Authors:Jan Kohlhaase Pages: 1 - 49 Abstract: Publication date: 7 September 2017 Source:Advances in Mathematics, Volume 317 Author(s): Jan Kohlhaase We develop a duality theory for admissible smooth representations of p-adic Lie groups on vector spaces over fields of characteristic p. To this end we introduce certain higher smooth duality functors and relate our construction to the Auslander duality of completed group rings. We study the behavior of smooth duality under tensor products, inflation and induction, and discuss the dimension theory of smooth mod-p representations of a p-adic reductive group. Finally, we compute the higher smooth duals of the irreducible smooth representations of GL 2 ( Q p ) in characteristic p and relate our results to the contragredient operation of Colmez.

Authors:Gufang Zhao; Changlong Zhong Pages: 50 - 90 Abstract: Publication date: 7 September 2017 Source:Advances in Mathematics, Volume 317 Author(s): Gufang Zhao, Changlong Zhong For any formal group law, there is a formal affine Hecke algebra defined by Hoffnung–Malagón-López–Savage–Zainoulline. Coming from this formal group law, there is also an oriented cohomology theory. We identify the formal affine Hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the Steinberg variety. As a consequence, this algebra acts on the corresponding cohomology of the Springer fibers. This generalizes the action of classical affine Hecke algebra on the K-theory of the Springer fibers constructed by Lusztig. We also give a residue interpretation of the formal affine Hecke algebra, which generalizes the residue construction of Ginzburg–Kapranov–Vasserot when the formal group law comes from a 1-dimensional algebraic group.

Authors:Dominic Descombes; Maël Pavón Pages: 91 - 107 Abstract: Publication date: 7 September 2017 Source:Advances in Mathematics, Volume 317 Author(s): Dominic Descombes, Maël Pavón We prove an explicit characterization of all injective subsets of the model space l ∞ ( I ) of all bounded real-valued functions defined on a non-empty set I, endowed with the supremum norm. Since the class of all injective metric spaces coincides with the one of all absolute 1-Lipschitz retracts, the present work yields a characterization of all the subsets of l ∞ ( I ) that are absolute 1-Lipschitz retracts.

Authors:Semyon Alesker; Egor Shelukhin Pages: 1 - 52 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Semyon Alesker, Egor Shelukhin We prove a C 0 a priori estimate on a solution of the quaternionic Calabi problem on an arbitrary compact connected HKT-manifold. This generalizes earlier results [9] and [7] where this result was proven under certain extra assumptions on the manifold.

Authors:Jason Metcalfe; Daniel Tataru; Mihai Tohaneanu Pages: 53 - 93 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Jason Metcalfe, Daniel Tataru, Mihai Tohaneanu In this article we study the pointwise decay properties of solutions to the Maxwell system on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time, we establish peeling estimates, as well as a t − 4 rate of decay on compact regions for all the components of the Maxwell tensor.

Authors:Christophe Eyral; Mutsuo Oka Pages: 94 - 113 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Christophe Eyral, Mutsuo Oka In an unpublished lecture note, J. Briançon observed that if { f t } is a family of isolated complex hypersurface singularities such that the Newton boundary of f t is independent of t and f t is non-degenerate, then the corresponding family of hypersurfaces { f t − 1 ( 0 ) } is Whitney equisingular (and hence topologically equisingular). A first generalization of this assertion to families with non-isolated singularities was given by the second author under a rather technical condition. In the present paper, we give a new generalization under a simpler condition.

Authors:Nathan Brownlowe; Alexander Mundey; David Pask; Jack Spielberg; Anne Thomas Pages: 114 - 186 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Nathan Brownlowe, Alexander Mundey, David Pask, Jack Spielberg, Anne Thomas To a large class of graphs of groups we associate a C ⁎ -algebra universal for generators and relations. We show that this C ⁎ -algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass–Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag–Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple C ⁎ -algebras associated to generalised Baumslag–Solitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce–Deddens algebra.

