Authors:Jingbo Dou; Qianqiao Guo; Meijun Zhu Pages: 1 - 45 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Jingbo Dou, Qianqiao Guo, Meijun Zhu In this paper we establish the reversed sharp Hardy–Littlewood–Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and Yan in [6]) by introducing a uniform approach. The extremal functions are classified via the method of moving spheres, and the best constants are computed. The new approach can also be applied to obtain the classical HLS inequality and other similar inequalities.

Authors:Nathan Pflueger Pages: 46 - 63 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Nathan Pflueger We consider a general curve of fixed gonality k and genus g. We propose an estimate ρ ‾ g , k ( d , r ) for the dimension of the variety W d r ( C ) of special linear series on C, by solving an analogous problem in tropical geometry. Using work of Coppens and Martens, we prove that this estimate is exactly correct if k ≥ 1 5 g + 2 , and is an upper bound in all other cases. We also completely characterize the cases in which W d r ( C ) has the same dimension as for a general curve of genus g.

Authors:Tanmay Deshpande Pages: 64 - 106 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Tanmay Deshpande Let k be the algebraic closure of a finite field F q of characteristic p. Let G be a connected unipotent group over k equipped with an F q -structure given by a Frobenius map F : G ⟶ G . We will denote the corresponding algebraic group defined over F q by G 0 . Character sheaves on G are certain special objects in the triangulated braided monoidal category D G ( G ) of bounded conjugation equivariant Q ‾ l -complexes (where l ≠ p is a prime number) on G. Boyarchenko has proved that the “trace of Frobenius” functions associated with F-stable character sheaves on G form an orthonormal basis of the space of class functions on G 0 ( F q ) and that the matrix relating this basis to the basis formed by the irreducible characters of G 0 ( F q ) is block diagonal with “small” blocks. In particular, there is a partition of the set of character sheaves as well as a partition of the set of irreducible characters of G 0 ( F q ) into “small” families known as L -packets. In this paper we describe these block matrices relating character sheaves and irreducible characters corresponding to each L -packet. We prove that these matrices can be described as certain “crossed S-matrices” associated with each L -packet. We will also derive a formula for the dimensions of the irreducible representations of G 0 ( F q ) in terms of certain modular categorical data associated with the corresponding L -packet. In fact we will formulate and prove more general results which hold for possibly disconnected groups G such that G ∘ is unipotent. To prove our results, we will establish a formula (which holds for any algebraic group G) which expresses the inner product of the “trace of Frobenius” function of any F-stable object of D G ( G ) with any character of G 0 ( F q ) (or of any of its pure inner forms) in terms of certain categorical operations.

Authors:Xin Fang; Ghislain Fourier; Peter Littelmann Pages: 107 - 149 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Xin Fang, Ghislain Fourier, Peter Littelmann We present a new approach to construct T-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton–Okounkov bodies with ideas originally stemming from PBW-filtrations. For each pair ( S , > ) consisting of a birational sequence and a monomial order, we attach to the affine variety G / / U a monoid Γ = Γ ( S , > ) . As a side effect we get a vector space basis B Γ of C [ G / / U ] , the elements being indexed by Γ. The basis B Γ has multiplicative properties very similar to those of the dual canonical basis. This makes it possible to transfer the methods of Alexeev and Brion [1] to this more general setting, once one knows that the monoid Γ is finitely generated and saturated.

Authors:Eva Bayer-Fluckiger; Uriya A. First Pages: 150 - 184 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Eva Bayer-Fluckiger, Uriya A. First Let R be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an R-algebra with involution, which are rationally isomorphic and have isomorphic semisimple coradicals, are in fact isomorphic. The same result is also obtained for quadratic forms equipped with an action of a finite group. The results have cohomological restatements that resemble the Grothendieck–Serre conjecture, except the group schemes involved are not reductive. We show that these group schemes are closely related to group schemes arising in Bruhat–Tits theory.

Authors:William H. Meeks; Joaquín Pérez Pages: 185 - 197 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): William H. Meeks, Joaquín Pérez We prove that any complete, embedded minimal surface M with finite topology in a homogeneous three-manifold N has positive injectivity radius. When one relaxes the condition that N be homogeneous to that of being locally homogeneous, then we show that the closure of M has the structure of a minimal lamination of N. As an application of this general result we prove that any complete, embedded minimal surface with finite genus and a countable number of ends is compact when the ambient space is S 3 equipped with a homogeneous metric of nonnegative scalar curvature.

