Authors:Marco Schlichting Pages: 1 - 81 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Marco Schlichting We improve homology stability ranges for elementary and special linear groups over rings with many units. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative local rings with infinite residue fields, we show that the obstruction to further stability is given by Milnor–Witt K-theory. As an application we construct Euler classes of projective modules with values in the cohomology of the Milnor–Witt K-theory sheaf. For d-dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the Euler class is necessary and sufficient for an oriented projective module P of rank d to split off a rank 1 free direct summand. Along the way we obtain a new presentation of Milnor–Witt K-theory and of symplectic K 2 simplifying the classical Matsumoto–Moore presentation.

Authors:Ivan Izmestiev; Steven Klee; Isabella Novik Pages: 82 - 114 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Ivan Izmestiev, Steven Klee, Isabella Novik We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly ( d + 1 ) -colored) triangulation of a combinatorial d-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial d-manifold can be connected by a sequence of cross-flips. Along the way we prove that for every m ≥ d + 2 and any closed combinatorial d-manifold M, two m-colored triangulations of M can be connected by a sequence of bistellar flips that preserve the vertex colorings.

Authors:Thomas Hudson; Takeshi Ikeda; Tomoo Matsumura; Hiroshi Naruse Pages: 115 - 156 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Thomas Hudson, Takeshi Ikeda, Tomoo Matsumura, Hiroshi Naruse We prove a determinantal formula that describes the K-theoretic degeneracy loci classes for Grassmann bundles. We further prove Pfaffian formulas for symplectic and odd orthogonal Grassmann bundles. The former generalizes Damon–Kempf–Laksov's determinantal formula, and the latter generalize Pragacz–Kazarian's formulas for the Chow ring.

Authors:Tomoyuki Arakawa; Anne Moreau Pages: 157 - 209 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Tomoyuki Arakawa, Anne Moreau We show that sheet closures appear as associated varieties of affine vertex algebras. Further, we give new examples of non-admissible affine vertex algebras whose associated variety is contained in the nilpotent cone. We also prove some conjectures from our previous paper and give new examples of lisse affine W-algebras.

Authors:Kenneth Ascher; Dori Bejleri Pages: 210 - 243 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Kenneth Ascher, Dori Bejleri We classify the log canonical models of elliptic surface pairs ( f : X → C , S + F A ) where f : X → C is an elliptic fibration, S is a section, and F A is a weighted sum of marked fibers. In particular, we show how the log canonical models depend on the choice of the weights. We describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes. In addition, we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers. This is the first step in constructing compactifications of moduli spaces of elliptic surfaces using the minimal model program.

Authors:Yann Pequignot Pages: 244 - 249 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Yann Pequignot The shift graph G S is defined on the space of infinite subsets of natural numbers by letting two sets be adjacent if one can be obtained from the other by removing its least element. We show that this graph is not a minimum among the graphs of the form G f defined on some Polish space X, where two distinct points are adjacent if one can be obtained from the other by a given Borel function f : X → X . This answers the primary outstanding question from [8].

Authors:David Glickenstein; Joseph Thomas Pages: 250 - 278 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): David Glickenstein, Joseph Thomas A piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating the simplex has an isometric embedding into a constant curvature background geometry (Euclidean, hyperbolic, or spherical) with the specified edge lengths. Additional geometric structure leads to a notion of discrete conformal structure, generalizing circle packings and their generalizations as studied by Thurston and others. We analyze discrete conformal variations of piecewise constant curvature 2-manifolds, giving particular attention to the variation of angles. Formulas are derived for the derivatives of angles in each background geometry, which yield formulas for the derivatives of curvatures and to curvature functionals. Finally, we provide a complete classification of possible definitions of discrete conformal structures in each of the background geometries.

Authors:Liran Shaul Pages: 279 - 328 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Liran Shaul Let K be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings A over K , such that H 0 ( A ) is essentially of finite type over K , and A has finite flat dimension over K . We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for f ! in this setting. As application, we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings. Our results generalize and solve some recent conjectures of Yekutieli.