Authors:Jeffrey Streets Pages: 187 - 215 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Jeffrey Streets We study the generalized Kähler–Ricci flow on complex surfaces with nondegenerate Poisson structure, proving long time existence and convergence of the flow to a weak hyperKähler structure.

Authors:Charles H. Conley; Dimitar Grantcharov Pages: 216 - 254 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Charles H. Conley, Dimitar Grantcharov The Lie algebra of vector fields on R m acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to sl m + 1 , and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor gl m . We prove two results. First, we realize explicitly all injective objects of the parabolic category O g l m ( sl m + 1 ) of gl m -finite sl m + 1 -modules, as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e., sl m + 1 -invariant splittings of their order filtrations. In the case of modules of differential operators from a tensor density module to an arbitrary tensor field module, we determine when there exists a unique projective quantization, when there exists no projective quantization, and when there exist multiple projective quantizations.

Authors:Lechao Xiao Pages: 255 - 291 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Lechao Xiao The one-dimensional oscillatory integral operator associated to a real analytic phase S is given by T λ f ( x ) = ∫ − ∞ ∞ e i λ S ( x , y ) χ ( x , y ) f ( y ) d y . In their fundamental work, Phong and Stein established sharp L 2 estimates for T λ . The goal of this paper is to extend their results to all endpoints. In particular, we obtain a complete characterization for the mapping properties for T λ on L p ( R ) . More precisely, we show that ‖ T λ f ‖ p ≲ λ − α ‖ f ‖ p holds for some α > 0 if and only if ( 1 α p , 1 α p ′ ) lies in the reduced Newton polygon of S.

Authors:Peng Qu; Zhouping Xin Pages: 292 - 355 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Peng Qu, Zhouping Xin Entropy weak solutions with bounded periodic initial data are considered for the system of weakly nonlinear gas dynamics. Through a modified Glimm scheme, an approximate solution sequence is constructed, and then a priori estimates are provided with the methods of approximate characteristics and approximate conservation laws, which gives not only the existence and uniqueness but also the uniform total variation bounds for the entropy solutions.

Authors:Mircea Petrache; Tristan Rivière Pages: 469 - 540 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Mircea Petrache, Tristan Rivière We study the minimization problem for the Yang–Mills energy under fixed boundary connection in supercritical dimension n ≥ 5 . We define the natural function space A G in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A G can be also interpreted as a space of weak connections on a “real measure theoretic version” of reflexive sheaves from complex geometry. We prove the existence of weak solutions to the Yang–Mills Plateau problem in the space A G . We then prove the optimal regularity result for solutions of this Plateau problem. On the way to prove this result we establish a Coulomb gauge extraction theorem for weak curvatures with small Yang–Mills density. This generalizes to the general framework of weak L 2 curvatures previous works of Meyer–Rivière and Tao–Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.

Authors:Deepam Patel; G.V. Ravindra Pages: 554 - 575 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Deepam Patel, G.V. Ravindra Let Y be a smooth projective variety over C , and X be a smooth hypersurface in Y. We prove that the natural restriction map on Chow groups of codimension two cycles is an isomorphism when restricted to the torsion subgroups provided dim Y ≥ 5 . We prove an analogous statement for a very general hypersurface X ⊂ P 4 of degree ≥5. In the more general setting of a very general hypersurface X of sufficiently high degree in a fixed smooth projective four-fold Y, under some additional hypothesis, we prove that the restriction map is an isomorphism on ℓ-primary torsion for almost all primes ℓ. As a consequence, we obtain a weak Lefschetz theorem for torsion in the Griffiths groups of codimension 2 cycles, and prove the injectivity of the Abel–Jacobi map when restricted to torsion in this Griffiths group, thereby providing a partial answer to a question of Nori.

Authors:Bhargav Bhatt; Daniel Halpern-Leistner Pages: 576 - 612 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Bhargav Bhatt, Daniel Halpern-Leistner We establish several strengthened versions of Lurie's Tannaka duality theorem for certain classes of spectral algebraic stacks. Our most general version of Tannaka duality identifies maps between stacks with exact symmetric monoidal functors between ∞-categories of quasi-coherent complexes which preserve connective and pseudo-coherent complexes.