Authors:Vladimir G. Pestov Pages: 1 - 17 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Vladimir G. Pestov There is a countable metrizable group acting continuously on the space of rationals in such a way that the only equivariant compactification of the space is a singleton. This is obtained by a recursive application of a construction due to Megrelishvili, which is a metric fan equipped with a certain group of homeomorphisms. The question of existence of a topological transformation group with the property in the title was asked by Yu.M. Smirnov in the 1980s.

Authors:Robin Ming Chen; Jilong Hu; Dehua Wang Pages: 18 - 60 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Robin Ming Chen, Jilong Hu, Dehua Wang The linear stability of rectilinear compressible vortex sheets is studied for two-dimensional isentropic elastic flows. This problem has a free boundary and the boundary is characteristic. A necessary and sufficient condition is obtained for the linear stability of the rectilinear vortex sheets. More precisely, it is shown that, besides the stable supersonic zone, the elasticity exerts an additional stable subsonic zone. We also find that there is a class of states in the interior of subsonic zone where the stability of such states is weaker than the stability of other states in the sense that there is an extra loss of tangential derivatives with respect to the source terms. This is a new feature that the Euler flow does not possess. One of the difficulties for the elastic flow is that the non-differentiable points of the eigenvalues may coincide with the roots of the Lopatinskii determinant. As a result, the Kreiss symmetrization cannot be applied directly. Instead, we perform an upper triangularization of the system to separate only the outgoing modes at all points in the frequency space, so that an exact estimate of the outgoing modes can be obtained. Moreover, all the outgoing modes are shown to be zero due to the L 2 -regularity of solutions. The estimates for the incoming modes can be derived directly from the Lopatinskii determinant. This new approach avoids the lengthy computation and estimates for the outgoing modes when Kreiss symmetrization is applied. This method can also be applied to the Euler flow and MHD flow.

Authors:Ehud Meir Pages: 61 - 90 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Ehud Meir We study Hopf algebras via tools from geometric invariant theory. We show that all the invariants we get can be constructed using the integrals of the Hopf algebra and its dual together with the multiplication and the comultiplication, and that these invariants determine the isomorphism class of the Hopf algebra. We then define certain canonical subspaces I n v i , j of tensor powers of H and H ⁎ , and use the invariant theory to prove that these subspaces satisfy a certain non-degeneracy condition. Using this non-degeneracy condition together with results on symmetric monoidal categories, we prove that the spaces I n v i , j can also be described as ( H ⊗ i ⊗ ( H ⁎ ) ⊗ j ) A , where A is the group of Hopf automorphisms of H. As a result we prove that the number of possible Hopf orders of any semisimple Hopf algebra over a given number ring is finite. We give some examples of these invariants arising from the theory of Frobenius–Schur Indicators, and from Reshetikhin–Turaev invariants of three manifolds. We give a complete description of the invariants for a group algebra, proving that they all encode the number of homomorphisms from some finitely presented group to the group. We also show that if all the invariants are algebraic integers, then the Hopf algebra satisfies Kaplansky's sixth conjecture: the dimensions of the irreducible representations of H divide the dimension of H.

Authors:Mikhail Bondarko; Frédéric Déglise Pages: 91 - 189 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Mikhail Bondarko, Frédéric Déglise The aim of this work is to construct certain homotopy t-structures on various categories of motivic homotopy theory, extending works of Voevodsky, Morel, Déglise and Ayoub. We prove these t-structures possess many good properties, some analogous to those of the perverse t-structure of Beilinson, Bernstein and Deligne. We compute the homology of certain motives, notably in the case of relative curves. We also show that the hearts of these t-structures provide convenient extensions of the theory of homotopy invariant sheaves with transfers, extending some of the main results of Voevodsky. These t-structures are closely related to Gersten weight structures as defined by Bondarko.

Authors:Guotai Deng; Sze-Man Ngai Pages: 190 - 237 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Guotai Deng, Sze-Man Ngai By constructing an infinite graph-directed iterated function system associated with a finite iterated function system, we develop a new approach for proving the differentiability of the L q -spectrum and establishing the multifractal formalism of certain self-similar measures with overlaps, especially those defined by similitudes with different contraction ratios. We apply our technique to a well-known class of self-similar measures of generalized finite type.