Authors:Roland Lötscher; Mark MacDonald Pages: 329 - 360 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Roland Lötscher, Mark MacDonald The notion of a ( G , N ) -slice of a G-variety was introduced by P.I. Katsylo in the early 80's for an algebraically closed base field of characteristic 0. Slices (also known under the name of relative sections) have ever since provided a fundamental tool in invariant theory, allowing reduction of rational or regular invariants of an algebraic group G to invariants of a “simpler” group. We refine this notion for a G-scheme over an arbitrary field, and use it to get reduction of structure group results for G-torsors. Namely we show that any ( G , N ) -slice of a versal G-scheme gives surjective maps H 1 ( L , N ) → H 1 ( L , G ) in fppf-cohomology for infinite fields L containing F. We show that every stabilizer in general position H for a geometrically irreducible G-variety V gives rise to a ( G , N G ( H ) ) -slice in our sense. The combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type E 7 .

Authors:Jungkai Chen; Zhi Jiang; Zhiyu Tian Pages: 361 - 390 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Jungkai Chen, Zhi Jiang, Zhiyu Tian We study the Albanese image of a compact Kähler manifold whose geometric genus is one. In particular, we prove that if the Albanese map is not surjective, then the manifold maps surjectively onto an ample divisor in some abelian variety, and in many cases the ample divisor is a theta divisor. With a further natural assumption on the topology of the manifold, we prove that the manifold is an algebraic fiber space over a genus two curve. Finally we apply these results to study the geometry of a compact Kähler manifold which has the same Hodge numbers as those of an abelian variety of the same dimension.

Authors:Jean-Marie Lescure; Stéphane Vassout Pages: 391 - 450 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Jean-Marie Lescure, Stéphane Vassout As announced in [36], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study Lagrangian conic submanifolds of the symplectic groupoid T ⁎ G . This includes their product, transposition and inversion. We also study the relationship between these Lagrangian submanifolds and the equivariant families of Lagrangian submanifolds of T ⁎ G x × T ⁎ G x parametrized by the units x ∈ G ( 0 ) of G. This allows us to select a subclass of Lagrangian distributions on any Lie groupoid G that deserve the name of Fourier integral G-operators (G-FIOs). By construction, the class of G-FIOs contains the class of equivariant families of ordinary Fourier integral operators on the manifolds G x , x ∈ G ( 0 ) . We then develop for G-FIOs the first stages of the calculus in the spirit of Hormander's work. Finally, we illustrate this calculus in the case of manifolds with boundary.

Authors:Qingchun Ji; Ke Zhu Pages: 451 - 474 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Qingchun Ji, Ke Zhu This paper develops a weighted L 2 -method for the (half) Dirac equation. For Dirac bundles over closed Riemann surfaces, we give a sufficient condition for the solvability of the (half) Dirac equation in terms of a curvature integral. Applying this to the Dolbeault–Dirac operator, we establish an automatic transversality criteria for holomorphic curves in Kähler manifolds. On compact Riemannian manifolds, we give a new perspective on some well-known results about the first eigenvalue of the Dirac operator, and improve the estimates when the Dirac bundle has a Z 2 -grading. On Riemannian manifolds with cylindrical ends, we obtain solvability in the L 2 -spaces with suitable exponential weights while allowing mild negativity of the curvature.

Authors:Motoo Tange Pages: 475 - 499 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Motoo Tange We consider a homology sphere M n ( K 1 , K 2 ) presented by two knots K 1 , K 2 with linking number 1 and framing ( 0 , n ) . We call the manifold Matsumoto's manifold. We show that M n ( T 2 , 3 , K 2 ) never bounds any contractible 4-manifold if n < 2 τ ( K 2 ) holds. We also give a formula of Ozsváth–Szabó's τ-invariant as the total sum of the Euler numbers of the reduced filtration. We compute the δ-invariants of the twisted Whitehead doubles of torus knots and correction terms of the branched covers of the Whitehead doubles. By using Owens and Strle's obstruction we show that the 12-twisted Whitehead double of the ( 2 , 7 ) -torus knot and the 20-twisted Whitehead double of the ( 3 , 7 ) -torus knot are not slice but the double branched covers bound rational homology 4-balls. These are new examples having a gap between what a knot is slice and what a double branched cover bounds a rational homology 4-ball.

