Dai; Guofang Wei; Zhenlei Zhang Pages: 1 - 33 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Xianzhe Dai, Guofang Wei, Zhenlei Zhang We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the L 2 Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.

Authors:Alexander Kiselev; Changhui Tan Pages: 34 - 55 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Alexander Kiselev, Changhui Tan In recent work of Luo and Hou [10], a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call “hyperbolic Boussinesq system”. This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot–Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of initial data.

Authors:Jacob Winding Pages: 56 - 86 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Jacob Winding We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, associated to good rational cones. We explain how good cones are related to collections of S L r ( Z ) -elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko, and the results about the usual multiple sine and elliptic gamma functions found by Narukawa.

Authors:O. Jenkinson; M. Pollicott Pages: 87 - 115 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): O. Jenkinson, M. Pollicott We prove that the algorithm of [19] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set E 2 (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25. The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension.

Authors:Moulay Tahar Benameur; Varghese Mathai Pages: 116 - 164 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Moulay Tahar Benameur, Varghese Mathai Given a constant magnetic field on Euclidean space R p determined by a skew-symmetric ( p × p ) matrix Θ, and a Z p -invariant probability measure μ on the disorder set Σ which is by hypothesis a Cantor set, where the action is assumed to be minimal, the corresponding Integrated Density of States of any self-adjoint operator affiliated to the twisted crossed product algebra C ( Σ ) ⋊ σ Z p , where σ is the multiplier on Z p associated to Θ, takes on values on spectral gaps in the magnetic gap-labelling group. The magnetic frequency group is defined as an explicit countable subgroup of R involving Pfaffians of Θ and its sub-matrices. We conjecture that the magnetic gap labelling group is a subgroup of the magnetic frequency group. We give evidence for the validity of our conjecture in 2D, 3D, the Jordan block diagonal case and the periodic case in all dimensions.

Authors:Jiayu Li; Chuanjing Zhang; Xi Zhang Pages: 165 - 214 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Jiayu Li, Chuanjing Zhang, Xi Zhang In this paper, we study the asymptotic behavior of the Hermitian–Yang–Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder–Narasimhan–Seshadri filtration, this answers a question by Bando and Siu.

Authors:Yinhe Peng; Liuzhen Wu Pages: 215 - 242 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Yinhe Peng, Liuzhen Wu We show that the Lindelöf property of a topological group is not inherited by its square, thus solving an old problem from [2]. Our construction introduces a new method to this area of mathematics and is of independent interest as it may have some further applications. Our method builds on ideas of [11] and techniques from [20] and give us also an optimal result about the partition properties on ω 1 .

Authors:Sheng Meng; De-Qi Zhang Pages: 243 - 273 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Sheng Meng, De-Qi Zhang An endomorphism f of a projective variety X is polarized (resp. quasi-polarized) if f ⁎ H ∼ q H (linear equivalence) for some ample (resp. nef and big) Cartier divisor H and integer q > 1 . First, we use cone analysis to show that a quasi-polarized endomorphism is always polarized, and the polarized property descends via any equivariant dominant rational map. Next, we show that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi-étale quotient of an abelian variety). Finally, we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one. As a consequence, the building blocks of polarized endomorphisms are those of Q-abelian varieties and those of Fano varieties of Picard number one. Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb ( X ) of X, and that the pullback of a power of f acts as a scalar multiplication on the Néron–Severi group of X (modulo torsion) when X is smooth and rationally connected. Partial answers about X being of Calabi–Yau type, or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.

Authors:Kei Yuen Chan Pages: 274 - 311 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Kei Yuen Chan This paper studies the Dirac cohomology of standard modules in the setting of graded Hecke algebras with geometric parameters. We prove that the Dirac cohomology of a standard module vanishes if and only if the module is not twisted-elliptic tempered. The proof makes use of two deep results. One is some structural information from the generalized Springer correspondence obtained by S. Kato and Lusztig. Another one is a computation of the Dirac cohomology of tempered modules by Barbasch–Ciubotaru–Trapa and Ciubotaru. We apply our result to compute the Dirac cohomology of ladder representations for type A n . For each of such representations with non-zero Dirac cohomology, we associate to a canonical Weyl group representation. We use the Dirac cohomology to conclude that such representations appear with multiplicity one.