Authors:Gabriele Di Cerbo Pages: 238 - 248 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Gabriele Di Cerbo We prove Fujita's spectrum conjecture on the discreteness of pseudo-effective thresholds for polarized varieties.

Authors:Tongzhu Li; Jie Qing; Changping Wang Pages: 249 - 294 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Tongzhu Li, Jie Qing, Changping Wang In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hypersurface. These results solve the major issues related to the conjectures of Cecil et al. on the classification of Dupin hypersurfaces.

Authors:David E. Evans; Terry Gannon Pages: 1 - 43 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): David E. Evans, Terry Gannon We realise non-unitary fusion categories using subfactor-like methods, and compute their quantum doubles and modular data. For concreteness we focus on generalising the Haagerup–Izumi family of Q-systems. For example, we construct endomorphism realisations of the (non-unitary) Yang–Lee model, and non-unitary analogues of one of the even subsystems of the Haagerup subfactor and of the Grossman–Snyder system. We supplement Izumi's equations for identifying the half-braidings, which were incomplete even in his Q-system setting. We conjecture a remarkably simple form for the modular S and T matrices of the doubles of these fusion categories. We would expect all of these doubles to be realised as the category of modules of a rational VOA and conformal net of factors. We expect our approach will also suffice to realise the non-semisimple tensor categories arising in logarithmic conformal field theories.

Authors:Harm Derksen; Visu Makam Pages: 44 - 63 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Harm Derksen, Visu Makam We study the left–right action of SL n × SL n on m-tuples of n × n matrices with entries in an infinite field K. We show that invariants of degree n 2 − n define the null cone. Consequently, invariants of degree ≤ n 6 generate the ring of invariants if char ( K ) = 0 . We also prove that for m ≫ 0 , invariants of degree at least n ⌊ n + 1 ⌋ are required to define the null cone. We generalize our results to matrix invariants of m-tuples of p × q matrices, and to rings of semi-invariants for quivers. For the proofs, we use new techniques such as the regularity lemma by Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor blow-ups of matrix spaces. We will discuss several applications to algebraic complexity theory, such as a deterministic polynomial time algorithm for non-commutative rational identity testing, and the existence of small division-free formulas for non-commutative polynomials.

Authors:Mathieu Duckerts-Antoine Pages: 64 - 120 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Mathieu Duckerts-Antoine We study the notion of a fundamental group in the framework of descent-exact homological categories. This setting is sufficiently large to include several categories of “algebraic” nature such as almost abelian categories, semi-abelian categories, and categories of topological semi-abelian algebras. For many adjunctions in this context, we describe the fundamental groups by generalised Brown–Ellis–Hopf formulae for the integral homology of groups.

Authors:Rita Jiménez Rolland; Jennifer C.H. Wilson Pages: 121 - 158 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Rita Jiménez Rolland, Jennifer C.H. Wilson A result of Lehrer describes a beautiful relationship between topological and combinatorial data on certain families of varieties with actions of finite reflection groups. His formula relates the cohomology of complex varieties to point counts on associated varieties over finite fields. Church, Ellenberg, and Farb use their representation stability results on the cohomology of flag manifolds, together with classical results on the cohomology rings, to prove asymptotic stability for “polynomial” statistics on associated varieties over finite fields. In this paper we investigate the underlying algebraic structure of these families' cohomology rings that makes the formulas convergent. We prove that asymptotic stability holds in general for subquotients of FI W -algebras finitely generated in degree at most one, a result that is in a sense sharp. As a consequence, we obtain convergence results for polynomial statistics on the set of maximal tori in Sp 2 n ( F q ‾ ) and SO 2 n + 1 ( F q ‾ ) that are invariant under the Frobenius morphism. Our results also give a new proof of the stability theorem for invariant maximal tori in GL n ( F q ‾ ) due to Church–Ellenberg–Farb.