Authors:Scott Mullane Pages: 500 - 519 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Scott Mullane For every g ≥ 2 and n ≥ g + 1 we exhibit infinitely many extremal effective divisors in M ‾ g , n coming from the strata of abelian differentials.

Authors:Zakhar Kabluchko; Vladislav Vysotsky; Dmitry Zaporozhets Pages: 595 - 629 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Zakhar Kabluchko, Vladislav Vysotsky, Dmitry Zaporozhets Consider a sequence of partial sums S i = ξ 1 + … + ξ i , 1 ≤ i ≤ n , starting at S 0 = 0 , whose increments ξ 1 , … , ξ n are random vectors in R d , d ≤ n . We are interested in the properties of the convex hull C n : = Conv ( S 0 , S 1 , … , S n ) . Assuming that the tuple ( ξ 1 , … , ξ n ) is exchangeable and a certain general position condition holds, we prove that the expected number of k-dimensional faces of C n is given by the formula E [ f k ( C n ) ] = 2 ⋅ k ! n ! ∑ l = 0 ∞ [ n + 1 d − 2 l ] { d − 2 l k + 1 } , for all 0 ≤ k ≤ d − 1 , where [ n m ] and { n m } are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0 ≤ i 1 < … < i k + 1 ≤ n , the points S i 1 , … , S i k + 1 form a k-dimensional face of Conv ( S 0 , S 1 , … , S n ) . This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments PubDate: 2017-09-17T11:43:03Z DOI: 10.1016/j.aim.2017.09.002 Issue No:Vol. 320 (2017)

Authors:Konstantin M. Dyakonov Pages: 630 - 651 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Konstantin M. Dyakonov Let θ be an inner function on the unit disk, and let K θ p : = H p ∩ θ H 0 p ‾ be the associated star-invariant subspace of the Hardy space H p , with p ≥ 1 . While a nontrivial function f ∈ K θ p is never divisible by θ, it may have a factor h which is ‘‘not too different” from θ in the sense that the ratio h / θ (or just the anti-analytic part thereof) is smooth on the circle. In this case, f is shown to have additional integrability and/or smoothness properties, much in the spirit of the Hardy–Littlewood–Sobolev embedding theorem. The appropriate norm estimates are established, and their sharpness is discussed.

Authors:Juan Dávila; Manuel del Pino; Xuan Hien Nguyen Pages: 674 - 729 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Juan Dávila, Manuel del Pino, Xuan Hien Nguyen Finite topology self-translating surfaces for the mean curvature flow constitute a key element in the analysis of Type II singularities from a compact surface because they arise as limits after suitable blow-up scalings around the singularity. We prove the existence of such a surface M ⊂ R 3 that is orientable, embedded, complete, and with three ends asymptotically paraboloidal. The fact that M is self-translating means that the moving surface S ( t ) = M + t e z evolves by mean curvature flow, or equivalently, that M satisfies the equation H M = ν ⋅ e z where H M denotes mean curvature, ν is a choice of unit normal to M, and e z is a unit vector along the z-axis. This surface M is in correspondence with the classical three-end Costa–Hoffman–Meeks minimal surface with large genus, which has two asymptotically catenoidal ends and one planar end, and a long array of small tunnels in the intersection region resembling a periodic Scherk surface. This example is the first non-trivial one of its kind, and it suggests a strong connection between this problem and the theory of embedded complete minimal surfaces with finite total curvature.

Authors:Graeme Wilkin Pages: 730 - 794 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Graeme Wilkin In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical point. Therefore we can classify all of the isomorphism classes which contain an initial condition that flows up to a given critical point. As an application, we then show that Nakajima's Hecke correspondence for quivers has a Morse-theoretic interpretation as pairs of critical points connected by flow lines for the norm-square of a moment map. The results are valid in the general setting of finite quivers with relations.

Authors:J.P. Pridham Pages: 795 - 826 Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): J.P. Pridham We show that real Deligne cohomology of a complex manifold X arises locally as a topological vector space completion of the analytic Lie groupoid of holomorphic vector bundles. Thus Beilinson's regulator arises naturally as a comparison map between K-theory groups of different types.