Authors:Pedro Resende Pages: 312 - 374 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Pedro Resende We study Fell bundles on groupoids from the viewpoint of quantale theory. Given any saturated upper semicontinuous Fell bundle π : E → G on an étale groupoid G with G 0 locally compact Hausdorff, equipped with a suitable completion C*-algebra A of its convolution algebra, we obtain a map of involutive quantales p : Max A → Ω ( G ) , where Max A consists of the closed linear subspaces of A and Ω ( G ) is the topology of G. We study various properties of p which mimick, to various degrees, those of open maps of topological spaces. These are closely related to properties of G, π, and A, such as G being Hausdorff, principal, or topological principal, or π being a line bundle. Under suitable conditions, which include G being Hausdorff, but without requiring saturation of the Fell bundle, A is an algebra of sections of the bundle if and only if it is the reduced C*-algebra C r ⁎ ( G , E ) . We also prove that Max A is stably Gelfand. This implies the existence of a pseudogroup I B and of an étale groupoid B associated canonically to any sub-C*-algebra B ⊂ A . We study a correspondence between Fell bundles and sub-C*-algebras based on these constructions, and compare it to the construction of Weyl groupoids from Cartan subalgebras.

Authors:Michel Broué; Ruth Corran; Jean Michel Pages: 375 - 458 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Michel Broué, Ruth Corran, Jean Michel We generalize the definition and properties of root systems to complex reflection groups — roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation. In the case of spetsial reflection groups, we generalize as well the definition and properties of bad primes.

Authors:Sergii Myroshnychenko; Dmitry Ryabogin Pages: 482 - 504 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Sergii Myroshnychenko, Dmitry Ryabogin Let 2 ≤ k ≤ d − 1 and let P and Q be two convex polytopes in E d . Assume that their projections, P H , Q H , onto every k-dimensional subspace H, are congruent. In this paper we show that P and Q or P and −Q are translates of each other. We also prove an analogous result for sections by showing that P = Q or P = − Q , provided the polytopes contain the origin in their interior and their sections, P ∩ H , Q ∩ H , by every k-dimensional subspace H, are congruent.

Authors:Kathrin Bringmann; Chris Jennings-Shaffer; Karl Mahlburg Pages: 505 - 532 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Kathrin Bringmann, Chris Jennings-Shaffer, Karl Mahlburg We analyze the mock modular behavior of P ¯ ω ( q ) , a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta functions, one of which is ω ( q ) . We find that the modular completion of P ¯ ω ( q ) is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions. We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies in a relatively simpler space.

Authors:Avy Soffer; Minh-Binh Tran Pages: 533 - 607 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Avy Soffer, Minh-Binh Tran The system that describes the dynamics of a Bose–Einstein Condensate (BEC) and the thermal cloud at finite temperature consists of a nonlinear Schrodinger (NLS) and a quantum Boltzmann (QB) equations. In such a system of trapped Bose gases at finite temperature, the QB equation corresponds to the evolution of the density distribution function of the thermal cloud and the NLS is the equation of the condensate. The quantum Boltzmann collision operator in this temperature regime is the sum of two operators C 12 and C 22 , which describe collisions of the condensate and the non-condensate atoms and collisions between non-condensate atoms. Above the BEC critical temperature, the system is reduced to an equation containing only a collision operator similar to C 22 , which possesses a blow-up positive radial solution with respect to the L ∞ norm (cf. [29]). On the other hand, at the very low temperature regime (only a portion of the transition temperature T B E C ), the system can be simplified into an equation of C 12 , with a different (much higher order) transition probability, which has a unique global classical positive radial solution with weighted L 1 norm (cf. [3]). In our model, we first decouple the QB, which contains C 12 + C 22 , and the NLS equations, then show a global existence and uniqueness result for classical positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in [29], due to the presence of the BEC, the collision integrals are associated to sophisticated energy manifolds rather than spheres, since the particle energy is approximated by the Bogoliubov dispersion law. Moreover, the mass of the full system is not conserved while it is conserved for the case considered in [29]. A new theory is then supplied.