Authors:Steven Boyer; Adam Clay Pages: 159 - 234 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Steven Boyer, Adam Clay We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology 3-sphere graph manifolds and relate them to the property of not being a Heegaard–Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions—using representations, using left-orders, using foliations, and using Heegaard–Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next we provide necessary and sufficient conditions for the existence of a co-oriented taut foliation on a graph manifold rational homology 3-sphere, respectively a left-order on its fundamental group, which depend solely on families of detected slopes on the boundaries of its pieces. The fact that Heegaard–Floer methods can be used to detect families of slopes on the boundary of a Seifert fibred manifold combines with certain conjectures in the literature to suggest an L-space gluing theorem for rational homology 3-sphere graph manifolds as well as other interesting problems in Heegaard–Floer theory.

Authors:Nikolaos Tziolas Pages: 235 - 289 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Nikolaos Tziolas This paper investigates the geometry of a smooth canonically polarized surface X defined over an algebraically closed field of characteristic p > 0 in the case when the automorphism scheme of X is not smooth. In particular, it is shown that a smooth canonically polarized surface X with 1 ≤ K X 2 ≤ 2 and non-smooth automorphism scheme tends to be uniruled and simply connected and is the purely inseparable quotient of a ruled or rational surface by a rational vector field. Moreover, restrictions on certain numerical invariants of X are obtained in order for Aut ( X ) to be smooth.

Authors:Plamen Iliev; Yuan Xu Pages: 290 - 326 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Plamen Iliev, Yuan Xu Connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves. For classical orthogonal polynomials whose weights are invariant under the action of the symmetric group, connection coefficients between a basis consisting of products of hypergeometric functions and another basis obtained from the first one by applying a permutation are studied. For the Jacobi polynomials on the simplex, it is shown that the connection coefficients can be expressed in terms of Tratnik's multivariable Racah polynomials and their weights. This gives, in particular, a new interpretation of the hidden duality between the variables and the degree indices of the Racah polynomials, which lies at the heart of their bispectral properties. These techniques also lead to explicit formulas for connection coefficients of Hahn and Krawtchouk polynomials of several variables, as well as for orthogonal polynomials on balls and spheres.

Authors:Takahiro Moteki; Yoshitsugu Takei Pages: 327 - 376 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Takahiro Moteki, Yoshitsugu Takei The middle convolution, introduced by Katz and developed by Dettweiler–Reiter, Oshima and others, defines an operation of reduction of linear ordinary differential equations with polynomial coefficients. In this paper, employing an idea of the exact steepest descent method proposed by Aoki–Kawai–Takei, we study how to determine the complete Stokes geometry and how to obtain the Borel summability of WKB solutions of a higher order linear ordinary differential equation with a large parameter when it is reduced to a second order equation via middle convolution. To show the practical usefulness of the method, we also investigate some concrete examples numerically.

Authors:Gautam Bharali; Andrew Zimmer Pages: 377 - 425 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Gautam Bharali, Andrew Zimmer In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary. Strongly pseudoconvex domains and weakly pseudoconvex domains of finite type always satisfy this Goldilocks condition, but we also present families of Goldilocks domains that have low boundary regularity or have boundary points of infinite type. We will show that the Kobayashi metric on these domains behaves, in some sense, like a negatively curved Riemannian metric. In particular, it satisfies a visibility condition in the sense of Eberlein and O'Neill. This behavior allows us to prove a variety of results concerning boundary extension of maps and to establish Wolff–Denjoy theorems for a wide collection of domains.

Authors:Anirban Basak; Mark Rudelson Pages: 426 - 483 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Anirban Basak, Mark Rudelson We consider a class of sparse random matrices of the form A n = ( ξ i , j δ i , j ) i , j = 1 n , where { ξ i , j } are i.i.d. centered random variables, and { δ i , j } are i.i.d. Bernoulli random variables taking value 1 with probability p n , and prove a quantitative estimate on the smallest singular value for p n = Ω ( log n n ) , under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse matrices. For p n = Ω ( n − α ) with some α ∈ ( 0 , 1 ) , we deduce that the condition number of A n is of order n with probability tending to one under the optimal moment assumption on { ξ i , j } . This in particular, extends a conjecture of von Neumann about the condition number to sparse random matrices with heavy-tailed entries. In the case that the random variables { ξ i , j } are i.i.d. sub-Gaussian, we further show that a sparse random matrix is singular with probability at most exp ( − c n p n ) whenever p n is above the critical threshold p n = Ω ( log n n ) . The results also extend to the case when { ξ i , j } have a non-zero mean. We further find quantitative estimates on the smallest singular value of the adjacency matrix of a directed Erdős–Réyni graph whenever its edge connectivity probability is above the critical threshold Ω ( log n n ) .