Authors:Alejandro Rivera Pages: 1 - 39 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Alejandro Rivera Let ( Σ , g ) be a closed connected surface equipped with a riemannian metric. Let ( λ n ) n ∈ N and ( ψ n ) n ∈ N be the increasing sequence of eigenvalues and the sequence of corresponding L 2 -normalized eigenfunctions of the laplacian on Σ. For each L > 0 , we consider ϕ L = ∑ 0 < λ n ≤ L ξ n λ n ψ n where the ξ n are i.i.d centered gaussians with variance 1. As L → ∞ , ϕ L converges a.s. to the Gaussian Free Field on Σ in the sense of distributions. We first compute the asymptotic behavior of the covariance function for this family of fields as L → ∞ . We then use this result to obtain the asymptotics of the probability that ϕ L is positive on a given open proper subset with smooth boundary. In doing so, we also prove the concentration of the supremum of ϕ L around 1 2 π ln L .

Authors:Christian Ikenmeyer; Greta Panova Pages: 40 - 66 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Christian Ikenmeyer, Greta Panova We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial. Moreover, we prove the positivity of rectangular Kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition. We also compare rectangular Kronecker coefficients with their corresponding plethysm coefficients, which leads to a new lower bound for rectangular Kronecker coefficients. Moreover, we prove that the saturation of the rectangular Kronecker semigroup is trivial, we show that the rectangular Kronecker positivity stretching factor is 2 for a long first row, and we completely classify the positivity of rectangular limit Kronecker coefficients that were introduced by Manivel in 2011.

Authors:Jonas Hirsch; Elena Mäder-Baumdicker Pages: 67 - 75 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Jonas Hirsch, Elena Mäder-Baumdicker We show that φ ∘ τ ˜ 3 , 1 : K → R 4 × { 0 } n − 4 is the unique minimizer among immersed Klein bottles in its conformal class, where φ : S 4 → R 4 is a stereographic projection and τ ˜ 3 , 1 is the bipolar surface of Lawson's τ 3 , 1 -surface [11]. We conjecture that φ ∘ τ ˜ 3 , 1 is the unique minimizer among immersed Klein bottles into R n , n ≥ 4 , whose existence the authors and P. Breuning proved in [2].

Authors:Monika Ludwig; Laura Silverstein Pages: 76 - 110 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Monika Ludwig, Laura Silverstein The Ehrhart polynomial and the reciprocity theorems by Ehrhart & Macdonald are extended to tensor valuations on lattice polytopes. A complete classification is established of tensor valuations of rank up to eight that are equivariant with respect to the special linear group over the integers and translation covariant. Every such valuation is a linear combination of the Ehrhart tensors which is shown to no longer hold true for rank nine.

Authors:Andrei K. Lerner; Sheldy Ombrosi; Israel P. Rivera-Ríos Pages: 153 - 181 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Andrei K. Lerner, Sheldy Ombrosi, Israel P. Rivera-Ríos In recent years, it has been well understood that a Calderón–Zygmund operator T is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator [ b , T ] with a locally integrable function b. This result is applied into two directions. If b ∈ B M O , we improve several weighted weak type bounds for [ b , T ] . If b belongs to the weighted BMO, we obtain a quantitative form of the two-weighted bound for [ b , T ] due to Bloom–Holmes–Lacey–Wick.

Authors:Matthew R. Mills Pages: 182 - 210 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Matthew R. Mills In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers of finite mutation type. In particular, we show that a mutation finite cluster quiver has a maximal green sequence unless it arises from a once-punctured closed marked surface, or one of the two quivers in the mutation class of X 7 . We develop a procedure to explicitly find maximal green sequences for cluster quivers associated to arbitrary triangulations of closed marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with boundary has a maximal green sequence. We also compute explicit maximal green sequences for exceptional quivers of finite mutation type.

Authors:Luca Giorgetti; Karl-Henning Rehren Pages: 211 - 223 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Luca Giorgetti, Karl-Henning Rehren We give a proof of a formula for the trace of self-braidings (in an arbitrary channel) in UMTCs which first appeared in the context of rational conformal field theories (CFTs) [2]. The trace is another invariant for UMTCs which depends only on modular data, and contains the expression of the Frobenius–Schur indicator [21] as a special case. Furthermore, we discuss some applications of the trace formula to the realizability problem of modular data and to the classification of UMTCs.

