Authors:Thomas Church; Jeremy Miller; Rohit Nagpal; Jens Reinhold Pages: 1 - 40 Abstract: Publication date: 31 July 2018 Source:Advances in Mathematics, Volume 333 Author(s): Thomas Church, Jeremy Miller, Rohit Nagpal, Jens Reinhold We prove two general results concerning spectral sequences of FI-modules. These results can be used to significantly improve stable ranges in a large portion of the stability theorems for FI-modules currently in the literature. We work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range. Previously, the best stable ranges known in these examples were exponential. Up to an additive constant, our work on congruence subgroups verifies a conjecture of Djament.
Authors:Ulrich Bunke; Thomas Nikolaus; Georg Tamme Pages: 41 - 86 Abstract: Publication date: 31 July 2018 Source:Advances in Mathematics, Volume 333 Author(s): Ulrich Bunke, Thomas Nikolaus, Georg Tamme We prove that the Beilinson regulator, which is a map from K-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E ∞ -ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R . The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K-theory spectrum is the source. To prove this result we compute the space of maps from the motivic K-theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E ∞ -refinement and prove a uniqueness result for these refinements.
Authors:Alexandra Kjuchukova Pages: 1 - 33 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Alexandra Kjuchukova Given a closed oriented PL four-manifold X and a closed surface B embedded in X with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of X branched along B. For X simply-connected, we deduce a necessary condition on the intersection form of a simply-connected irregular dihedral branched cover of ( X , B ) . When the singularities on B are two-bridge slice, we prove that the necessary condition on the intersection form of the cover is sharp. For X a simply-connected PL four-manifold with non-zero second Betti number, we construct infinite families of simply-connected PL manifolds which are irregular dihedral branched coverings of X. Given two four-manifolds X and Y whose intersection forms are odd, we obtain a necessary and sufficient condition for Y to be homeomorphic to an irregular dihedral p-fold cover of X, branched over a surface with a two-bridge slice singularity.
Authors:José Burillo; Yash Lodha; Lawrence Reeves Pages: 34 - 56 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): José Burillo, Yash Lodha, Lawrence Reeves In [7] Monod introduced examples of groups of piecewise projective homeomorphisms which are not amenable and which do not contain free subgroups, and in [6] Lodha and Moore introduced examples of finitely presented groups with the same property. In this article we examine the normal subgroup structure of these groups. Two important cases of our results are the groups H and G 0 . We show that the group H of piecewise projective homeomorphisms of R has the property that H ″ is simple and that every proper quotient of H is metabelian. We establish simplicity of the commutator subgroup of the group G 0 , which admits a presentation with 3 generators and 9 relations. Further, we show that every proper quotient of G 0 is abelian. It follows that the normal subgroups of these groups are in bijective correspondence with those of the abelian (or metabelian) quotient.
Authors:Ira M. Gessel; Yan Zhuang Pages: 85 - 141 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Ira M. Gessel, Yan Zhuang Since the early work of Richard Stanley, it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations. We formalize this notion of a shuffle-compatible permutation statistic and introduce the shuffle algebra of a shuffle-compatible permutation statistic, which encodes the distribution of the statistic over shuffles of permutations. This paper develops a theory of shuffle-compatibility for descent statistics—statistics that depend only on the descent set and length—which has close connections to the theory of P-partitions, quasisymmetric functions, and noncommutative symmetric functions. We use our framework to prove that many descent statistics are shuffle-compatible and to give explicit descriptions of their shuffle algebras, thus unifying past results of Stanley, Gessel, Stembridge, Aguiar–Bergeron–Nyman, and Petersen.
Authors:Eli Aljadeff; Yaakov Karasik Pages: 142 - 175 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Eli Aljadeff, Yaakov Karasik Let F be an algebraically closed field of characteristic zero and let G be a finite group. We consider graded Verbally prime T-ideals in the free G-graded algebra. It turns out that equivalent definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime G-graded T-ideals and strongly verbally prime T-ideals. At first, following Kemer's ideas, we classify G-graded verbally prime T-ideals. The main bulk of the paper is devoted to the stronger notion. We classify G-graded strongly verbally prime T-ideals which are T-ideal of affine G-graded algebras or equivalently G-graded T-ideals that contain a Capelli polynomial. It turns out that these are precisely the T-ideal of G-graded identities of finite dimensional G-graded, central over F (i.e. Z ( A ) e = F ) which admit a G-graded division algebra twisted form over a field k which contains F or equivalently over a field k which contains enough roots of unity (e.g. a primitive n-root of unity where n = o r d ( G ) ).