Authors:Wei He; Yeneng Sun Pages: 608 - 639 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Wei He, Yeneng Sun We prove some regularity properties (convexity, closedness, compactness and preservation of upper hemicontinuity) for distribution and regular conditional distribution of correspondences under the nowhere equivalence condition. We show the necessity of such a condition for any of these properties to hold. As an application, we demonstrate that the nowhere equivalence condition is satisfied on the underlying agent space if and only if pure-strategy Nash equilibria exist in general large games with any fixed uncountable compact action space.

Authors:Julien Barral; De-Jun Feng Pages: 640 - 718 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Julien Barral, De-Jun Feng Let μ be a planar Mandelbrot measure and π ⁎ μ its orthogonal projection on one of the principal axes. We study the thermodynamic and geometric properties of π ⁎ μ . We first show that π ⁎ μ is exact dimensional, with dim ( π ⁎ μ ) = min ( dim ( μ ) , dim ( ν ) ) , where ν is the Bernoulli product measure obtained as the expectation of π ⁎ μ . We also prove that π ⁎ μ is absolutely continuous with respect to ν if and only if dim ( μ ) > dim ( ν ) . Our results provides a new proof of Dekking–Grimmett–Falconer formula for the Hausdorff and box dimension of the topological support of π ⁎ μ , as well as a new variational interpretation. We obtain the free energy function τ π ⁎ μ of π ⁎ μ on a wide subinterval [ 0 , q c ) of R + . For q ∈ [ 0 , 1 ] , it is given by a variational formula which sometimes yields phase transitions of order larger than 1. For q > 1 , it is given by min ( τ ν , τ μ ) , which can exhibit first order phase transitions. This is in contrast with the analyticity of τ μ over [ 0 , q c ) . Also, we prove the validity of the multifractal formalism for π ⁎ μ at each α ∈ ( τ π ⁎ μ ′ ( q c − ) , τ π ⁎ μ ′ ( 0 + ) ] .

Authors:Alexandru D. Ionescu; Victor Lie Pages: 719 - 769 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Alexandru D. Ionescu, Victor Lie A basic model for describing plasma dynamics is given by the “one-fluid” Euler–Maxwell system, in which a compressible electron fluid interacts with its own self-consistent electromagnetic field. In this paper we prove long-term regularity of solutions of this system in 3 spatial dimensions, in the case of small initial data with nontrivial vorticity. Our main conclusion is that the time of existence of solutions depends only on the size of the vorticity of the initial data, as long as the initial data is sufficiently close to a constant stationary solution.

Authors:Sayan Bagchi Pages: 814 - 823 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Sayan Bagchi In this paper, we study the Heisenberg uniqueness pairs corresponding to a finite number of parallel lines Γ. We give a necessary condition and a sufficient condition for a subset Λ of R 2 so that ( Γ , Λ ) becomes a HUP.

Authors:Manuel Ritoré; Jesús Yepes Nicolás Pages: 824 - 863 Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Manuel Ritoré, Jesús Yepes Nicolás Given one metric measure space X satisfying a linear Brunn–Minkowski inequality, and a second one Y satisfying a Brunn–Minkowski inequality with exponent p ≥ − 1 , we prove that the product X × Y with the standard product distance and measure satisfies a Brunn–Minkowski inequality of order 1 / ( 1 + p − 1 ) under mild conditions on the measures and the assumption that the distances are strictly intrinsic. The same result holds when we consider restricted classes of sets. We also prove that a linear Brunn–Minkowski inequality is obtained in X × Y when Y satisfies a Prékopa–Leindler inequality. In particular, we show that the classical Brunn–Minkowski inequality holds for any pair of weakly unconditional sets in R n (i.e., those containing the projection of every point in the set onto every coordinate subspace) when we consider the standard distance and the product measure of n one-dimensional real measures with positively decreasing densities. This yields an improvement of the class of sets satisfying the Gaussian Brunn–Minkowski inequality. Furthermore, associated isoperimetric inequalities as well as recently obtained Brunn–Minkowski's inequalities are derived from our results.