Authors:Rui Han; Svetlana Jitomirskaya Pages: 224 - 250 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Rui Han, Svetlana Jitomirskaya We establish a topological criterion for connection between reducibility to constant rotations and dual localization, for the general family of analytic quasiperiodic Jacobi operators. As a corollary, we obtain the sharp arithmetic phase transition for the extended Harper's model in the positive Lyapunov exponent region.

Authors:Zhigang Bao; László Erdős; Kevin Schnelli Pages: 251 - 291 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Zhigang Bao, László Erdős, Kevin Schnelli Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + U B U ⁎ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate 1 N in the bulk of the spectrum.

Authors:Afonso S. Bandeira; Asaf Ferber; Matthew Kwan Pages: 292 - 312 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Afonso S. Bandeira, Asaf Ferber, Matthew Kwan Consider the sum X ( ξ ) = ∑ i = 1 n a i ξ i , where a = ( a i ) i = 1 n is a sequence of non-zero reals and ξ = ( ξ i ) i = 1 n is a sequence of i.i.d. Rademacher random variables (that is, Pr [ ξ i = 1 ] = Pr [ ξ i = − 1 ] = 1 / 2 ). The classical Littlewood–Offord problem asks for the best possible upper bound on the concentration probabilities Pr [ X = x ] . In this paper we study a resilience version of the Littlewood–Offord problem: how many of the ξ i is an adversary typically allowed to change without being able to force concentration on a particular value' We solve this problem asymptotically, and present a few interesting open problems.

Authors:Bhaswar B. Bhattacharya; Shirshendu Ganguly; Eyal Lubetzky; Yufei Zhao Pages: 313 - 347 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Bhaswar B. Bhattacharya, Shirshendu Ganguly, Eyal Lubetzky, Yufei Zhao The upper tail problem in the Erdős–Rényi random graph G ∼ G n , p asks to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1 + δ . Chatterjee and Dembo showed that in the sparse regime of p → 0 as n → ∞ with p ≥ n − α for an explicit α = α H > 0 , this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where H is a clique. Here we extend the latter work to any fixed graph H and determine a function c H ( δ ) such that, for p as above and any fixed δ > 0 , the upper tail probability is exp [ − ( c H ( δ ) + o ( 1 ) ) n 2 p Δ log ( 1 / p ) ] , where Δ is the maximum degree of H. As it turns out, the leading order constant in the large deviation rate function, c H ( δ ) , is governed by the independence polynomial of H, defined as P H ( x ) = ∑ i H ( k ) x k where i H ( k ) is the number of independent sets of size k in H. For instance, if H is a regular graph on m vertices, then c H ( δ ) is the minimum between 1 2 δ 2 / m and the unique positive solution of P H ( x ) = 1 + δ .

Authors:Feimin Huang; Tian-Yi Wang; Yong Wang Pages: 348 - 395 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Feimin Huang, Tian-Yi Wang, Yong Wang The low Mach limit for 1D non-isentropic compressible Navier–Stokes flow, whose density and temperature have different asymptotic states at infinity, is rigorously justified. The problems are considered on both well-prepared and ill-prepared data. For the well-prepared data, the solutions of compressible Navier–Stokes equations are shown to converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero when the difference between the states at ±∞ is suitably small. In particular, the velocity of diffusion wave is only driven by the variation of temperature. It is further shown that the solution of compressible Navier–Stokes system also has the same property when Mach number is small, which has never been observed before. The convergence rates on both Mach number and time are also obtained for the well-prepared data. For the ill-prepared data, the limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma. And the difference between the states at ±∞ can be arbitrary large in this case.