Authors:Igor Belegradek Pages: 176 - 198 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Igor Belegradek We determine the homeomorphism type of the hyperspace of positively curved C ∞ convex bodies in R n , and derive various properties of its quotient by the group of Euclidean isometries. We make a systematic study of hyperspaces of convex bodies that are at least C 1 . We show how to destroy the symmetry of a family of convex bodies, and prove that this cannot be done modulo congruence.
Authors:Rolf Schneider Pages: 199 - 234 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Rolf Schneider Let C be a closed convex cone in R n , pointed and with interior points. We consider sets of the form A = C ∖ K , where K ⊂ C is a closed convex set. If A has finite volume (Lebesgue measure), then A is called a C-coconvex set, and K is called C-close. The family of C-coconvex sets is closed under the addition ⊕ defined by C ∖ ( A 1 ⊕ A 2 ) = ( C ∖ A 1 ) + ( C ∖ A 2 ) . We develop first steps of a Brunn–Minkowski theory for C-coconvex sets, which relates this addition to the notion of volume. In particular, we establish the equality condition for a Brunn–Minkowski type inequality (with reversed inequality sign) and introduce mixed volumes and their integral representations. For C-close sets, surface area measures and cone-volume measures can be defined, in analogy to similar notions for convex bodies. They are Borel measures on the intersection of the unit sphere with the interior of the polar cone of C. We prove a Minkowski-type uniqueness theorem for C-close sets with equal surface area measures. Concerning Minkowski-type existence problems, we give conditions for a Borel measure to be either the surface area measure or the cone-volume measure of a C-close set. These conditions are sufficient in the first case, and necessary and sufficient in the second case.
Authors:Liuquan Wang; Cheng Zhang Pages: 311 - 348 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Liuquan Wang, Cheng Zhang Let f ( x ) = ∑ n = 0 ∞ 1 n ! q n ( n − 1 ) / 2 x n ( 0 < q < 1 ) be the deformed exponential function. It is known that the zeros of f ( x ) are real and form a negative decreasing sequence ( x k ) ( k ≥ 1 ). We investigate the complete asymptotic expansion for x k and prove that for any n ≥ 1 , as k → ∞ , x k = − k q 1 − k ( 1 + ∑ i = 1 n C i ( q ) k − 1 − i + o ( k − 1 − n ) ) , where C i ( q ) are some q series which can be determined recursively. We show that each C i ( q ) ∈ Q [ A 0 , A 1 , A 2 ] , where A i = ∑ m = 1 ∞ m i σ ( m ) q m and σ ( m ) denotes the sum of positive divisors of m. When writing C i as a polynomial in A 0 , A 1 and A 2 , we find explicit formulas for the coefficients of the linear terms by using Bernoulli numbers. Moreover, we also prove that C i ( q ) ∈ Q [ E 2 , E 4 , E 6 ] , where E 2 , E 4 and E ... PubDate: 2018-05-31T11:10:48Z DOI: 10.1016/j.aim.2018.05.006 Issue No:Vol. 332 (2018)
Authors:Hyenho Lho; Rahul Pandharipande Pages: 349 - 402 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Hyenho Lho, Rahul Pandharipande We study the fundamental relationship between stable quotient invariants and the B-model for local P 2 in all genera. Our main result is a direct geometric proof of the holomorphic anomaly equation in the precise form predicted by B-model physics. The method yields new holomorphic anomaly equations for an infinite class of twisted theories on projective spaces. An example of such a twisted theory is the formal quintic defined by a hyperplane section of P 4 in all genera via the Euler class of a complex. The formal quintic theory is found to satisfy the holomorphic anomaly equations conjectured for the true quintic theory. Therefore, the formal quintic theory and the true quintic theory should be related by transformations which respect the holomorphic anomaly equations.
Authors:Shenhui Liu Pages: 403 - 437 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Shenhui Liu With the method of moments and the mollification method, we study the central L-values of GL(2) Maass forms of weight 0 and level 1 and establish a positive-proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals, which is qualitatively optimal in view of Weyl's law. As an application of this result and a formula of Katok–Sarnak, we give a nonvanishing result on the first Fourier coefficients of Maass forms of weight 1 2 and level 4 in the Kohnen plus space.