Authors:Matthew Hedden; Thomas E. Mark Pages: 1 - 39 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Matthew Hedden, Thomas E. Mark We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of the fractional Dehn twist coefficient of the monodromy of any of its open book decompositions with connected binding. We prove this by showing that the rank of Floer homology gives bounds for the number of boundary parallel right or left Dehn twists necessary to add to a surface automorphism to guarantee that the associated contact manifold is tight or overtwisted, respectively. By examining branched double covers, we also show that the rank of the Khovanov homology of a link bounds the fractional Dehn twist coefficient of its odd-stranded braid representatives.

Authors:Elizaveta Rebrova; Roman Vershynin Pages: 40 - 83 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Elizaveta Rebrova, Roman Vershynin Can the behavior of a random matrix be improved by modifying a small fraction of its entries' Consider a random matrix A with i.i.d. entries. We show that the operator norm of A can be reduced to the optimal order O ( n ) by zeroing out a small submatrix of A if and only if the entries have zero mean and finite variance. Moreover, we obtain an almost optimal dependence between the size of the removed submatrix and the resulting operator norm. Our approach utilizes the cut norm and Grothendieck–Pietsch factorization for matrices, and it combines the methods developed recently by C. Le and R. Vershynin and by E. Rebrova and K. Tikhomirov.

Authors:Mauro Di Nasso; Lorenzo Luperi Baglini Pages: 84 - 117 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Mauro Di Nasso, Lorenzo Luperi Baglini We present general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of nonlinear equations. The goal is to contribute to an overall theory of Ramsey properties of (nonlinear) Diophantine equations that encompasses the known results in this area under a unified framework. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters β N , grounding on combinatorial properties of positive density sets and IP sets. Necessary conditions are proved by a new technique in nonstandard analysis, based on the use of the relation of u-equivalence for the hypernatural numbers N ⁎ .

Authors:Jinpeng An; Victor Beresnevich; Sanju Velani Pages: 148 - 202 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Jinpeng An, Victor Beresnevich, Sanju Velani For any i , j > 0 with i + j = 1 , let Bad ( i , j ) denote the set of points ( x , y ) ∈ R 2 such that max { ‖ q x ‖ 1 / i , ‖ q y ‖ 1 / j } > c / q for some positive constant c = c ( x , y ) and all q ∈ N . We show that Bad ( i , j ) ∩ C is winning in the sense of Schmidt games for a large class of planar curves C , namely, everywhere non-degenerate planar curves and straight lines satisfying a natural Diophantine condition. This strengthens recent results solving a problem of Davenport from the sixties. In short, within the context of Davenport's problem, the winning statement is best possible. Furthermore, we obtain the inhomogeneous generalisations of the winning results for planar curves and lines and also show that the inhomogeneous form of Bad ( i , j ) is winning for two dimensional Schmidt games.

Authors:Gilles Bonnet; Pierre Calka; Matthias Reitzner Pages: 203 - 240 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Gilles Bonnet, Pierre Calka, Matthias Reitzner Let Z be the typical cell of a stationary Poisson hyperplane tessellation in R d . The distribution of the number of facets f ( Z ) of the typical cell is investigated. It is shown, that under a well-spread condition on the directional distribution, the quantity n 2 d − 1 P ( f ( Z ) = n ) n is bounded from above and from below. When f ( Z ) is large, the isoperimetric ratio of Z is bounded away from zero with high probability. These results rely on one hand on the Complementary Theorem which provides a precise decomposition of the distribution of Z and on the other hand on several geometric estimates related to the approximation of polytopes by polytopes with fewer facets. From the asymptotics of the distribution of f ( Z ) , tail estimates for the so-called Φ content of Z are derived as well as results on the conditional distribution of Z when its Φ content is large.

