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:Dipendra Prasad Pages: 198 - 208 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Dipendra Prasad We construct examples of number fields which are not isomorphic but for which their adele groups, the idele groups, and the idele class groups are isomorphic. We also construct examples of projective algebraic curves which are not isomorphic but for which their Jacobian varieties are isomorphic. Both are constructed using an example in group theory provided by Leonard Scott of a finite group G and subgroups H 1 and H 2 which are not conjugate in G but for which the G-module Z [ G / H 1 ] is isomorphic to Z [ G / H 2 ] .

Authors:Jacek Brodzki; Chris Cave; Kang Li Pages: 209 - 233 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Jacek Brodzki, Chris Cave, Kang Li We give some new characterizations of exactness for locally compact second countable groups. In particular, we prove that a locally compact second countable group is exact if and only if it admits a topologically amenable action on a compact Hausdorff space. This answers an open question by Anantharaman-Delaroche.

Authors:Karsten Bohlen Pages: 234 - 285 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Karsten Bohlen We introduce and study a general pseudodifferential calculus for boundary value problems on a class of non-compact manifolds with boundary (so-called Lie manifolds with boundary). This is accomplished by constructing a suitable generalization of the Boutet de Monvel calculus for boundary value problems. The data consists of a compact manifold with corners M that is endowed with a Lie structure of vector fields 2 V , a so-called Lie manifold. The manifold M is split into two equal parts X + and X − which intersect in an embedded hypersurface Y ⊂ X ± . Our goal is to describe a transmission Boutet de Monvel calculus for boundary value problems compatible with the structure of Lie manifolds. Starting with the example of b-vector fields, we show that there are two groupoids integrating the Lie structures on M and on Y, respectively. These two groupoids form a bibundle (or a groupoid correspondence) and, under some mild assumptions, these groupoids are Morita equivalent. With the help of the bibundle structure and canonically defined manifolds with corners, which are blow-ups in particular cases, we define a class of Boutet de Monvel type operators. We then define the representation homomorphism for these operators and show closedness under composition with the help of a representation theorem. Finally, we consider appropriate Fredholm conditions and construct the parametrices for elliptic operators in the calculus.

Authors:Jaume Martí-Farré; Anna de Mier Pages: 286 - 314 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Jaume Martí-Farré, Anna de Mier A clutter is a family of mutually incomparable sets. The set of circuits of a matroid, its set of bases, and its set of hyperplanes are examples of clutters arising from matroids. In this paper we address the question of determining which are the matroidal clutters that best approximate an arbitrary clutter Λ. For this, we first define two orders under which to compare clutters, which give a total of four possibilities for approximating Λ (i.e., above or below with respect to each order); in fact, we actually consider the problem of approximating Λ with clutters from any collection of clutters Σ, not necessarily arising from matroids. We show that, under some mild conditions, there is a finite non-empty set of clutters from Σ that are the closest to Λ and, moreover, that Λ is uniquely determined by them, in the sense that it can be recovered using a suitable clutter operation. We then particularize these results to the case where Σ is a collection of matroidal clutters and give algorithmic procedures to compute these clutters.

Authors:Renjun Duan; Hongjun Yu Pages: 315 - 373 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Renjun Duan, Hongjun Yu The paper concerns the Cauchy problem on the relativistic Boltzmann equation for soft potentials in a periodic box. We show that the global-in-time solutions around relativistic Maxwellians exist in the weighted L ∞ perturbation framework and also approach equilibrium states in large time in the weighted L 2 framework at the rate of exp ( − λ t β ) for some λ > 0 and β ∈ ( 0 , 1 ) . The proof is based on the nonlinear L 2 energy method and nonlinear L ∞ pointwise estimate with appropriate exponential weights in momentum. The results extend those on the classical Boltzmann equation by Caflisch [2,3] and Strain and Guo [31] to the relativistic version, and also improve the recent result on almost exponential time-decay by Strain [28] to the exponential rate. Moreover, we study the propagation of spatial regularity for the obtained solutions and also the large time behavior in the corresponding regular Sobolev space, provided that the spatial derivatives of initial data are bounded, not necessarily small.