Authors:Grigoris Paouris; Petros Valettas Pages: 438 - 464 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Grigoris Paouris, Petros Valettas We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space X = ( R n , ‖ ⋅ ‖ ) there exists an invertible linear map T : R n → R n with P ( ‖ T G ‖ − E ‖ T G ‖ > ε E ‖ T G ‖ ) ≤ C exp ( − c max { ε 2 , ε } log n ) , ε > 0 , where G is the standard n-dimensional Gaussian vector and C , c > 0 are universal constants. It follows that for every ε ∈ ( 0 , 1 ) and for every normed space X = ( R n , ‖ ⋅ ‖ ) there exists a k-dimensional subspace of X which is ( 1 + ε ) -Euclidean and k ≥ c ε log n / log 1 ε . This improves by a logarithmic on ε term the best previously known result due to G. Schechtman.
Authors:Alex Fink; Karola Mészáros; Avery St. Dizier Pages: 465 - 475 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Alex Fink, Karola Mészáros, Avery St. Dizier We show that the dual character of the flagged Weyl module of any diagram is a positively weighted integer point transform of a generalized permutahedron. In particular, Schubert and key polynomials are positively weighted integer point transforms of generalized permutahedra. This implies several recent conjectures of Monical, Tokcan and Yong.
Authors:Pieter Spaas Pages: 510 - 552 Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Pieter Spaas We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II1 factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II1 factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II1 factor up to unitary conjugacy are not classifiable by countable structures, and deduce that the same holds for any McDuff II1 factor with at least one Cartan subalgebra.
Authors:Imre Bárány; Julien Bureaux; Ben Lund Pages: 143 - 169 Abstract: Publication date: 20 June 2018 Source:Advances in Mathematics, Volume 331 Author(s): Imre Bárány, Julien Bureaux, Ben Lund Given a convex cone C in R d , an integral zonotope T is the sum of segments [ 0 , v i ] ( i = 1 , … , m ) where each v i ∈ C is a vector with integer coordinates. The endpoint of T is k = ∑ 1 m v i . Let T ( C , k ) be the family of all integral zonotopes in C whose endpoint is k ∈ C . We prove that, for large k, the zonotopes in T ( C , k ) have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes in T ( C , k ) are very close to a fixed convex set. We also establish several combinatorial properties of a typical zonotope in T ( C , k ) .
Authors:Saiei-Jaeyeong Matsubara-Heo Pages: 170 - 208 Abstract: Publication date: 20 June 2018 Source:Advances in Mathematics, Volume 331 Author(s): Saiei-Jaeyeong Matsubara-Heo We introduce a ring H of partial difference-differential operators with constant coefficients initially defined by H. Glüsing-Lürßen for ordinary difference-differential operators and investigate its cohomological properties. Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last section, we deduce from this injectivity properties of various function modules over H as well as the density results of exponential polynomial solutions for partial difference-differential equations.
Authors:Matt Szczesny Pages: 209 - 238 Abstract: Publication date: 20 June 2018 Source:Advances in Mathematics, Volume 331 Author(s): Matt Szczesny We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the representation possess a compatible grading, and conditions on the support of the module. Quotients by these ideals lead to combinatorial Hopf algebras which can be interpreted as Hall algebras of certain sub-categories of modules. In the case of the free commutative monoid on n generators, we obtain a co-commutative Hopf algebra structure on n-dimensional skew shapes, whose underlying associative product amounts to a “stacking” operation on the skew shapes. The primitive elements of this Hopf algebra correspond to connected skew shapes, and form a graded Lie algebra by anti-symmetrizing the associative product. We interpret this Hopf algebra as the Hall algebra of a certain category of coherent torsion sheaves on A / F 1 n supported at the origin, where F 1 denotes the field of one element. This Hopf algebra may be viewed as an n-dimensional generalization of the Hopf algebra of symmetric functions, which corresponds to the case n = 1 .