Authors:Tivadar Danka Pages: 613 - 666 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Tivadar Danka In this paper universality limits are studied in connection with measures which exhibit power-type singular behavior somewhere in their support. We extend the results of Lubinsky for Jacobi measures supported on [ − 1 , 1 ] to generalized Jacobi measures supported on a compact subset of the real line, where the singularity can be located in the interior or at an endpoint of the support. The analysis is based upon the Riemann–Hilbert method, Christoffel functions, the polynomial inverse image method of Totik and the normal family approach of Lubinsky.

Authors:Francisco L. Hernández; Evgeny M. Semenov; Pedro Tradacete Pages: 667 - 690 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Francisco L. Hernández, Evgeny M. Semenov, Pedro Tradacete We study the interpolation and extrapolation properties of strictly singular operators between different L p spaces. To this end, the structure of strictly singular non-compact operators between L p − L q spaces is analyzed. Among other things, we clarify the relation between strict singularity and the L-characteristic set of an operator. In particular, Krasnoselskii's interpolation theorem for compact operators is extended to the class of strictly singular operators.

Authors:Xavier Ros-Oton; Joaquim Serra Pages: 710 - 747 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Xavier Ros-Oton, Joaquim Serra We study the regularity of the free boundary in the fully nonlinear thin obstacle problem. Our main result establishes that the free boundary is C 1 near any regular point. This extends to the fully nonlinear setting the celebrated result of Athanasopoulos–Caffarelli–Salsa [1]. The proofs we present here are completely independent from those in [1], and do not rely on any monotonicity formula. Furthermore, an interesting and novel feature of our proofs is that we establish the regularity of the free boundary without classifying blow-ups, a priori they could be non-homogeneous and/or non-unique. We do not classify blow-ups but only prove that they are 1D on { x n = 0 } .

Authors:Selçuk Barlak; Xin Li Pages: 748 - 769 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Selçuk Barlak, Xin Li We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan subalgebras. This observation allows us to deduce, among other things, that a crossed product O 2 ⋊ α Z p satisfies the UCT if there is some automorphism γ of O 2 with the property that γ ( D 2 ) ⊆ O 2 ⋊ α Z p is regular, where D 2 denotes the canonical masa of O 2 . We prove that this condition is automatic if γ ( D 2 ) ⊆ O 2 ⋊ α Z p is not a masa or α ( γ ( D 2 ) ) is inner conjugate to γ ( D 2 ) . Finally, we relate the UCT problem for separable, nuclear, M 2 ∞ -absorbing C*-algebras to Cartan subalgebras and order two automorphisms of O 2 .

Authors:Reza Seyyedali Pages: 770 - 805 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Reza Seyyedali We prove that the existence of extremal metrics implies asymptotically relative Chow stability. An application of this is the uniqueness, up to automorphisms, of extremal metrics in any polarization.

Authors:Christoph Haberl; Lukas Parapatits Pages: 806 - 865 Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Christoph Haberl, Lukas Parapatits All measurable SL ( n ) -covariant symmetric tensor valuations on convex polytopes containing the origin in their interiors are completely classified. It is shown that essentially the only examples of such valuations are the moment tensor and a tensor derived from L p surface area measures. This generalizes and unifies earlier results for the scalar, vector and matrix valued case.

Authors:Nir Lev; Alexander Olevskii Pages: 1 - 26 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): Nir Lev, Alexander Olevskii We prove that a positive-definite measure in R n with uniformly discrete support and discrete closed spectrum, is representable as a finite linear combination of Dirac combs, translated and modulated. This extends our recent results where we proved this under the assumption that also the spectrum is uniformly discrete. As an application we obtain that Hof's quasicrystals with uniformly discrete diffraction spectra must have a periodic diffraction structure.