Authors:Richard H. Bamler; Qi S. Zhang Pages: 396 - 450 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Richard H. Bamler, Qi S. Zhang In this paper we analyze Ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time-dependent constant. On such Ricci flows we establish a new time-derivative bound for solutions to the heat equation. Based on this bound, we solve several open problems: 1. distance distortion estimates, 2. the existence of a cutoff function, 3. Gaussian bounds for heat kernels, and, 4. a backward pseudolocality theorem, which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time. Using the backward pseudolocality theorem, we next establish a uniform L 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity. We also obtain a stronger ε-regularity theorem for Ricci flows. This result is particularly useful in the study of Kähler Ricci flows on Fano manifolds, where it can be used to derive certain convergence results.

Authors:Gabriel Khan; Bo Yang; Fangyang Zheng Pages: 451 - 471 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Gabriel Khan, Bo Yang, Fangyang Zheng In [2], Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori T R 2 n for any n ≥ 3 . We will call these examples BSV-tori. In this note, we show that on a flat 6-torus, all the orthogonal complex structures are either the complex tori or the BSV-tori. This solves the classification problem for compact Hermitian manifolds with flat Riemannian connection in the case of complex dimension three.

Authors:Gi-Sang Cheon; Ana Luzón; Manuel A. Morón; L. Felipe Prieto-Martinez; Minho Song Pages: 522 - 566 Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Gi-Sang Cheon, Ana Luzón, Manuel A. Morón, L. Felipe Prieto-Martinez, Minho Song We introduce a Frechet Lie group structure on the Riordan group. We give a description of the corresponding Lie algebra as a vector space of infinite lower triangular matrices. We describe a natural linear action induced on the Frechet space K N by any element in the Lie algebra. We relate this to a certain family of bivariate linear partial differential equations. We obtain the solutions of such equations using one-parameter groups in the Riordan group. We show how a particular semidirect product decomposition in the Riordan group is reflected in the Lie algebra. We study the stabilizer of a formal power series under the action induced by Riordan matrices. We get stabilizers in the fraction field K ( ( x ) ) using bi-infinite representations. We provide some examples. The main tool to get our results is the paper [18] where the Riordan group was described using inverse sequences of groups of finite matrices.

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Alexander Koldobsky, Alexander S. Merkurjev, Vladyslav Yaskin An infinitely smooth convex body in R n is called polynomially integrable of degree N if its parallel section functions are polynomials of degree N. We prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids, if N ≥ n − 1 . This is in contrast with the case of even dimensions and the case of odd dimensions with N < n − 1 , where such bodies do not exist, as it was recently shown by Agranovsky.

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Gunter Malle We investigate the action of outer automorphisms of finite groups of Lie type on their irreducible characters. We obtain a definite result for cuspidal characters. As an application we verify the inductive McKay condition for some further infinite families of simple groups at certain primes.

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Raphaël Danchin, Piotr Bogusław Mucha Here we prove the existence of global in time regular solutions to the two-dimensional compressible Navier–Stokes equations supplemented with arbitrary large initial velocity v 0 and almost constant density ϱ 0 , for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible Navier–Stokes equations supplemented with the divergence-free projection of v 0 , is global. The systems are examined in R d with d ≥ 2 , in the critical B ˙ 2 , 1 s Besov spaces framework.

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Changfeng Gui, Yong Liu, Juncheng Wei In this paper, we study axially symmetric solutions of Allen–Cahn equation in the three dimensional Euclidean space. Using a sophisticated continuation method, we show the existence of a complete family of axially symmetric solutions with a range of logarithmic growth rates, which may be regarded as the analogue of the family of catenoids and hence called two-end solutions. Nonexistence of two-end solution with a small growth rate is also shown, which differs from the theory of minimal surfaces.

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Brad Rodgers We characterize the limiting distribution of Rudin–Shapiro polynomials, showing that, normalized, their values become uniformly distributed in the disc. This resolves conjectures of Saffari and Montgomery. Our proof proceeds by relating the polynomials' distribution to that of a product of weakly dependent random matrices, which we analyze using the representation theory of S U ( 2 ) . Our approach leads us to a non-commutative analogue of the classical central limit theorem of Salem and Zygmund, which may be of independent interest.