Authors:Zeng Lian; Yi Wang Pages: 374 - 424 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Zeng Lian, Yi Wang For linear random dynamical systems in a separable Banach space X, we derived a series of Krein–Rutman type Theorems with respect to co-invariant cone family with rank-k, which present a (quasi)-equivalence relation between the measurably co-invariant cone family and the measurably dominated splitting of X. Moreover, such (quasi)-equivalence relation turns out to be an equivalence relation whenever (i) k = 1 ; or (ii) in the frame of the Multiplicative Ergodic Theorem with certain Lyapunov exponent being greater than the negative infinity. For the second case, we thoroughly investigated the relations between the Lyapunov exponents, the co-invariant cone family and the measurably dominated splitting for linear random dynamical systems in X.

Authors:Avinash J. Dalal; Jennifer Morse Pages: 425 - 458 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Avinash J. Dalal, Jennifer Morse We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on the type-A affine Weyl group. We construct two one-parameter families of functions that respectively transition positively with Hall–Littlewood and Macdonald's P-functions, and specialize to the representatives for Schubert classes of homology and cohomology of the affine Grassmannian. Our approach leads us to conjecture that all elements in a defining set of 3-point genus 0 Gromov–Witten invariants for flag manifolds can be formulated as strong covers.

Authors:Dmitrii V. Pasechnik Pages: 459 - 472 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Dmitrii V. Pasechnik We augment the list of finite universal locally toroidal regular polytopes of type { 3 , 3 , 4 , 3 , 3 } due to P. McMullen and E. Schulte, adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal polytope is related to a well-known Y-shaped presentation for the sporadic simple group Fi 22 , and admits S 4 × O 8 + ( 2 ) : S 3 as the automorphism group. We also discuss further extensions of its quotients in the context of Y-shaped presentations. As well, we note that two known examples of finite universal polytopes of type { 3 , 3 , 4 , 3 , 3 } are related to Y-shaped presentations of orthogonal groups over F 2 . Mixing construction is used in a number of places to describe covers and 2-covers.

Authors:M. Junge; F. Sukochev; D. Zanin Pages: 473 - 546 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): M. Junge, F. Sukochev, D. Zanin Let L ( H ) be the ⁎-algebra of all bounded operators on an infinite dimensional Hilbert space H and let ( I , ‖ ⋅ ‖ I ) be an ideal in L ( H ) equipped with a Banach norm which is distinct from the Schatten–von Neumann ideal L p ( H ) , 1 ≤ p < 2 . We prove that I isomorphically embeds into an L p -space L p ( R ) , 1 ≤ p < 2 (here, R is the hyperfinite II1-factor) if its commutative core (that is, Calkin space for I ) isomorphically embeds into L p ( 0 , 1 ) . Furthermore, we prove that an Orlicz ideal L M ( H ) ≠ L p ( H ) isomorphically embeds into L p ( R ) , 1 ≤ p < 2 , if and only if it is an interpolation space for the Banach couple ( L p ( H ) , L 2 ( H ) ) . Finally, we consider isomorphic embeddings of ( I , ‖ ⋅ ‖ I ) into L p -spaces associated with arbitrary finite von Neumann algebras.

Authors:Paul M.N. Feehan Pages: 547 - 587 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Paul M.N. Feehan We prove an L d / 2 energy gap result for Yang–Mills connections on principal G-bundles, P, over arbitrary, closed, Riemannian, smooth manifolds of dimension d ≥ 2 . We apply our version of the Łojasiewicz–Simon gradient inequality [16,19] to remove a positivity constraint on a combination of the Ricci and Riemannian curvatures in a previous L d / 2 -energy gap result due to Gerhardt [23, Theorem 1.2] and a previous L ∞ -energy gap result due to Bourguignon, Lawson, and Simons [10, Theorem C], [11, Theorem 5.3], as well as an L 2 -energy gap result due to Nakajima [42, Corollary 1.2] for a Yang–Mills connection over the sphere, S d , but with an arbitrary Riemannian metric.

Authors:Daniel Chan; Boris Lerner Pages: 588 - 635 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Daniel Chan, Boris Lerner We introduce a new moduli stack, called the Serre stable moduli stack, which corresponds to studying families of point objects in an abelian category with a Serre functor. This allows us in particular, to re-interpret the classical derived equivalence between most concealed-canonical algebras and weighted projective lines by showing they are induced by the universal sheaf on the Serre stable moduli stack. We explain why the method works by showing that the Serre stable moduli stack is the tautological moduli problem that allows one to recover certain nice stacks such as weighted projective lines from their moduli of sheaves. As a result, this new stack should be of interest in both representation theory and algebraic geometry.