Authors:Matt Kerr; Colleen Robles Pages: 27 - 87 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): Matt Kerr, Colleen Robles Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford–Tate domains, arise as open G R -orbits in flag varieties G / P . We investigate Hodge-theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths–Yukawa coupling to the variety of lines on G / P (under a minimal homogeneous embedding), construct a large class of polarized G R -orbits in G / P , and compute the associated Hodge-theoretic boundary components. An emphasis is placed throughout on adjoint flag varieties and the corresponding families of Hodge structures of levels two and four.

Authors:Chongying Dong; Li Ren Pages: 88 - 101 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): Chongying Dong, Li Ren The rationality of the parafermion vertex operator algebra K ( g , k ) associated to any finite dimensional simple Lie algebra g and any nonnegative integer k is established and the irreducible modules are determined.

Authors:David Conlon; Joonkyung Lee Pages: 130 - 165 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): David Conlon, Joonkyung Lee Given a graph H on vertex set { 1 , 2 , ⋯ , n } and a function f : [ 0 , 1 ] 2 → R , define ‖ f ‖ H : = ∫ ∏ i j ∈ E ( H ) f ( x i , x j ) d μ V ( H ) 1 / E ( H ) , where μ is the Lebesgue measure on [ 0 , 1 ] . We say that H is norming if ‖ ⋅ ‖ H is a semi-norm. A similar notion ‖ ⋅ ‖ r ( H ) is defined by ‖ f ‖ r ( H ) : = ‖ f ‖ H and H is said to be weakly norming if ‖ ⋅ ‖ r ( H ) is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. We demonstrate that any graph whose edges percolate in an appropriate way under the action of a certain natural family of automorphisms is weakly norming. This result includes all previously known examples of weakly norming graphs, but also allows us to identify a much broader class arising from finite reflection groups. We include several applications of our results. In particular, we define and compare a number of generalisations of Gowers' octahedral norms and we prove some new instances of Sidorenko's conjecture.

Authors:V. Gol'dshtein; A. Ukhlov Pages: 166 - 193 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): V. Gol'dshtein, A. Ukhlov In this paper we prove discreteness of the spectrum of the Neumann–Laplacian (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial Neumann eigenvalue are obtained in terms of geometric characteristics of Sobolev mappings. The suggested approach is based on Sobolev–Poincaré inequalities that are obtained with the help of a geometric theory of composition operators on Sobolev spaces. These composition operators are induced by generalizations of conformal mappings that are called as mappings of bounded 2-distortion (weak 2-quasiconformal mappings).

Authors:Christine Bessenrodt; Thorsten Holm; Peter Jørgensen Pages: 194 - 245 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): Christine Bessenrodt, Thorsten Holm, Peter Jørgensen An SL 2 -tiling is a bi-infinite matrix of positive integers such that each adjacent 2 × 2 -submatrix has determinant 1. Such tilings are infinite analogues of Conway–Coxeter friezes, and they have strong links to cluster algebras, combinatorics, mathematical physics, and representation theory. We show that, by means of so-called Conway–Coxeter counting, every SL 2 -tiling arises from a triangulation of the disc with two, three or four accumulation points. This improves earlier results which only discovered SL 2 -tilings with infinitely many entries equal to 1. Indeed, our methods show that there are large classes of tilings with only finitely many entries equal to 1, including a class of tilings with no 1's at all. In the latter case, we show that the minimal entry of a tiling is unique.

Authors:Marta Casanellas; Jesús Fernández-Sánchez; Mateusz Michałek Pages: 285 - 323 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): Marta Casanellas, Jesús Fernández-Sánchez, Mateusz Michałek Phylogenetic varieties related to equivariant substitution models have been studied largely in the last years. One of the main objectives has been finding a set of generators of the ideal of these varieties, but this has not yet been achieved in some cases (for example, for the general Markov model this involves the open “salmon conjecture”, see [2]) and it is not clear how to use all generators in practice. Motivated by applications in biology, we tackle the problem from another point of view. The elements of the ideal that could be useful for applications in phylogenetics only need to describe the variety around certain points of no evolution (see [13]). We produce a collection of explicit equations that describe the variety on a Zariski open neighborhood of these points (see Theorem 5.4). Namely, for any tree T on any number of leaves (and any degrees at the interior nodes) and for any equivariant model on any set of states κ, we compute the codimension of the corresponding phylogenetic variety. We prove that this variety is smooth at general points of no evolution and, if a mild technical condition is satisfied (“d-claw tree hypothesis”), we provide an algorithm to produce a complete intersection that describes the variety around these points.