Authors:Piotr Budzyński; Zenon Jan Jabłoński; Il Bong Jung; Jan Stochel Pages: 484 - 556 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Piotr Budzyński, Zenon Jan Jabłoński, Il Bong Jung, Jan Stochel A recent example of a non-hyponormal injective composition operator in an L 2 -space generating Stieltjes moment sequences, invented by three of the present authors, was built over a non-locally finite directed tree. The main goal of this paper is to solve the problem of whether there exists such an operator over a locally finite directed graph and, in the affirmative case, to find the simplest possible graph with these properties (simplicity refers to local valency). The problem is solved affirmatively for the locally finite directed graph G 2 , 0 , which consists of two branches and one loop. The only simpler directed graph for which the problem remains unsolved consists of one branch and one loop. The consistency condition, the only efficient tool for verifying subnormality of unbounded composition operators, is intensively studied in the context of G 2 , 0 , which leads to a constructive method of solving the problem. The method itself is partly based on transforming the Krein and the Friedrichs measures coming either from shifted Al-Salam–Carlitz q-polynomials or from a quartic birth and death process.

Authors:Sergio Estrada; Simone Virili Pages: 557 - 609 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Sergio Estrada, Simone Virili We introduce the new concept of cartesian module over a pseudofunctor R from a small category to the category of small preadditive categories. Already the case when R is a (strict) functor taking values in the category of commutative rings is sufficient to cover the classical construction of quasi-coherent sheaves of modules over a scheme. On the other hand, our general setting allows for a good theory of contravariant additive locally flat functors, providing a geometrically meaningful extension of a classical Representation Theorem of Makkai and Paré. As an application, we relate and extend some previous constructions of the pure derived category of a scheme.

Authors:Mohamed Saïdi; Akio Tamagawa Pages: 610 - 662 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Mohamed Saïdi, Akio Tamagawa In this paper we prove a refined version of Uchida's theorem on isomorphisms between absolute Galois groups of global fields in positive characteristics, where one “ignores” the information provided by a “small” set of primes.

Authors:Daniele Faenzi; Francesco Malaspina Pages: 663 - 695 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Daniele Faenzi, Francesco Malaspina We provide two examples of smooth projective surfaces of tame CM type, by showing that the parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in P 5 is either a single point or a projective line. These turn out to be the only smooth projective ACM varieties of tame CM type besides elliptic curves, [1]. For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For F 0 and F 1 , embedded as quintic or sextic scrolls, a complete classification of rigid ACM bundles is given.

Authors:Jiahong Wu; Yifei Wu Pages: 759 - 888 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Jiahong Wu, Yifei Wu This paper establishes the global existence and uniqueness of smooth solutions to the two-dimensional compressible magnetohydrodynamic system when the initial data is close to an equilibrium state. In addition, explicit large-time decay rates for various Sobolev norms of the solutions are also obtained. These results are achieved through a new approach of diagonalizing a system of coupled linearized equations. The standard method of diagonalization via the eigenvalues and eigenvectors of the matrix symbol is very difficult to implement here. This new process allows us to obtain an integral representation of the full system through explicit kernels. In addition, in order to overcome various difficulties such as the anisotropicity and criticality, we fully exploit the structure of the integral representation and employ extremely delicate Fourier analysis and associated estimates.

Authors:Yanir A. Rubinstein; Jake P. Solomon Pages: 889 - 939 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Yanir A. Rubinstein, Jake P. Solomon This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in C n . Existence of geodesics in the space of positive Lagrangians is an important step in a program for proving existence and uniqueness of special Lagrangians. Moreover, it would imply certain cases of the strong Arnold conjecture from Hamiltonian dynamics. We show the DSL is degenerate elliptic. We introduce a space–time Lagrangian angle for one-parameter families of graph Lagrangians, and construct its regularized lift. The superlevel sets of the regularized lift define subequations for the DSL in the sense of Harvey–Lawson. We extend the existence theory of Harvey–Lawson for subequations to the setting of domains with corners, and thus obtain solutions to the Dirichlet problem for the DSL in all branches. Moreover, we introduce the calibration measure, which plays a rôle similar to that of the Monge–Ampère measure in convex and complex geometry. The existence of this measure and regularity estimates allow us to prove that the solutions we obtain in the outer branches of the DSL have a well-defined length in the space of positive Lagrangians.

