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: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:Gui-Qiang Chen; Beixiang Fang Pages: 493 - 539 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Gui-Qiang Chen, Beixiang Fang We are concerned with the stability of multidimensional (M-D) transonic shocks in steady supersonic flow past multidimensional wedges. One of our motivations is that the global stability issue for the M-D case is much more sensitive than that for the 2-D case, which requires more careful rigorous mathematical analysis. In this paper, we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock-front for potential flow with respect to the M-D perturbation of the wedge boundary in appropriate function spaces. To achieve this, we first formulate the stability problem as a free boundary problem for nonlinear elliptic equations. Then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second-order nonlinear elliptic equations in an unbounded domain. To solve this reduced problem, we linearize the nonlinear problem on the background shock solution and then, after solving this linearized elliptic problem, develop a nonlinear iteration scheme that is proved to be contractive.

Authors:Volker Bach; Miguel Ballesteros; Alessandro Pizzo Pages: 540 - 572 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Volker Bach, Miguel Ballesteros, Alessandro Pizzo The processes of emission and absorption of photons by atoms can be rigorously understood in the low-energy limit if we neglect the creation and annihilation of electrons. They are related to eigenvalues of the atomic Hamiltonian that are embedded in the continuous spectrum of the free Hamiltonian of the electromagnetic field. The mathematical analysis of the perturbation of these eigenvalues due to the electromagnetic interaction relies on a complex deformation technique relating the original Hamiltonian to a non-selfadjoint operator. We develop a new technique to analyze the spectrum of operators used in non-relativistic quantum electrodynamics. Our method can be applied to prove most of the results that previously required an involved renormalization group construction. We use analytic perturbation theory of operators in Hilbert spaces instead. More precisely, we extend the multi-scale analysis introduced by one of us in 2003 [15], which was used so far only for the study of selfadjoint operators, to non-selfadjoint operators. Compared to the selfadjoint case (see, for example, [3]) the analysis of these non-selfadjoint operators is more difficult, because we cannot make use of the functional calculus (spectral theorem) and the min–max principle in some crucial estimates.

Authors:Daniel Erman Pages: 573 - 582 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Daniel Erman On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a divergent series. But there are alternate geometric approaches for assigning (often fractional) intersection multiplicities in some singular settings. Our motivating question comes from Fulton, who asks whether an analytic continuation of the divergent series from Serre's formula can be related to these fractional multiplicities. We apply work of Avramov and Buchweitz to answer Fulton's question in the context of graded rings.

Authors:Jiguang Bao; Hongjie Ju; Haigang Li Pages: 583 - 629 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Jiguang Bao, Hongjie Ju, Haigang Li In this paper, we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the Lamé systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary. When the distance tends to zero, the optimal blow-up rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions.

Authors:Enrico Carlini; Maria Virginia Catalisano; Alessandro Oneto Pages: 630 - 662 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Enrico Carlini, Maria Virginia Catalisano, Alessandro Oneto The Waring locus of a form F is the collection of the degree one forms appearing in some minimal sum of powers decomposition of F. In this paper, we give a complete description of Waring loci for several families of forms, such as quadrics, monomials, binary forms and plane cubics. We also introduce a Waring loci version of Strassen's Conjecture, which implies the original conjecture, and we prove it in many cases.

Authors:Chikara Nakayama Pages: 663 - 725 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Chikara Nakayama We introduce the log étale cohomology in the second (full log étale) sense after K. Kato and prove several fundamental results including base change theorems. To prove them, we use some corresponding results in the log étale cohomology in the first (kummer log étale) sense. Conversely, they are used to show some new results in the kummer log étale cohomology.

Authors:James Freitag; Rahim Moosa Pages: 726 - 755 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): James Freitag, Rahim Moosa Hrushovski's generalization and application of Jouanolou (1978) [9] is here refined and extended to the partial differential setting with possibly nonconstant coefficient fields. In particular, it is shown that if X is a differential-algebraic variety over a partial differential field F that is finitely generated over its constant field F 0 , then there exists a dominant differential-rational map from X to the constant points of an algebraic variety V over F 0 , such that all but finitely many codimension one subvarieties of X over F arise as pull-backs of algebraic subvarieties of V over F 0 . As an application, it is shown that the algebraic solutions to a first order algebraic differential equation over C ( t ) are of bounded height, answering a question of Eremenko. Two expected model-theoretic applications to DCF 0 , m are also given: 1) Lascar rank and Morley rank agree in dimension two, and 2) dimension one strongly minimal sets orthogonal to the constants are ℵ 0 -categorical. A detailed exposition of Hrushovski's original (unpublished) theorem is included, influenced by Ghys (2000) [5].