Authors:Zili Zhang Pages: 636 - 679 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Zili Zhang Let S → C be a smooth projective surface with numerically trivial canonical bundle fibered onto a curve. We prove the multiplicativity of the perverse filtration with respect to the cup product on H ⁎ ( S [ n ] , Q ) for the natural morphism S [ n ] → C ( n ) . We also prove the multiplicativity for five families of Hitchin systems obtained in a similar way and compute the perverse numbers of the Hitchin moduli spaces. We show that for small values of n the perverse numbers match the predictions of the numerical version of the de Cataldo–Hausel–Migliorini P = W conjecture and of the conjecture by Hausel, Letellier and Rodriguez-Villegas.

Authors:Guido Pezzini Pages: 680 - 736 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): Guido Pezzini We define and study spherical subgroups of finite type of a Kac–Moody group. In analogy with the standard theory of spherical varieties, we introduce a combinatorial object associated with such a subgroup, its homogeneous spherical datum, and we prove that it satisfies the same axioms as in the finite-dimensional case. Our main tool is a study of varieties that are spherical under the action of a connected reductive group L, and come equipped with a transitive action of a group containing L as a Levi subgroup.

Authors:I. Akbarbaglu; S. Gła̧b; S. Maghsoudi; F. Strobin Pages: 737 - 763 Abstract: Publication date: 25 May 2017 Source:Advances in Mathematics, Volume 312 Author(s): I. Akbarbaglu, S. Gła̧b, S. Maghsoudi, F. Strobin For i = 1 , 2 , 3 , let φ i be Young functions, ( Ω , μ ) a (topological) measure space, E an ideal of μ-measurable complex-valued functions defined on Ω and E φ i be the corresponding Calderón–Lozanowskiĭ space. Our aim in this paper is to give, under mild conditions, several results on topological size (in the sense of Baire) of the sets { ( f , g ) ∈ E φ 1 × E φ 2 : f ⨀ g ∈ E φ 3 } and { ( f , g ) ∈ E φ 1 × E φ 2 : ∃ x ∈ V , ( f ⨀ g ) ( x ) is well defined } where ⨀ denotes the convolution or pointwise product of functions and V a compact neighborhood. Our results sharpen and unify the related results obtained in diverse areas during recent thirty years.

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:Willem Veys; W. A. Zúñiga-Galindo Pages: 295 - 337 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Willem Veys, W. A. Zúñiga-Galindo In the 70's Igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero. In the present article this theory is extended to the case of rational functions, or, more generally, meromorphic functions f / g , with coefficients in a local field of characteristic zero. This generalization is far from being straightforward due to the fact that several new geometric phenomena appear. Also, the oscillatory integrals have two different asymptotic expansions: the ‘usual’ one when the norm of the parameter tends to infinity, and another one when the norm of the parameter tends to zero. The first asymptotic expansion is controlled by the poles (with negative real parts) of all the twisted local zeta functions associated to the meromorphic functions f / g − c , for certain special values c. The second expansion is controlled by the poles (with positive real parts) of all the twisted local zeta functions associated to f / g .

Authors:Cho-Ho Chu; Michael Rigby Pages: 338 - 377 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Cho-Ho Chu, Michael Rigby Given a fixed-point free compact holomorphic self-map f on a bounded symmetric domain D, which may be infinite dimensional, we establish the existence of a family { H ( ξ , λ ) } λ > 0 of convex f-invariant domains at a point ξ in the boundary ∂D of D, which generalises completely Wolff's theorem for the open unit disc in C . Further, we construct horoballs at ξ and show that they are exactly the f-invariant domains when D is of finite rank. Consequently, we show in the latter case that the limit functions of the iterates ( f n ) with weakly closed range all accumulate in one single boundary component of ∂D.

Authors:Ganna Kudryavtseva; Mark V. Lawson Pages: 378 - 468 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Ganna Kudryavtseva, Mark V. Lawson This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by étale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We prove several main results. To start with, we establish a duality between the category of complete restriction monoids and the category of étale localic categories. The relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames. This result builds on the work of Pedro Resende on the connection between pseudogroups and étale localic groupoids but in the process we both generalize and simplify: for example, we do not require involutions and, in addition, we render his result functorial. A wider class of quantales, called multiplicative Ehresmann quantal frames, is put into a correspondence with those localic categories where the multiplication structure map is semiopen, and all the other structure maps are open. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction between étale localic categories and étale topological categories. In fact, varying morphisms, we obtain several adjunctions. Just as in the commutative case, we restrict these adjunctions to spatial-sober and coherent-spectral equivalences. The classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups. Consequently, we deduce several dualities between distributive restriction semigroups and spectral étale topological categories. We also specialize these dualities for the setting where the topological categories are cancellative or are groupoids. Our approach thus links, unifies and extends the approaches taken in the work by Lawson and Lenz and by Resende.