Authors:Roger E. Behrend; Ilse Fischer; Matjaž Konvalinka Pages: 324 - 365 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): Roger E. Behrend, Ilse Fischer, Matjaž Konvalinka We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang–Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of ( 2 n + 1 ) × ( 2 n + 1 ) DASASMs is ∏ i = 0 n ( 3 i ) ! ( n + i ) ! , and a conjecture of Stroganov from 2008 that the ratio between the numbers of ( 2 n + 1 ) × ( 2 n + 1 ) DASASMs with central entry −1 and 1 is n / ( n + 1 ) . Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.

Authors:Peter S. Ozsváth; András I. Stipsicz; Zoltán Szabó Pages: 366 - 426 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó We modify the construction of knot Floer homology to produce a one-parameter family of homologies tHFK for knots in S 3 . These invariants can be used to give homomorphisms from the smooth concordance group C to Z , giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.

Authors:David Dumas Pages: 427 - 473 Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): David Dumas We study the limits of holonomy representations of complex projective structures on a compact Riemann surface in the Morgan–Shalen compactification of the character variety. We show that the dual R -trees of the quadratic differentials associated to a divergent sequence of projective structures determine the Morgan–Shalen limit points up to a natural folding operation. For quadratic differentials with simple zeros, no folding is possible and the limit of holonomy representations is isometric to the dual tree. We also derive an estimate for the growth rate of the holonomy map in terms of a norm on the space of quadratic differentials.

Authors:Yalong Cao; Naichung Conan Leung Pages: 48 - 70 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Yalong Cao, Naichung Conan Leung We study orientability issues of moduli spaces from gauge theories on Calabi–Yau manifolds. Our results generalize and strengthen those for Donaldson–Thomas theory on Calabi–Yau manifolds of dimensions 3 and 4. We also prove a corresponding result in the relative situation which is relevant to the gluing formula in DT theory.

Authors:Olivier Blondeau-Fournier; Pierre Mathieu; David Ridout; Simon Wood Pages: 71 - 123 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Olivier Blondeau-Fournier, Pierre Mathieu, David Ridout, Simon Wood We give new proofs of the rationality of the N = 1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu–Schwarz and Ramond sectors. For this, we combine the standard free field realisation with the theory of Jack symmetric functions. A key role is played by Jack symmetric polynomials with a certain negative parameter that are labelled by admissible partitions. These polynomials are shown to describe free fermion correlators, suitably dressed by a symmetrising factor. The classification proofs concentrate on explicitly identifying Zhu's algebra and its twisted analogue. Interestingly, these identifications do not use an explicit expression for the non-trivial vacuum singular vector. While the latter is known to be expressible in terms of an Uglov symmetric polynomial or a linear combination of Jack superpolynomials, it turns out that standard Jack polynomials (and functions) suffice to prove the classification.

Authors:Wayne Smith; Dmitriy M. Stolyarov; Alexander Volberg Pages: 185 - 202 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Wayne Smith, Dmitriy M. Stolyarov, Alexander Volberg We obtain a necessary and sufficient condition for the operator of integration to be bounded on H ∞ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions.

Authors:Thomas Creutzig; Antun Milas Pages: 203 - 227 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Thomas Creutzig, Antun Milas We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's partial theta functions coming from ADE root lattices, which are then linked to representation theory of W-algebras. We derive modular transformation properties of regularized higher rank partial and false theta functions as well as Kostant's version of these. Modulo conjectures in representation theory, as an application, we compute regularized quantum dimensions of atypical and typical modules of “narrow” logarithmic W-algebras associated to rescaled root lattices. This paper substantially generalize our previous work [19] pertaining to ( 1 , p ) -singlet W-algebras (the sl 2 case). Results in this paper are very general and are applicable in a variety of situations.