Authors:José Carlos Díaz-Ramos; Miguel Domínguez-Vázquez; Víctor Sanmartín-López Pages: 756 - 805 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): José Carlos Díaz-Ramos, Miguel Domínguez-Vázquez, Víctor Sanmartín-López We classify isoparametric hypersurfaces in complex hyperbolic spaces.

Authors:Ana Khukhro; Alain Valette Pages: 806 - 834 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Ana Khukhro, Alain Valette We consider box spaces of finitely generated, residually finite groups G, and try to distinguish them up to coarse equivalence. We show that, for n ≥ 2 , the group S L n ( Z ) has a continuum of box spaces which are pairwise non-coarsely equivalent expanders. Moreover, varying the integer n ≥ 3 , expanders given as box spaces of S L n ( Z ) are pairwise not coarsely equivalent; similarly, varying the prime p, expanders given as box spaces of S L 2 ( Z [ p ] ) are pairwise not coarsely equivalent. We also show that, for prime p, the family of finite groups ( P S L 2 ( Z / p n Z ) ) n ≥ 1 can be turned in infinitely many ways (up to coarse equivalence) into a 6-regular expander. A strong form of non-expansion for a box space is the existence of α ∈ ] 0 , 1 ] such that the diameter of each component X n satisfies d i a m ( X n ) = Ω ( X n α ) . By [2], the existence of such a box space implies that G virtually maps onto Z : we establish the converse. For the lamplighter group ( Z / 2 Z ) ≀ Z and for a semi-direct product Z 2 ⋊ Z , such box spaces are explicitly constructed using specific congruence subgroups. We finally introduce the full box space of G, i.e. the metric disjoint union of all finite quotients of G. We prove that the full box space of a group mapping onto the free group F 2 is not coarsely equivalent to the full box space of an S-arithmetic group satisfying the Congruence Subgroup Property.

Authors:Max Engelstein Pages: 835 - 947 Abstract: Publication date: 9 July 2017 Source:Advances in Mathematics, Volume 314 Author(s): Max Engelstein We study parabolic chord arc domains, introduced by Hofmann, Lewis and Nyström [14], and prove a free boundary regularity result below the continuous threshold. More precisely, we show that a Reifenberg flat, parabolic chord arc domain whose Poisson kernel has logarithm in VMO must in fact be a vanishing chord arc domain (i.e. satisfies a vanishing Carleson measure condition). This generalizes, to the parabolic setting, a result of Kenig and Toro [26] and answers in the affirmative a question left open in the aforementioned paper of Hofmann et al. A key step in this proof is a classification of “flat” blowups for the parabolic problem.

Authors:Ilke Canakci; Sibylle Schroll Pages: 1 - 49 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Ilke Canakci, Sibylle Schroll In the context of representation theory of finite dimensional algebras, string algebras have been extensively studied and most aspects of their representation theory are well-understood. One exception to this is the classification of extensions between indecomposable modules. In this paper we explicitly describe such extensions for a class of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. We relate the extension spaces of indecomposable modules to crossings of generalised arcs in the surface and give explicit bases of the extension spaces for indecomposable modules in almost all cases. We show that the dimensions of these extension spaces are given in terms of crossing arcs in the surface. Our approach is new and consists of interpreting snake graphs as indecomposable modules. In order to show that our basis is a spanning set, we need to work in the associated cluster category where we explicitly calculate the middle terms of extensions and give bases of their extension spaces. We note that not all extensions in the cluster category give rise to extensions for the Jacobian algebra.

Authors:Dimitris Vlitas Pages: 50 - 61 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Dimitris Vlitas In a recent paper [5] S. Solecki proved a finite self-dual Ramsey theorem that extends simultaneously the classical finite Ramsey theorem [4] and the Graham–Rothschild theorem [2]. In this paper we prove the corresponding infinite dimensional version of the self-dual theorem. As a consequence, we extend the classical infinite Ramsey theorem [4] and the Carlson–Simpson theorem [1].

Authors:Caucher Birkar; Joe Waldron Pages: 62 - 101 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Caucher Birkar, Joe Waldron We prove the following results for projective klt pairs of dimension 3 over an algebraically closed field of characteristic p > 5 : the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc.

Authors:Misha Bialy; Andrey E. Mironov Pages: 102 - 126 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Misha Bialy, Andrey E. Mironov In this paper we introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff billiard. Using this system we get new results on algebraic Birkhoff conjecture on integrable billiards.