Authors:Ilya Kossovskiy; Ming Xiao Pages: 239 - 267 Abstract: Publication date: 20 June 2018 Source:Advances in Mathematics, Volume 331 Author(s): Ilya Kossovskiy, Ming Xiao A well known result of Forstnerić [15] states that most real-analytic strictly pseudoconvex hypersurfaces in complex space are not holomorphically embeddable into spheres of higher dimension. A more recent result by Forstnerić [16] states even more: most real-analytic hypersurfaces do not admit a holomorphic embedding even into a merely algebraic hypersurface of higher dimension, in particular, a hyperquadric. We emphasize that both cited theorems are proved by showing that the set of embeddable hypersurfaces is a set of first Baire category. At the same time, the classical theorem of Webster [30] asserts that every real-algebraic Levi-nondegenerate hypersurface admits a transverse holomorphic embedding into a nondegenerate real hyperquadric in complex space. In this paper, we provide effective results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, under the codimension restriction N ≤ 2 n , the defining functions φ ( z , z ¯ , u ) of all real-analytic hypersurfaces M = { v = φ ( z , z ¯ , u ) } ⊂ C n + 1 containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric Q ⊂ C N + 1 satisfy an universal algebraic partial differential equation D ( φ ) = 0 , where the algebraic-differential operator D = D ( n , N ) depends on n ≥ 1 , n < N ≤ 2 n only. To the best of our knowledge, this is the first effective result characterizing real-analytic hypersurfaces embeddable into a hyperquadric of higher dimension. As an application, we show that for every n , N as above there exists μ = μ ( n , N ) such that a Zariski generic real-analytic hypersurface M ⊂ C n + 1 of degree ≥μ is not transversally holomorphically embeddable into any hyperquadric Q ⊂ C N + 1 . We also provide an explicit upper bound for μ in terms of n , N . To the best of our knowledge, this gives the first effective lower bound for the CR-complexity of a Zariski generic real-algebraic hypersurface in complex space of a fixed degree.
Authors:Marcelo Laca; Iain Raeburn; Jacqui Ramagge; Michael F. Whittaker Pages: 268 - 325 Abstract: Publication date: 20 June 2018 Source:Advances in Mathematics, Volume 331 Author(s): Marcelo Laca, Iain Raeburn, Jacqui Ramagge, Michael F. Whittaker We consider self-similar actions of groupoids on the path spaces of finite directed graphs, and construct examples of such self-similar actions using a suitable notion of graph automaton. Self-similar groupoid actions have a Cuntz–Pimsner algebra and a Toeplitz algebra, both of which carry natural dynamics lifted from the gauge actions. We study the equilibrium states (the KMS states) on the resulting dynamical systems. Above a critical inverse temperature, the KMS states on the Toeplitz algebra are parametrised by the traces on the full C ⁎ -algebra of the groupoid, and we describe a program for finding such traces. The critical inverse temperature is the logarithm of the spectral radius of the incidence matrix of the graph, and at the critical temperature the KMS states on the Toeplitz algebra factor through states of the Cuntz–Pimsner algebra. Under a verifiable hypothesis on the self-similar action, there is a unique KMS state on the Cuntz–Pimsner algebra. We discuss an explicit method of computing the values of this KMS state, and illustrate with examples.
Authors:Mrinal Kanti Das Pages: 326 - 338 Abstract: Publication date: 20 June 2018 Source:Advances in Mathematics, Volume 331 Author(s): Mrinal Kanti Das This article concerns Murthy's conjecture on complete intersections, made in 1975. The sole breakthrough on this conjecture has still been the result proved by Mohan Kumar in 1978. The conjecture is open in general. In this article we improve Mohan Kumar's bound when the base field is F ‾ p . As an application, we prove that any local complete intersection surface in the affine space A F ‾ p d is a set-theoretic complete intersection, generalizing a result of Bloch–Murthy–Szpiro.
Authors:Alexander Engström; Florian Kohl Pages: 1 - 37 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Alexander Engström, Florian Kohl Transfer-matrix methods originated in physics where they were used to count the number of allowed particle states on a structure whose width n is a parameter. Typically, the number of states is exponential in n. One mathematical instance of this methodology is to enumerate the proper vertex colorings of a graph of growing size by a fixed number of colors. In Ehrhart theory, lattice points in the dilation of a fixed polytope by a factor k are enumerated. By inclusion–exclusion, relevant conditions on how the lattice points interact with hyperplanes are enforced. Typically, the number of points are (quasi-) polynomial in k. The text-book example is that for a fixed graph, the number of proper vertex colorings with k colors is polynomial in k. This paper investigates the joint enumeration problem with both parameters n and k free. We start off with the classical graph colorings and then explore common situations in combinatorics related to Ehrhart theory. We show how symmetries can be explored to reduce calculations and explain the interactions with Discrete Geometry.
Authors:Erik Carlsson; Fernando Rodriguez Villegas Pages: 38 - 60 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Erik Carlsson, Fernando Rodriguez Villegas We prove some combinatorial conjectures extending those proposed in [13,14]. The proof uses a vertex operator due to Nekrasov, Okounkov, and the first author [4] to obtain a “gluing formula” for the relevant generating series, essentially reducing the computation to the case of CP 1 with three punctures.
Authors:Bo'az Klartag Pages: 74 - 108 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Bo'az Klartag This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar.