Authors:ChiaKuei Peng; Chao Qian Pages: 469 - 480 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): ChiaKuei Peng, Chao Qian Given a closed manifold M and a vector bundle ξ of rank n over M, by gluing two copies of the disc bundle of ξ, we can obtain a closed manifold D ( ξ , M ) , the so-called double manifold. In this paper, we firstly prove that each sphere bundle S r ( ξ ) of radius r > 0 is an isoparametric hypersurface in the total space of ξ equipped with a connection metric, and for r > 0 small enough, the induced metric of S r ( ξ ) has positive Ricci curvature under the additional assumptions that M has a metric with positive Ricci curvature and n ≥ 3 . As an application, if M admits a metric with positive Ricci curvature and n ≥ 2 , then we construct a metric with positive Ricci curvature on D ( ξ , M ) . Moreover, under the same metric, D ( ξ , M ) admits a natural isoparametric foliation. For a compact minimal isoparametric hypersurface Y n in S n + 1 ( 1 ) , which separates S n + 1 ( 1 ) into S + n + 1 and S − n + 1 , one can get double manifolds D ( S + n + 1 ) and D ( S − n + 1 ) . Inspired by Tang, Xie and Yan's work on scalar curvature of such manifolds with isoparametric foliations (cf. [25]), we study Ricci curvature of them with isoparametric foliations in the last part.

Authors:Anna Skripka Pages: 481 - 509 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Anna Skripka We prove estimates for Schatten norms of multiple operator integrals with symbols arising from Fourier series and, as a consequence, derive higher order trace formulas for functions given by absolutely convergent Fourier series of unitary operators and for rational functions bounded on the real line of resolvent comparable self-adjoint operators. These results substantially extend the main results of [12,14,13], where the estimates were proved for multiple operator integrals with symbols arising from polynomials, the trace formulas in the unitary case for analytic functions on the unit disc, and the trace formulas in the resolvent comparable self-adjoint case for rational functions bounded and analytic in the lower half-plane. Our results rely crucially on an algebraic approach to symbols of multiple operator integrals developed in this paper to replace the prior restrictive analytic approach of [12].

Authors:Assaf Rinot Pages: 510 - 531 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Assaf Rinot It is proved that for every uncountable cardinal λ, GCH + □ ( λ + ) entails the existence of a cf ( λ ) -complete λ + -Souslin tree. In particular, if GCH holds and there are no ℵ 2 -Souslin trees, then ℵ 2 is weakly compact in Gödel's constructible universe, improving Gregory's 1976 lower bound. Furthermore, it follows that if GCH holds and there are no ℵ 2 and ℵ 3 Souslin trees, then the Axiom of Determinacy holds in L ( R ) .

Authors:Chul-hee Lee Pages: 532 - 568 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Chul-hee Lee We study properties of solutions of Q-systems in the WZW fusion ring obtained by the Kirillov–Reshetikhin modules. We make a conjecture about their positivity and periodicity and give a proof of it in some cases. We also construct a positive solution of the level k restricted Q-system of classical types in the fusion rings. As an application, we prove some conjectures of Kirillov and Kuniba–Nakanishi–Suzuki on the level k restricted Q-systems.

Authors:Fritz Gesztesy; Maxim Zinchenko Pages: 569 - 597 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Fritz Gesztesy, Maxim Zinchenko We extend a result on renormalized oscillation theory, originally derived for Sturm–Liouville and Dirac-type operators on arbitrary intervals in the context of scalar coefficients, to the case of general Hamiltonian systems with block matrix coefficients. In particular, this contains the cases of general Sturm–Liouville and Dirac-type operators with block matrix-valued coefficients as special cases. The principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate Wronskians of solutions, oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails.

Authors:Grégory Chatel; Vincent Pilaud Pages: 598 - 633 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Grégory Chatel, Vincent Pilaud Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the classical conditions for binary search trees. Similar to binary trees for the Tamari lattice, Cambrian trees provide convenient combinatorial models for N. Reading's Cambrian lattices of type A. Based on a natural surjection from signed permutations to Cambrian trees, we define the Cambrian Hopf algebra extending J.-L. Loday and M. Ronco's algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. We then construct the Baxter–Cambrian algebra which extends S. Law and N. Reading's Baxter Hopf algebra on rectangulations and S. Giraudo's equivalent Hopf algebra on twin binary trees.