Authors:Pham Hung Quy; Kazuma Shimomoto Pages: 127 - 166 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Pham Hung Quy, Kazuma Shimomoto The main aim of this article is to study the relation between F-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many other topics. (1) We prove that if every parameter ideal of a Noetherian local ring of prime characteristic p > 0 is Frobenius closed, then it is F-injective. (2) We prove a necessary and sufficient condition for the injectivity of the Frobenius action on H m i ( R ) for all i ≤ f m ( R ) , where f m ( R ) is the finiteness dimension of R. As applications, we prove the following results. (a) If the ring is F-injective, then every ideal generated by a filter regular sequence, whose length is equal to the finiteness dimension of the ring, is Frobenius closed. It is a generalization of a recent result of Ma and which is stated for generalized Cohen–Macaulay local rings. (b) Let ( R , m , k ) be a generalized Cohen–Macaulay ring of characteristic p > 0 . If the Frobenius action is injective on the local cohomology H m i ( R ) for all i < dim R , then R is Buchsbaum. This gives an answer to a question of Takagi. (3) We consider the problem when the union of two F-injective closed subschemes of a Noetherian F p -scheme is F-injective. Using this idea, we construct an F-injective local ring R such that R has a parameter ideal that is not Frobenius closed. This result adds a new member to the family of F-singularities. (4) We give the first ideal-theoretic characterization of F-injectivity in terms the Frobenius closure and the limit closure. We also give an answer to the question about when the Frobenius action on the top local cohomology is injective.

Authors:Benjamin Harris; Tobias Weich Pages: 176 - 236 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Benjamin Harris, Tobias Weich Let G be a real, reductive algebraic group, and let X be a homogeneous space for G with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of L 2 ( X ) . Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on X. These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition.

Authors:Garrett Ervin Pages: 237 - 281 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Garrett Ervin In 1958, Sierpiński asked whether there exists a linear order X that is isomorphic to its lexicographically ordered cube but is not isomorphic to its square. The main result of this paper is that the answer is negative. More generally, if X is isomorphic to any one of its finite powers X n , n > 1 , it is isomorphic to all of them.

Authors:Vassily Gorbounov; Christian Korff Pages: 282 - 356 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Vassily Gorbounov, Christian Korff We introduce and study a new mathematical structure in the generalised (quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six-vertex model. Our approach offers a new perspective on already established and well-studied special cases, for example equivariant K-theory, and in addition allows us to formulate a conjecture on the so-far unknown case of quantum equivariant K-theory.

Authors:Ka-Sing Lau; Xiang-Yang Wang Pages: 357 - 378 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Ka-Sing Lau, Xiang-Yang Wang For any contractive iterated function system (IFS, including the Moran systems), we show that there is a natural hyperbolic graph on the symbolic space, which yields the Hölder equivalence of the hyperbolic boundary and the invariant set of the IFS. This completes the previous studies ([16,20,30]) by eliminating superfluous conditions, and admits more classes of sets (e.g., the Moran sets). We also show that the bounded degree property of the graph can be used to characterize certain separation properties of the IFS (open set condition, weak separation condition); the bounded degree property is particularly important when we consider random walks on such graphs. This application and the other application to Lipschitz equivalence of self-similar sets will be discussed.

Authors:Pavel Shvartsman Pages: 379 - 469 Abstract: Publication date: 20 June 2017 Source:Advances in Mathematics, Volume 313 Author(s): Pavel Shvartsman Let L p m ( R n ) , 1 ≤ p ≤ ∞ , be the homogeneous Sobolev space, and let E ⊂ R n be a closed set. For each p > n and each non-negative integer m we give an intrinsic characterization of the restrictions { D α F E : α ≤ m } to E of m-jets generated by functions F ∈ L p m + 1 ( R n ) . Our trace criterion is expressed in terms of variations of corresponding Taylor remainders of m-jets evaluated on a certain family of “well separated” two point subsets of E. For p = ∞ this result coincides with the classical Whitney–Glaeser extension theorem for m-jets. Our approach is based on a representation of the Sobolev space L p m + 1 ( R n ) , p > n , as a union of C m , ( d ) ( R n ) -spaces where d belongs to a family of metrics on R n with certain “nice” properties. Here C m , ( d ) ( R n ) is the space of C m -functions on R n whose partial derivatives of order m are Lipschitz functions with respect to d. This enables us to show that, for every non-negative integer m and every p ∈ ( n , ∞ ) , the very same classical linear Whitney extension operator as in [31] provides an almost optimal extension of m-jets generated by L p m + 1 -functions.

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 .