Authors:Mário J. Edmundo; Marcello Mamino; Luca Prelli; Janak Ramakrishnan; Giuseppina Terzo Pages: 940 - 992 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Mário J. Edmundo, Marcello Mamino, Luca Prelli, Janak Ramakrishnan, Giuseppina Terzo Let M be an arbitrary o-minimal structure. Let G be a definably compact, definably connected, abelian definable group of dimension n. Here we compute: (i) the new intrinsic o-minimal fundamental group of G; (ii) for each k > 0 , the k-torsion subgroups of G; (iii) the o-minimal cohomology algebra over Q of G. As a corollary we obtain a new uniform proof of Pillay's conjecture, an o-minimal analogue of Hilbert's fifth problem, relating definably compact groups to compact real Lie groups, extending the proof already known in o-minimal expansions of ordered fields.

Authors:Deb Kumar Giri; R.K. Srivastava Pages: 993 - 1016 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Deb Kumar Giri, R.K. Srivastava A Heisenberg uniqueness pair is a pair ( Γ , Λ ) , where Γ is a curve and Λ is a set in R 2 such that whenever a finite Borel measure μ having support on Γ which is absolutely continuous with respect to the arc length on Γ satisfies μ ˆ Λ = 0 , then it is identically 0. In this article, we investigate the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and certain exponential curves. Further, we work out a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. In the latter case, we observe a phenomenon of interlacing of three trigonometric polynomials.

Authors:Robert M. Guralnick; Attila Maróti; László Pyber Pages: 1017 - 1063 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Robert M. Guralnick, Attila Maróti, László Pyber Let G be a transitive normal subgroup of a permutation group A of finite degree n. The factor group A / G can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that A / G < n if G is primitive unless n = 3 4 , 54, 38, 58, or 316. This bound is sharp when n is prime. In fact, when G is primitive, Out ( G ) < n unless G is a member of a given infinite sequence of primitive groups and n is different from the previously listed integers. Many other results of this flavor are established not only for permutation groups but also for linear groups and Galois groups.

Authors:Jiarui Fei Pages: 1064 - 1112 Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): Jiarui Fei We relate the m-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when m = 2 . Each g -vector cone G ◇ l of these cluster algebras controls the 2-truncated Kronecker products for all symmetric functions of degree no greater than l. As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all G ◇ l 's. As an application, we compute some invariant rings.

Authors:Emanuele Dotto Pages: 1 - 96 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Emanuele Dotto We develop a theory of Goodwillie calculus for functors between G-equivariant homotopy theories, where G is a finite group. We construct J-excisive approximations for any finite G-set J. These combine into a poset, the Goodwillie tree, that extends the classical Goodwillie tower. We prove convergence results for the tree of a functor on pointed G-spaces that commutes with fixed-points, and we reinterpret the Tom Dieck-splitting as an instance of a more general splitting phenomenon that occurs for the fixed-points of the equivariant derivatives of these functors. As our main example we describe the layers of the tree of the identity functor in terms of the equivariant Spanier–Whitehead duals of the partition complexes.

Authors:Marc Hoyois; Sarah Scherotzke; Nicolò Sibilla Pages: 97 - 154 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Marc Hoyois, Sarah Scherotzke, Nicolò Sibilla We propose a categorification of the Chern character that refines earlier work of Toën and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X, which we introduce, to S 1 -equivariant perfect complexes on the derived free loop stack L X . As an application of the theory, we show that Toën and Vezzosi's secondary Chern character factors through secondary K-theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal ( ∞ , n ) -categories, which is of independent interest.

Authors:Karl Liechty; Dong Wang Pages: 155 - 208 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Karl Liechty, Dong Wang We study a model of nonintersecting Brownian bridges on an interval with either absorbing or reflecting walls at the boundaries, focusing on the point in space-time at which the particles meet the wall. These processes are determinantal, and in different scaling limits when the particles approach the reflecting (resp. absorbing) walls we obtain hard-edge limiting kernels which are the even (resp. odd) parts of the Pearcey and tacnode kernels. We also show that in the single time case, our hard-edge tacnode kernels are equivalent to the ones studied by Delvaux [16], defined in terms of a 4 × 4 Lax pair for the inhomogeneous Painlevé II equation (PII). As a technical ingredient in the proof, we construct a Schlesinger transform for the 4 × 4 Lax pair in [16] which preserves the Hastings–McLeod solutions to PII.