Authors:Kyle Hambrook Pages: 634 - 648 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Kyle Hambrook We construct explicit (i.e., non-random) examples of Salem sets in R 2 of dimension s for every 0 ≤ s ≤ 2 . In particular, we give the first explicit examples of Salem sets in R 2 of dimension 0 < s < 1 . This extends a theorem of Kaufman.

Authors:Young-Hoon Kiem; In-Kyun Kim; Hwayoung Lee; Kyoung-Seog Lee Pages: 649 - 661 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Young-Hoon Kiem, In-Kyun Kim, Hwayoung Lee, Kyoung-Seog Lee We prove that the derived category of a smooth complete intersection variety is equivalent to a full subcategory of the derived category of a smooth projective Fano variety. This enables us to define a new invariant of smooth projective varieties and raise many interesting questions.

Authors:Salim Rostam Pages: 662 - 729 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Salim Rostam We prove that cyclotomic Yokonuma–Hecke algebras of type A are cyclotomic quiver Hecke algebras and we give an explicit isomorphism with its inverse, using a similar result of Brundan and Kleshchev on cyclotomic Hecke algebras. The quiver we use is given by disjoint copies of cyclic quivers. We relate this work to an isomorphism of Lusztig.

Authors:Ian J. Leary; Nansen Petrosyan Pages: 730 - 747 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Ian J. Leary, Nansen Petrosyan We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for E _ G , the classifying space for proper actions of G. They are the first examples that have these properties and also admit cocompact models for E _ G . We exhibit groups G whose virtual cohomological dimension and Bredon cohomological dimension are two that do not admit any 2-dimensional contractible proper G-CW-complex.

Authors:Yash Jhaveri; Robin Neumayer Pages: 748 - 795 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Yash Jhaveri, Robin Neumayer We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack estimate.

Authors:Joseph H.G. Fu; Dušan Pokorný; Jan Rataj Pages: 796 - 832 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Joseph H.G. Fu, Dušan Pokorný, Jan Rataj The class WDC ( M ) consists of all subsets of a smooth manifold M that may be expressed in local coordinates as sufficiently regular sublevel sets of DC (differences of convex) functions. If M is Riemannian and G is a group of isometries acting transitively on the sphere bundle SM, we define the invariant curvature measures of compact WDC subsets of M, and show that pairs of such subsets are subject to the array of kinematic formulas known to apply to smoother sets. Restricting to the case ( M , G ) = ( R d , S O ( d ) ‾ ) , this extends and subsumes Federer's theory of sets with positive reach in an essential way. The key technical point is equivalent to a sharpening of a classical theorem of Ewald, Larman, and Rogers characterizing the dimension of the set of directions of line segments lying in the boundary of a given convex body.

Authors:Alexander Premet Pages: 833 - 884 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Alexander Premet Let G be a simple algebraic group over an algebraically closed field of characteristic p > 0 and suppose that p is a very good prime for G. In this paper we prove that any maximal Lie subalgebra M of g = Lie ( G ) with rad ( M ) ≠ 0 has the form M = Lie ( P ) for some maximal parabolic subgroup P of G. This means that Morozov's theorem on maximal subalgebras is valid under mild assumptions on G. We show that such assumptions are necessary by providing a counterexample to Morozov's theorem for groups of type E 8 over fields of characteristic 5. Our proof relies on the main results and methods of the classification theory of finite dimensional simple Lie algebras over fields of prime characteristic.

Authors:Alexander I. Bobenko; Felix Günther Pages: 885 - 932 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Alexander I. Bobenko, Felix Günther Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian differentials. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann–Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann–Roch Theorem, and a discrete Abel–Jacobi map.

Authors:Christopher M. Drupieski Pages: 935 - 937 Abstract: Publication date: 30 April 2017 Source:Advances in Mathematics, Volume 311 Author(s): Christopher M. Drupieski The main theorem of the author's article Drupieski (2016) [2] is proved by way of verifying Conjecture 5.4.1 of the author's earlier work Drupieski (2013) [1]. However, the aforementioned conjecture does not make sense as written. The purpose of this note is to give a formulation of the conjecture that does make sense, and then to describe the required modifications to its proof.