Authors:Matthew Harrison-Trainor Pages: 109 - 147 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Matthew Harrison-Trainor The Scott rank of a countable structure is a measure, coming from the proof of Scott's isomorphism theorem, of the complexity of that structure. The Scott spectrum of a theory (by which we mean a sentence of L ω 1 ω ) is the set of Scott ranks of countable models of that theory. In Z F C + P D we give a descriptive-set-theoretic classification of the sets of ordinals which are the Scott spectrum of a theory: they are particular Σ 1 1 classes of ordinals. Our investigation of Scott spectra leads to the resolution (in ZFC) of a number of open problems about Scott ranks. We answer a question of Montalbán by showing, for each α < ω 1 , that there is a Π 2 in theory with no models of Scott rank less than α. We also answer a question of Knight and Calvert by showing that there are computable models of high Scott rank which are not computably approximable by models of low Scott rank. Finally, we answer a question of Sacks and Marker by showing that δ 2 1 is the least ordinal α such that if the models of a computable theory T have Scott rank bounded below ω 1 , then their Scott ranks are bounded below α.
Authors:Alexander Engel Pages: 148 - 172 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Alexander Engel We prove the Banach strong Novikov conjecture for groups having polynomially bounded higher-order combinatorial functions. This includes all automatic groups.
Authors:Hoang Dinh Van; Wendy Lowen Pages: 173 - 228 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Hoang Dinh Van, Wendy Lowen The aim of this work is to construct a complex which through its higher structure directly controlls deformations of general prestacks, building on the work of Gerstenhaber and Schack for presheaves of algebras. In defining a Gerstenhaber–Schack complex C GS • ( A ) for an arbitrary prestack A , we have to introduce a differential with an infinite sequence of components instead of just two as in the presheaf case. If A ˜ denotes the Grothendieck construction of A , which is a U -graded category, we explicitly construct inverse quasi-isomorphisms F and G between C GS • ( A ) and the Hochschild complex C U ( A ˜ ) , as well as a concrete homotopy T : F G ⟶ 1 , which had not been obtained even in the presheaf case. As a consequence, by applying the Homotopy Transfer Theorem, one can transfer the dg Lie structure present on the Hochschild complex in order to obtain an L ∞ -structure on C GS • ( A ) , which controlls the higher deformation theory of the prestack A . This answers the open problem about the higher structure on the Gerstenhaber–Schack complex at once in the general prestack case.
Authors:József Balogh; Adam Zsolt Wagner Pages: 229 - 252 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): József Balogh, Adam Zsolt Wagner A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family F ⊆ P ( n ) that does not contain a 2-chain F 1 ⊊ F 2 . Erdős later extended this result and determined the largest family not containing a k-chain F 1 ⊊ … ⊊ F k . Erdős and Katona and later Kleitman asked how many such chains must appear in families whose size is larger than the corresponding extremal result. This question was resolved for 2-chains by Kleitman in 1966, who showed that amongst families of size M in P ( n ) , the number of 2-chains is minimized by a family whose sets are taken as close to the middle layer as possible. He also conjectured that the same conclusion should hold for all k, not just 2. The best result on this question is due to Das, Gan and Sudakov who showed that Kleitman's conjecture holds for families whose size is at most the size of the k + 1 middle layers of P ( n ) , provided k ≤ n − 6 . Our main result is that for every fixed k and ε > 0 , if n is sufficiently large then Kleitman's conjecture holds for families of size at most ( 1 − ε ) 2 n , thereby establishing Kleitman's conjecture asymptotically. Our proof is based on ideas of Kleitman and Das, Gan and Sudakov. Several open problems are also given.
Authors:Boris Pittel Pages: 280 - 306 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Boris Pittel An integer partition of n is a decreasing sequence of positive integers that add up to [ n ] . Back in 1979 Macdonald posed a question about the limit value of the probability that two partitions chosen uniformly at random, and independently of each other, are comparable in terms of the dominance order. In 1982 Wilf conjectured that the uniformly random partition is a size-ordered degree sequence of a simple graph with the limit probability 0. In 1997 we showed that in both, seemingly unrelated, cases the limit probabilities are indeed zero, but our method left open the problem of convergence rates. The main result in this paper is that each of the probabilities is e − 0.11 log n / log log n , at most. A key element of the argument is a local limit theorem, with convergence rate, for the joint distribution of the [ n 1 / 4 − ε ] tallest columns and the [ n 1 / 4 − ε ] longest rows of the Young diagram representing the random partition.