Authors:Ronald Cramer; Chaoping Xing Pages: 238 - 253 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Ronald Cramer, Chaoping Xing The Hasse–Weil bound is a deep result in mathematics and has found wide applications in mathematics, theoretical computer science, information theory etc. In general, the bound is tight and cannot be improved. However, for some special families of curves the bound could be improved substantially. In this paper, we focus on the Hasse–Weil bound for the curve defined by y p − y = f ( x ) over the finite field F q , where p is the characteristic of F q . In 1993, Moreno–Moreno [7] gave an improvement to the Hasse–Weil bound for this family of curves. Recently, Kaufman and Lovett [4, FOCS2011] showed that the Hasse–Weil bound can be improved for this family of curves with f ( x ) = g ( x ) + h ( x ) , where g ( x ) is a polynomial of degree ≪ q and h ( x ) is a sparse polynomial of arbitrary degree but bounded weight degree. The other recent improvement by Rojas-Leon and Wan [9, Math. Ann. 2011] shows that an extra p can be removed for this family of curves if p is very large compared with polynomial degree of f ( x ) and log p q . In this paper, we focus on the most interesting case for applications, namely p = 2 . We show that the Hasse–Weil bound for this special family of curves can be improved if q = 2 n with odd n ⩾ 3 which is the same case where Serre [10] improved the Hasse–Weil bound. However, our improvement is greater than Serre's and Moreno–Morenao's improvements for this special family of curves. Furthermore, our improvement works for p = 2 compared with the requirement of large p by Rojas-Leon and Wan. In addition, our improvement finds interesting applications to character sums, cryptography and coding theory. The key idea behind is that this curve has the Hasse–Witt invariant 0 and we show that the Hasse–Weil bound can be improved for any curves with the Hasse–Witt invariant 0. The main tool used in our proof involves Newton polygon and some results in algebraic geometry.

Authors:Fang Wang Pages: 306 - 333 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Fang Wang The scattering operators associated to an ACHE metric of Bergman type on a strictly pseudoconvex domain are a one-parameter family of CR-conformally invariant pseudo-differential operators of Heisenberg class with respect to the induced CR structure on the boundary. In this paper, we mainly show that if the boundary Webster scalar curvature is positive, then for γ ∈ ( 0 , 1 ) the renormalised scattering operator P 2 γ has positive spectrum and satisfies the maximum principal; moreover, the fractional curvature Q 2 γ is also positive. This is parallel to the result of Guillarmou–Qing [16] for the real case. We also give two energy extension formulae for P 2 γ , which are parallel to the energy extension given by Chang–Case [2] for the real case.

Authors:Klaus Thomsen Pages: 334 - 391 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Klaus Thomsen The paper contains a study of the gauge invariant KMS weights for a generalized gauge action on a graph C ⁎ -algebra. When the graph is irreducible and has the property that there are at most countably many roads to infinity in the graph, a complete description is given of the structure of KMS weights for the gauge action. The structure is very rich and is identical with the structure of KMS states for the restriction of the action to any corner defined by a projection in the fixed point algebra.

Authors:Irakli Patchkoria Pages: 392 - 435 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Irakli Patchkoria We prove that for an odd prime p, the derived category D ( K U ( p ) ) of the p-local complex periodic K-theory spectrum K U ( p ) is triangulated equivalent to the derived category of its homotopy ring π ⁎ K U ( p ) . This implies that if p is an odd prime, the triangulated category D ( K U ( p ) ) is algebraic.

Authors:Kathrin Bringmann; Ben Kane; Daniel Parry; Robert Rhoades Pages: 436 - 451 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Kathrin Bringmann, Ben Kane, Daniel Parry, Robert Rhoades In this paper, we establish asymptotics of radial limits for certain functions of Wright. These functions appear in bootstrap percolation and the generating function for partitions without sequences of k consecutive part sizes. We specifically establish asymptotics numerically obtained by Zagier in the case k = 3 .