Authors:Ryo Fujita Pages: 241 - 266 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Ryo Fujita We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa–Suzuki functor (Arakawa–Suzuki, 1998) gives a fully faithful embedding of a block of the deformed BGG category of gl m into the module category of a suitable completion of degenerate affine Hecke algebra of G L n .

Authors:Toke Meier Carlsen Pages: 326 - 335 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Toke Meier Carlsen We characterise when the Leavitt path algebras over Z of two arbitrary countable directed graphs are ⁎-isomorphic by showing that two Leavitt path algebras over Z are ⁎-isomorphic if and only if the corresponding graph groupoids are isomorphic (if and only if there is a diagonal preserving isomorphism between the corresponding graph C ⁎ -algebras). We also prove that any ⁎-homomorphism between two Leavitt path algebras over Z maps the diagonal to the diagonal. Both results hold for more general subrings of C than just Z .

Authors:Xin Lu; Kang Zuo Pages: 336 - 354 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Xin Lu, Kang Zuo Let f : X → B be a locally non-trivial relatively minimal fibration of curves of genus g ≥ 2 . We obtain a lower bound of the slope λ ( f ) increasing with the gonality of the general fiber of f. In particular, we show that λ ( f ) ≥ 4 provided that f is non-hyperelliptic and g ≥ 16 .

Authors:Yonghwa Cho; Yongnam Lee Pages: 394 - 436 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Yonghwa Cho, Yongnam Lee Dolgachev surfaces are simply connected minimal elliptic surfaces with p g = q = 0 and of Kodaira dimension 1. These surfaces are constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via Q -Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper [25]. Also, some exceptional bundles on Dolgachev surfaces associated with Q -Gorenstein smoothing have been constructed based on the idea of Hacking [12]. In the case if Dolgachev surfaces were of type ( 2 , 3 ) , we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exists a nontrivial phantom category in the derived category.

Authors:Christoph Bandt Pages: 437 - 485 Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Christoph Bandt Bernoulli convolutions are certain measures on the unit interval depending on a parameter β between 1 and 2. In spite of their simple definition, they are not yet well understood. We study their two-dimensional density which exists by a theorem of Solomyak. To each Bernoulli convolution, there is an interval D called the overlap region, and a map which assigns two values to each point of D and one value to all other points of [ 0 , 1 ] . There are two types of finite orbits of these multivalued maps which correspond to zeros and potential singularities of the density, respectively. Orbits which do not meet D belong to an ordinary map called β-transformation and exist for all β > 1.6182 . They were studied by Erdös, Jóo, Komornik, Sidorov, de Vries and others as points with unique addresses, and by Jordan, Shmerkin and Solomyak as points with maximal local dimension. In the two-dimensional view, these orbits form address curves related to the Milnor–Thurston itineraries in one-dimensional dynamics. The curves depend smoothly on the parameter and represent quantiles of all corresponding Bernoulli convolutions. Finite orbits which intersect D have a network-like structure and can exist only at Perron parameters β. Their points are intersections of extended address curves, and can have finite or countable number of addresses, as found by Sidorov. For an uncountable number of parameters, the central point 1 2 has only two addresses. The intersection of periodic address curves can lead to singularities of the measures. We give examples which are not Pisot or Salem parameters.

Authors:Anssi Lahtinen; David Sprehn Pages: 1 - 37 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Anssi Lahtinen, David Sprehn The cohomology of the degree-n general linear group over a finite field of characteristic p, with coefficients also in characteristic p, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in n. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in n. We also construct nontrivial elements in the mod p homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain mysterious.