Authors:Dirk Hofmann; Pedro Nora Pages: 307 - 360 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Dirk Hofmann, Pedro Nora A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces, the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval [ 0 , 1 ] and present duality theory for ordered and metric compact Hausdorff spaces and (suitably defined) finitely cocomplete categories enriched in [ 0 , 1 ] .
Authors:Gwyn Bellamy; Ulrich Thiel Pages: 361 - 419 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Gwyn Bellamy, Ulrich Thiel We show that the category of graded modules over a finite-dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category. When the algebra is self-injective, we show furthermore that this highest weight category has tilting modules in the sense of Ringel. This provides a new perspective on the representation theory of such algebras, and leads to several new structures attached to them. There are a wide variety of examples in algebraic Lie theory to which this applies: restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras.
Authors:Andreas Rosenschon; Anand Sawant Pages: 420 - 432 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Andreas Rosenschon, Anand Sawant A smooth projective scheme X over a field k is said to satisfy the Rost nilpotence principle if any endomorphism of X in the category of Chow motives that vanishes on an extension of the base field k is nilpotent. We show that an étale motivic analogue of the Rost nilpotence principle holds for all smooth projective schemes over a perfect field. This provides a new approach to the question of Rost nilpotence and allows us to obtain an elegant proof of Rost nilpotence for surfaces, as well as for birationally ruled threefolds over a field of characteristic 0.
Authors:Alexander Grigor'yan; Eryan Hu; Jiaxin Hu Pages: 433 - 515 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Alexander Grigor'yan, Eryan Hu, Jiaxin Hu We prove necessary and sufficient conditions for stable-like estimates of the heat kernel for jump type Dirichlet forms on metric measure spaces. The conditions are given in terms of the volume growth function, jump kernel and a generalized capacity.
Authors:Pär Kurlberg; Igor Wigman Pages: 516 - 552 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Pär Kurlberg, Igor Wigman This is a manuscript containing the full proofs of results announced in [10], together with some recent updates. We prove that the Nazarov–Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for “arithmetic random waves”, i.e. random toral Laplace eigenfunctions.
Authors:Antonio Auffinger; Wei-Kuo Chen Pages: 553 - 588 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Antonio Auffinger, Wei-Kuo Chen We investigate the energy landscape of the spherical mixed even p-spin model near its maximum energy. We relate the distance between pairs of near maxima to the support of the Parisi measure at zero temperature. We then provide an algebraic relation that characterizes one-step replica symmetric breaking Parisi measures. For these measures, we show that any two nonparallel spin configurations around the maximum energy are asymptotically orthogonal to each other. In sharp contrast, we study models with full replica symmetry breaking and show that all possible values of the asymptotic distance are attained near the maximum energy.
Authors:Marek Biskup; Oren Louidor Pages: 589 - 687 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Marek Biskup, Oren Louidor We study the local structure of the extremal process associated with the Discrete Gaussian Free Field (DGFF) in scaled-up (square-)lattice versions of bounded open planar domains subject to mild regularity conditions on the boundary. We prove that, in the scaling limit, this process tends to a Cox process decorated by independent, correlated clusters whose distribution is completely characterized. As an application, we control the scaling limit of the discrete supercritical Liouville measure, extract a Poisson–Dirichlet statistics for the limit of the Gibbs measure associated with the DGFF and establish the “freezing phenomenon” conjectured to occur in the “glassy” phase. In addition, we prove a local limit theorem for the position and value of the absolute maximum. The proofs are based on a concentric, finite-range decomposition of the DGFF and entropic-repulsion arguments for an associated random walk. Although we naturally build on our earlier work on this problem, the methods developed here are largely independent.
Authors:Óscar Ciaurri; Luz Roncal; Pablo Raúl Stinga; José L. Torrea; Juan Luis Varona Pages: 688 - 738 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Óscar Ciaurri, Luz Roncal, Pablo Raúl Stinga, José L. Torrea, Juan Luis Varona The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size h > 0 ( − Δ h ) s u = f , for u , f : Z h → R , 0 < s < 1 , is performed. The pointwise nonlocal formula for ( − Δ h ) s u and the nonlocal discrete mean value property for discrete s-harmonic functions are obtained. We observe that a characterization of ( − Δ h ) s as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Hölder spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincaré inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution) u = ( − Δ h ) − s f , which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy–Littlewood–Sobolev inequality for ( − Δ h ) − s . As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as h → 0 in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of h under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian ( − Δ ) s U = F , in R , can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of h.