Authors:Alberto Enciso; Renato Lucà; Daniel Peralta-Salas Pages: 452 - 486 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Alberto Enciso, Renato Lucà, Daniel Peralta-Salas We prove that the vortex structures of solutions to the 3D Navier–Stokes equations can change their topology without any loss of regularity. More precisely, we construct smooth high-frequency solutions to the Navier–Stokes equations where vortex lines and vortex tubes of arbitrarily complicated topologies are created and destroyed in arbitrarily small times. This instance of vortex reconnection is structurally stable and in perfect agreement with the existing computer simulations and experiments. We also provide a (non-structurally stable) scenario where the destruction of vortex structures is instantaneous.

Authors:Eli Aljadeff; Geoffrey Janssens; Yakov Karasik Pages: 487 - 511 Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Eli Aljadeff, Geoffrey Janssens, Yakov Karasik Let F be a field of characteristic zero and W an associative affine F-algebra satisfying a polynomial identity (PI). The codimension sequence { c n ( W ) } associated to W is known to be of the form Θ ( n t d n ) , where d is the well known PI-exponent of W. In this paper we establish an algebraic interpretation of the polynomial part (the constant t) by means of Kemer's theory. In particular, we show that in case W is a basic algebra (hence finite dimensional), t = q − d 2 + s , where q is the number of simple component in W / J ( W ) and s + 1 is the nilpotency degree of J ( W ) (the Jacobson radical of W). Thus proving a conjecture of Giambruno.

Authors:Plebanek Abstract: Publication date: 13 April 2017 Source:Advances in Mathematics, Volume 310 Author(s): A. Avilés, G. Plebanek, J. Rodríguez We study the lattice structure of the family of weakly compact subsets of the unit ball B X of a separable Banach space X, equipped with the inclusion relation (this structure is denoted by K ( B X ) ) and also with the parametrized family of “almost inclusion” relations K ⊆ L + ε B X , where ε > 0 (this structure is denoted by AK ( B X ) ). Tukey equivalence between partially ordered sets and a suitable extension to deal with AK ( B X ) are used. Assuming the axiom of analytic determinacy, we prove that separable Banach spaces fall into four categories, namely: K ( B X ) is equivalent either to a singleton, or to ω ω , or to the family K ( Q ) of compact subsets of the rational numbers, or to the family [ c ] < ω of all finite subsets of the continuum. Also under the axiom of analytic determinacy, a similar classification of AK ( B X ) is obtained. For separable Banach spaces not containing ℓ 1 , we prove in ZFC that K ( B X ) ∼ AK ( B X ) are equivalent to either {0}, ω ω , K ( Q ) or [ c ] < ω . The lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied.

Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Ergün Yalçın Let G be a finite p-group and k be a field of characteristic p. A topological space X is called an n-Moore space if its reduced homology is nonzero only in dimension n. We call a G-CW-complex X an n _ -Moore G-space over k if for every subgroup H of G, the fixed point set X H is an n _ ( H ) -Moore space with coefficients in k, where n _ ( H ) is a function of H. We show that if X is a finite n _ -Moore G-space, then the reduced homology module of X is an endo-permutation kG-module generated by relative syzygies. A kG-module M is an endo-permutation module if End k ( M ) = M ⊗ k M ⁎ is a permutation kG-module. We consider the Grothendieck group of finite Moore G-spaces M ( G ) , with addition given by the join operation, and relate this group to the Dade group generated by relative syzygies.

Authors:Adam Abstract: Publication date: 17 March 2017 Source:Advances in Mathematics, Volume 309 Author(s): Adam Parusiński, Laurenţiu Păunescu In this paper we show Whitney's fibering conjecture in the real and complex, local analytic and global algebraic cases. For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of the Zariski equisingularity, we show the existence of Whitney's stratified fibration, satisfying the conditions (b) of Whitney and (w) of Verdier. Our construction is based on the Puiseux with parameter theorem and a generalization of Whitney's interpolation. For algebraic sets our construction gives a global stratification. We also present several applications of the arc-wise analytic trivialization, mainly to the stratification theory and the equisingularity of analytic set and function germs. In the real algebraic case, for an algebraic family of projective varieties, we show that the Zariski equisingularity implies local constancy of the associated weight filtration.