Authors:Ved Datar; Bin Guo; Jian Song; Xiaowei Wang Pages: 38 - 83 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Ved Datar, Bin Guo, Jian Song, Xiaowei Wang We give criterions for the existence of toric conical Kähler–Einstein and Kähler–Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry–Emery–Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical Kähler–Einstein metrics in the Gromov–Hausdorff topology.

Authors:Eric M. Friedlander Pages: 84 - 113 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Eric M. Friedlander We investigate rational G-modules M for a linear algebraic group G over an algebraically closed field k of characteristic p > 0 using filtrations by sub-coalgebras of the coordinate algebra k [ G ] of G. Even in the special case of the additive group G a , interesting structures and examples are revealed. The “degree” filtration we consider for unipotent algebraic groups leads to a “filtration by exponential degree” applicable to rational G modules for any linear algebraic group G of exponential type; this filtration is defined in terms of 1-parameter subgroups and is related to support varieties introduced recently by the author for such rational G-modules. We formulate in terms of this filtration a necessary and sufficient condition for rational injectivity for rational G-modules. Our investigation leads to the consideration of two new classes of rational G-modules: those that are “mock injective” and those that are “mock trivial”.

Authors:Martin Henk; Hannes Pollehn Pages: 114 - 141 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Martin Henk, Hannes Pollehn We prove tight subspace concentration inequalities for the dual curvature measures C ˜ q ( K , ⋅ ) of an n-dimensional origin-symmetric convex body for q ≥ n + 1 . This supplements former results obtained in the range q ≤ n .

Authors:Jaiung Jun Pages: 142 - 192 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Jaiung Jun We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is ‘multi-valued’. This paper largely consists of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme ( X , O X ) and prove that Γ ( X , O X ) ≃ R for any integral affine hyperring scheme X = Spec R .

Authors:Joachim Toft; Elmira Nabizadeh Pages: 193 - 225 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Joachim Toft, Elmira Nabizadeh We characterize periodic elements in Gevrey classes, Gelfand–Shilov distribution spaces and modulation spaces, in terms of estimates of involved Fourier coefficients, and by estimates of their short-time Fourier transforms. If q ∈ [ 1 , ∞ ) , ω is a suitable weight and ( E 0 E ) ′ is the set of all E-periodic elements, then we prove that the dual of M ( ω ) ∞ , q ∩ ( E 0 E ) ′ equals M ( 1 / ω ) ∞ , q ′ ∩ ( E 0 E ) ′ by suitable extensions of Bessel's identity.

Authors:Agnieszka Bodzenta; Alexey Bondal Pages: 226 - 278 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Agnieszka Bodzenta, Alexey Bondal Given a relatively projective birational morphism f : X → Y of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over Y) generators T X , f and S X , f in D b ( X ) . We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that D b ( X ) has such a filtration L where the lattice is the set of all birational decompositions f : X → g Z → h Y with smooth Z. The t-structures related to T X , f and S X , f are proved to be glued via filtrations left and right dual to L . We realise all such Z as the fine moduli spaces of simple quotients of O X in the heart of the t-structure for which S X , g is a relative projective generator over Y. This implements the program of interpreting relevant smooth contractions of X in terms of a suitable system of t-structures on D b ( X ) .

Authors:Kyungkeun Kang; Hideyuki Miura; Tai-Peng Tsai Pages: 326 - 366 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Kyungkeun Kang, Hideyuki Miura, Tai-Peng Tsai We derive refined estimates of the Green tensor of the stationary Stokes system in the half space. We then investigate the spatial asymptotics of stationary solutions of the incompressible Navier–Stokes equations in the half space. We also discuss the asymptotics of fast decaying flows in the whole space and exterior domains. In the Appendix we consider axisymmetric self-similar solutions.