Authors:Xiaowei Xu; Ling Yang; Yongsheng Zhang Pages: 739 - 762 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Xiaowei Xu, Ling Yang, Yongsheng Zhang It has been 40 years since Lawson and Osserman introduced the three minimal cones associated with Dirichlet problems in their 1977 Acta paper [13]. The first cone was shown area-minimizing by Harvey and Lawson in the celebrated paper [10]. In this paper, we confirm that the other two are also area-minimizing. In fact, we show that every Lawson–Osserman cone of type ( n , p , 2 ) constructed in [26] is area-minimizing.
Authors:Alexey Glazyrin; Wei-Hsuan Yu Pages: 810 - 833 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Alexey Glazyrin, Wei-Hsuan Yu The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only s distinct values. Two-distance spherical sets with the set of scalar products { α , − α } , α ∈ [ 0 , 1 ) , are called equiangular. The problem of determining the maximum size of s-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an s-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in R n , n ≥ 7 , is n ( n + 1 ) 2 with possible exceptions for some n = ( 2 k + 1 ) 2 − 3 , k ∈ N . We also prove the universal upper bound ∼ 2 3 n a 2 for equiangular sets with α = 1 a and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound.
Authors:Piotr Biler; Grzegorz Karch; Jacek Zienkiewicz Pages: 834 - 875 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz We consider the parabolic–elliptic model for the chemotaxis with fractional (anomalous) diffusion. Global-in-time solutions are constructed under (nearly) optimal assumptions on the size of radial initial data. Moreover, criteria for blowup of radial solutions in terms of suitable Morrey spaces norms are derived.
Authors:Marco Mackaay; Ben Webster Pages: 876 - 945 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Marco Mackaay, Ben Webster In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin–Turaev invariants for sl n . Over the past decade, such invariants have been constructed in a variety of different ways, using matrix factorizations, category O , affine Grassmannians, and diagrammatic categorifications of tensor products. While the definitions of these theories are quite different, there is a key commonality between them which makes it possible to prove that they are all isomorphic: they arise from a skew Howe dual action of gl ℓ for some ℓ. In this paper, we show that the construction of knot homology based on categorifying tensor products (from earlier work of the second author) fits into this framework, and thus agrees with other such homologies, such as Khovanov–Rozansky homology. We accomplish this by categorifying the action of gl ℓ × gl n on ⋀ p ( C ℓ ⊗ C n ) using diagrammatic bimodules. In this action, the functors corresponding to gl ℓ and gl n are quite different in nature, but they will switch roles under Koszul duality.
Authors:Sebastián Donoso; Wenbo Sun Pages: 946 - 996 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Sebastián Donoso, Wenbo Sun We show that for every ergodic system ( X , μ , T 1 , … , T d ) with commuting transformations, the average 1 N d + 1 ∑ 0 ≤ n 1 , … , n d ≤ N − 1 ∑ 0 ≤ n ≤ N − 1 f 1 ( T 1 n ∏ j = 1 d T j n j x ) f 2 ( T 2 n ∏ j = 1 d T j n j x ) ⋯ f d ( T d n ∏ j = 1 d T j n j x ) converges for μ-a.e. x ∈ X as N → ∞ . If X is distal, we prove that the average 1 N ∑ n = 0 N − 1 f 1 ( T 1 n x ) f 2 ( T 2 n x ) ⋯ f d ( T d n x ) converges for μ-a.e. x ∈ X as N → ∞ . We also establish the pointwise convergence of averages along cubical configurations arising from a system with commuting transformations. Our methods combine the existence of sated and magic extensions introduced by Austin and Host respectively with ideas on topological models by Huang, Shao and Ye.
Authors:Thomas Scanlon Pages: 1071 - 1100 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Thomas Scanlon Let Y be a complex algebraic variety, G ↷ Y an action of an algebraic group on Y, U ⊆ Y ( C ) a complex submanifold, Γ < G ( C ) a discrete, Zariski dense subgroup of G ( C ) which preserves U, and π : U → X ( C ) an analytic covering map of the complex algebraic variety X expressing X ( C ) as Γ \ U . We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative χ ˜ : Y → Z (where Z is some algebraic variety) expressing the quotient of Y by the action of the constant points of G. Under the additional hypothesis that the restriction of π to some set containing a fundamental domain is definable in an o-minimal expansion of the real field, we show as a consequence of the Peterzil–Starchenko o-minimal GAGA theorem that the prima facie differentially analytic relation χ : = χ ˜ ∘ π − 1 is a well-defined, differential constructible function. The function χ nearly inverts π in the sense that for any differential field K of meromorphic functions, if a , b ∈ X ( K ) then χ ( a ) = χ ( b ) if and only if after suitable restriction there is some γ ∈ G ( C ) with π ( γ ⋅ π − 1 ( a ) ) = b .