Authors:Brian D. Boe; Jonathan R. Kujawa; Daniel K. Nakano Pages: 228 - 277 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Brian D. Boe, Jonathan R. Kujawa, Daniel K. Nakano Tensor triangular geometry as introduced by Balmer [3] is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category. In this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the Balmer spectrum. For the general linear Lie superalgebra g = g 0 ¯ ⊕ g 1 ¯ we construct a Zariski space from a detecting subalgebra of g and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional g -modules which are semisimple over g 0 ¯ .

Authors:Qintao Deng; Huiling Gu; Qiaoyu Wei Pages: 278 - 305 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Qintao Deng, Huiling Gu, Qiaoyu Wei In this paper, we will prove that any closed minimal Willmore hypersurface M 4 of S 5 with constant scalar curvature must be isoparametric. To be precise, M 4 is either an equatorial 4 sphere, a product of sphere S 2 ( 2 2 ) × S 2 ( 2 2 ) or a Cartan's minimal hypersurface. In particular, the value of the second fundamental form S can only be 0, 4, 12. This result strongly supports Chern's Conjecture.

Authors:J. Elias; M.E. Rossi Pages: 306 - 327 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): J. Elias, M.E. Rossi Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given.

Authors:Gabriele Grillo; Matteo Muratori; Juan Luis Vázquez Pages: 328 - 377 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Gabriele Grillo, Matteo Muratori, Juan Luis Vázquez We consider nonnegative solutions of the porous medium equation (PME) on Cartan–Hadamard manifolds whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we estimate the location of the free boundary. A global Harnack principle follows. We also present a change of variables that allows to transform radially symmetric solutions of the PME on model manifolds into radially symmetric solutions of a corresponding weighted PME on Euclidean space and back. This equivalence turns out to be an important tool of the theory.

Authors:Piotr Beben; Jelena Grbić Pages: 378 - 425 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Piotr Beben, Jelena Grbić This paper aims to find the most general combinatorial conditions under which a moment–angle complex ( D 2 , S 1 ) K is a co-H-space, thus splitting unstably in terms of its full subcomplexes. In this way we study to which extent the conjecture holds that a moment–angle complex over a Golod simplicial complex is a co-H-space. Our main tool is a certain generalisation of the theory of labelled configuration spaces.

Authors:Stewart Wilcox Pages: 426 - 492 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Stewart Wilcox We first consider the rational Cherednik algebra corresponding to the action of a finite group on a complex variety, as defined by Etingof. We define a category of representations of this algebra which is analogous to “category O ” for the rational Cherednik algebra of a vector space. We generalise to this setting Bezrukavnikov and Etingof's results about the possible support sets of such representations. Then we focus on the case of S n acting on C n , determining which irreducible modules in this category have which support sets. We also show that the category of representations with a given support, modulo those with smaller support, is equivalent to the category of finite dimensional representations of a certain Hecke algebra.

Authors:Eduardo Teixeira; Miguel Urbano Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): Damião J. Araújo, Eduardo V. Teixeira, José Miguel Urbano We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where ellipticity degenerates. As a consequence, we are able to prove the planar counterpart of the longstanding conjecture that solutions of the degenerate p-Poisson equation with a bounded source are locally of class C p ′ = C 1 , 1 p − 1 ; this regularity is optimal.