Authors:Chris Hall; Doron Puder; William F. Sawin Pages: 367 - 410 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Chris Hall, Doron Puder, William F. Sawin Let G be a finite connected graph, and let ρ be the spectral radius of its universal cover. For example, if G is k-regular then ρ = 2 k − 1 . We show that for every r, there is an r-covering (a.k.a. an r-lift) of G where all the new eigenvalues are bounded from above by ρ. It follows that a bipartite Ramanujan graph has a Ramanujan r-covering for every r. This generalizes the r = 2 case due to Marcus, Spielman and Srivastava [26]. Every r-covering of G corresponds to a labeling of the edges of G by elements of the symmetric group S r . We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by [26], a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from [27]. Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the r-th matching polynomial of G to be the average matching polynomial of all r-coverings of G. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside [ − ρ , ρ ] .

Authors:Liang Kong; Hao Zheng Pages: 411 - 426 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Liang Kong, Hao Zheng We define the Drinfeld center of a monoidal category enriched over a braided monoidal category, and show that every modular tensor category can be realized in a canonical way as the Drinfeld center of a self-enriched monoidal category. We also give a generalization of this result for important applications in physics.

Authors:Sun Kim Abstract: Publication date: 5 February 2018 Source:Advances in Mathematics, Volume 325 Author(s): Sun Kim In 1980, D. M. Bressoud obtained an analytic generalization of the Rogers–Ramanujan–Gordon identities. He then tried to establish a combinatorial interpretation of his identity, which specializes to many well-known Rogers–Ramanujan type identities. He proved that a certain partition identity follows from his identity in a very restrictive case and conjectured that the partition identity holds true in general. In this paper, we prove Bressoud's conjecture for the general case by providing bijective proofs.

Authors:Adam Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Adam Osękowski The paper contains the study of sharp logarithmic estimates for positive dyadic shifts A given on probability spaces ( X , μ ) equipped with a tree-like structure. For any K > 0 we determine the smallest constant L = L ( K ) such that ∫ E A f d μ ≤ K ∫ R Ψ ( f ) d μ + L ( K ) ⋅ μ ( E ) , where Ψ ( t ) = ( t + 1 ) log ( t + 1 ) − t , E is an arbitrary measurable subset of X and f is an integrable function on X. The proof exploits Bellman function method: we extract the above estimate from the existence of an appropriate special function, enjoying certain size and concavity-type conditions. As a corollary, a dual exponential bound is obtained.

Authors:Kaledin Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): D. Kaledin We give a simple construction of the correspondence between square-zero extensions R ′ of a ring R by an R-bimodule M and second MacLane cohomology classes of R with coefficients in M (the simplest non-trivial case of the construction is R = M = Z / p , R ′ = Z / p 2 , thus the Bokstein homomorphism of the title). Following Jibladze and Pirashvili, we treat MacLane cohomology as cohomology of non-additive endofunctors of the category of projective R-modules. We explain how to describe liftings of R-modules and complexes of R-modules to R ′ in terms of data purely over R. We show that if R is commutative, then commutative square-zero extensions R ′ correspond to multiplicative extensions of endofunctors. We then explore in detail one particular multiplicative non-additive endofunctor constructed from cyclic powers of a module V over a commutative ring R annihilated by a prime p. In this case, R ′ is the second Witt vectors ring W 2 ( R ) considered as a square-zero extension of R by the Frobenius twist R ( 1 ) .

Authors:Daniel Nam; Trang Abstract: Publication date: 14 January 2018 Source:Advances in Mathematics, Volume 324 Author(s): Daniel Rodríguez, Nam Trang Under various appropriate hypotheses it is shown that there is only one determinacy model of the form L ( R , μ ) in which μ is a supercompact measure on P ω 1 ( R ) . In particular, this gives a positive answer to a question asked by W.H. Woodin in 1983.

Authors:Weiwei Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Weiwei Wu We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group G. Combining this with some generalizations of Seidel's algebraic frameworks from [35], we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call special isogenous tori. This extends the work of Abouzaid–Smith [2]. We also show that derived Fukaya categories are complete invariants of special isogenous tori.