Authors:Ionel Popescu Pages: 1101 - 1159 Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Ionel Popescu In this paper we deal with free functional inequalities on the circle. There are some interesting changes from their classical counterparts. For example, the free Poincaré inequality has a slight change which seems to account for the lack of invariance under rotations of the base measure. Another instance is the modified Wasserstein distance on the circle which provides the tools for analyzing transportation, Log-Sobolev, and HWI inequalities. These new phenomena also indicate that they have classical counterparts, which does not seem to have been investigated thus far.
Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): François Le Maître We initiate the study of a measurable analogue of small topological full groups that we call L 1 full groups. These groups are endowed with a Polish group topology which admits a natural complete right invariant metric. We mostly focus on L 1 full groups of measure-preserving Z -actions which are actually a complete invariant of flip conjugacy. We prove that for ergodic actions the closure of the derived group is topologically simple although it can fail to be simple. We also show that the closure of the derived group is connected, and that for measure-preserving free actions of non-amenable groups the closure of the derived group and the L 1 full group itself are never amenable. In the case of a measure-preserving ergodic Z -action, the closure of the derived group is shown to be the kernel of the index map. If such an action is moreover by homeomorphism on the Cantor space, we show that the topological full group is dense in the L 1 full group. Using Juschenko–Monod and Matui's results on topological full groups, we conclude that L 1 full groups of ergodic Z -actions are amenable as topological groups, and that they are topologically finitely generated if and only if the Z -action has finite entropy.
Authors:Peter Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): György Pál Gehér, Peter Šemrl Botelho, Jamison, and Molnár [1], and Gehér and Šemrl [4] have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space H. As a straightforward consequence one can characterize surjective isometries of Grassmann spaces of projections of a fixed finite corank. In this paper we solve the remaining structural problem for surjective isometries on the set P ∞ ( H ) of all projections of infinite rank and infinite corank when H is separable. The proof technique is entirely different from the previous ones and is based on the study of geodesics in the Grassmannian P ∞ ( H ) . However, the same method gives an alternative proof in the case of finite rank projections.
Abstract: Publication date: 9 July 2018 Source:Advances in Mathematics, Volume 332 Author(s): Xu Xu Inversive distance circle packing metric was introduced by P Bowers and K Stephenson [7] as a generalization of Thurston's circle packing metric [34]. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo [22] proved the infinitesimal rigidity and then Luo [27] proved the global rigidity. In this paper, based on an observation of Zhou [37], we prove this conjecture for inversive distance in ( − 1 , + ∞ ) by variational principles. We also study the global rigidity of a combinatorial curvature introduced in [14,16,19] with respect to the inversive distance circle packing metrics where the inversive distance is in ( − 1 , + ∞ ) .
Authors:Asger Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Asger Törnquist We show that there are no infinite maximal almost disjoint (“mad”) families in Solovay's model, thus solving a long-standing problem posed by A.R.D. Mathias in 1969. We also give a new proof of Mathias' theorem that no analytic infinite almost disjoint family can be maximal, and show more generally that if Martin's Axiom holds at κ < 2 ℵ 0 , then no κ-Souslin infinite almost disjoint family can be maximal. Finally we show that if ℵ 1 L [ a ] < ℵ 1 , then there are no Σ 2 1 [ a ] infinite mad families.
Authors:Boban Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): Žikica Perović, Boban Veličković Solving a well-known problem of Maharam, Talagrand [18] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure one can canonically assign a certain countable ordinal, its exhaustivity rank. In this paper, we use carefully constructed Schreier families and norms derived from them to provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This gives rise to uncountably many non isomorphic separable atomless Maharam algebras.
Authors:Liberati Abstract: Publication date: 25 May 2018 Source:Advances in Mathematics, Volume 330 Author(s): José I. Liberati We find a necessary and sufficient condition for the existence of the tensor product of modules over a vertex algebra. We define the notion of vertex bilinear map and provide two algebraic constructions of the tensor product, where one of them is of ring theoretical type. We show the relation between tensor product and vertex homomorphisms. We prove commutativity of the tensor product. We also prove associativity of the tensor product of modules under certain necessary and sufficient condition. Finally, we show certain functorial properties of vertex homomorphism and the tensor product.
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