Abstract: Publication date: 20 August 2017 Source:Advances in Mathematics, Volume 316 Author(s): András Máthé The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree d, we construct a compact set E ⊂ R n of Hausdorff dimension n / d which does not contain finite point configurations corresponding to the zero sets of the given polynomials. Given a set E ⊂ R n , we study the angles determined by three-point subsets of E. The main result implies the existence of a compact set in R n of Hausdorff dimension n / 2 which does not contain the angle π / 2 . (This is known to be sharp if n is even.) We show that there is a compact set of Hausdorff dimension n / 8 which does not contain an angle in any given countable set. We also construct a compact set E ⊂ R n of Hausdorff dimension n / 6 for which the set of angles determined by E is Lebesgue null. In the other direction, we present a result that every set of sufficiently large dimension contains an angle ε close to any given angle. The main result can also be applied to distance sets. As a corollary we obtain a compact set E ⊂ R n ( n ≥ 2 ) of Hausdorff dimension n / 2 which does not contain rational distances nor collinear points, for which the distance set is Lebesgue null, moreover, every distance and direction is realised only at most once by E.

Authors:Chao Xia Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): Chao Xia In this paper, we show that the inverse anisotropic mean curvature flow in R n + 1 , initiating from a star-shaped, strictly F-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the C ∞ topology. As an application, we prove a Minkowski type inequality for star-shaped, F-mean convex hypersurfaces.

Authors:J.J. Abstract: Publication date: 31 July 2017 Source:Advances in Mathematics, Volume 315 Author(s): J.J. Sánchez-Gabites Suppose that a closed surface S ⊆ R 3 is an attractor, not necessarily global, for a discrete dynamical system. Assuming that its set of wild points W is totally disconnected, we prove that (up to an ambient homeomorphism) it has to be contained in a straight line. As a corollary we show that there exist uncountably many different 2-spheres in R 3 none of which can be realized as an attractor for a homeomorphism. Our techniques hinge on a quantity r ( K ) that can be defined for any compact set K ⊆ R 3 and is related to “how wildly” it sits in R 3 . We establish the topological results that (i) r ( W ) ≤ r ( S ) and (ii) any totally disconnected set having a finite r must be contained in a straight line (up to an ambient homeomorphism). The main result follows from these and the fact that attractors have a finite r.

Authors:Ugo Boscain; Robert Neel Luca Rizzi Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Ugo Boscain, Robert Neel, Luca Rizzi On a sub-Riemannian manifold we define two types of Laplacians. The macroscopic Laplacian Δ ω , as the divergence of the horizontal gradient, once a volume ω is fixed, and the microscopic Laplacian, as the operator associated with a sequence of geodesic random walks. We consider a general class of random walks, where all sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement c to the sub-Riemannian distribution, and is denoted by L c . We address the problem of equivalence of the two operators. This problem is interesting since, on equiregular sub-Riemannian manifolds, there is always an intrinsic volume (e.g. Popp's one P ) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation: • On contact structures, for every volume ω, there exists a unique complement c such that Δ ω = L c . • On Carnot groups, if H is the Haar volume, then there always exists a complement c such that Δ H = L c . However this complement is not unique in general. • For quasi-contact structures, in general, Δ P ≠ L c for any choice of c. In particular, L c is not symmetric with respect to Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, Δ P is the unique intrinsic macroscopic Laplacian. A crucial notion that we introduce here is the N-intrinsic volume, i.e. a volume that depends only on the set of parameters of the nilpotent approximation. When the nilpotent approximation does not depend on the point, a N-intrinsic volume is unique up to a scaling by a constant and the corresponding N-intrinsic sub-Laplacian is unique. This is what happens for dimension less than or equal to 4, and in particular in the 4-dimensional quasi-contact structure mentioned above. Finally, we prove a general theorem on the convergence of families of random walks to a diffusion, that gives, in particular, the convergence of the random walks mentioned above to the diffusion generated by L c .

Authors:Alessio Figalli; David Jerison Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Alessio Figalli, David Jerison We prove a quantitative stability result for the Brunn–Minkowski inequality: if A = B = 1 , t ∈ [ τ , 1 − τ ] with τ > 0 , and t A + ( 1 − t ) B 1 / n ≤ 1 + δ for some small δ, then, up to a translation, both A and B are quantitatively close (in terms of δ) to a convex set K .