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Denis Potapov, Anna Skripka, Fedor Sukochev, Anna Tomskova We establish sharp conditions on scalar functions and perturbations that guarantee Schatten summability of nth order operator Taylor remainders. In the special case of dimension one, our estimates of these remainders deliver well known classical estimates of scalar Taylor remainders. We prove that if a scalar function f is in the set C n and a perturbation is in the pth Schatten class S p , p > n , then the respective nth order operator Taylor remainder is an element of S p / n and has an estimate like the one in [16]. We construct examples of f ∈ C n and perturbations in S n such that the nth order Taylor remainder of the respective operator function is not in S 1 . Our construction relies, in particular, on novel dimension dependent estimates for Schatten norms of multilinear Schur multipliers from below that are of interest in their own right. Our results apply to both self-adjoint and unitary operators.

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Shi-Lei Kong, Ka-Sing Lau, Ting-Kam Leonard Wong Let K be a self-similar set satisfying the open set condition. Following Kaimanovich's elegant idea [25], it has been proved that on the symbolic space X of K a natural augmented tree structure E exists; it is hyperbolic, and the hyperbolic boundary ∂ H X with the Gromov metric is Hölder equivalent to K. In this paper we consider certain reversible random walks with return ratio 0 < λ < 1 on ( X , E ) . We show that the Martin boundary M can be identified with ∂ H X and K. With this setup and a device of Silverstein [41], we obtain precise estimates of the Martin kernel and the Naïm kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate Θ ( ξ , η ) ≍ ξ − η − ( α + β ) , where α is the Hausdorff dimension of K and β depends on λ. For suitable β, the kernel defines a regular non-local Dirichlet form on K. This extends the results of Kigami [27] concerning random walks on certain trees with Cantor-type sets as boundaries (see also [5]).

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Gueo Grantcharov, Mehdi Lejmi, Misha Verbitsky A hypercomplex manifold M is a manifold equipped with three complex structures I , J , K satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called the Obata connection. A quaternionic Hermitian metric is a Riemannian metric which is invariant with respect to unitary quaternions. Such a metric is called hyperkähler with torsion (HKT for short) if it is locally obtained as the Hessian of a function averaged with quaternions. An HKT metric is a natural analogue of a Kähler metric on a complex manifold. We push this analogy further, proving a quaternionic analogue of the result of Buchdahl and of Lamari that a compact complex surface M admits a Kähler structure if and only if b 1 ( M ) is even. We show that a hypercomplex manifold M with the Obata holonomy contained in S L ( 2 , H ) admits an HKT structure if and only if H 1 ( O ( M , I ) ) is even-dimensional.

Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Jean-Paul Brasselet, Maurício Corrêa, Fernando Lourenço In this work we prove a Baum–Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum–Bott type rationality conjecture in this context.

Authors:Fan Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Fan Ge Assume the Riemann Hypothesis. We establish a local structure theorem for zeros of the Riemann zeta-function ζ ( s ) and its derivative ζ ′ ( s ) . As an application, we prove a stronger form of half of a conjecture of Radziwiłł [18] concerning the global statistics of these zeros. Roughly speaking, we show that on the Riemann Hypothesis, if there occurs a small gap between consecutive zeta zeros, then there is exactly one zero of ζ ′ ( s ) lying not only very close to the critical line but also between that pair of zeta zeros. This refines a result of Zhang [22]. Some related results are also shown. For example, we prove a weak form of a conjecture of Soundararajan, and suggest a repulsion phenomena for zeros of ζ ′ ( s ) .

Authors:Marcin Abstract: Publication date: 7 November 2017 Source:Advances in Mathematics, Volume 320 Author(s): Marcin Chałupnik We introduce the notion of an affine strict polynomial functor. We show how this concept helps to understand homological behavior of the operation of Frobenius twist in the category of strict polynomial functors over a field of positive characteristic. We also point out for an analogy between our category and the category of representations of the group of algebraic loops on G L n .

Authors:Leonor Godinho; Frederik von Heymann Silvia Sabatini Abstract: Publication date: 15 October 2017 Source:Advances in Mathematics, Volume 319 Author(s): Leonor Godinho, Frederik von Heymann, Silvia Sabatini We generalize the well-known “12 and 24” theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a wider category of objects, here called reflexive GKM graphs, that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action. As an application, we provide bounds on the Betti numbers for certain monotone Hamiltonian spaces which depend on the minimal Chern number of the manifold.