Authors:Anssi Lahtinen; David Sprehn Pages: 1 - 37 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Anssi Lahtinen, David Sprehn The cohomology of the degree-n general linear group over a finite field of characteristic p, with coefficients also in characteristic p, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in n. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in n. We also construct nontrivial elements in the mod p homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain mysterious.

Authors:Ved Datar; Bin Guo; Jian Song; Xiaowei Wang Pages: 38 - 83 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Ved Datar, Bin Guo, Jian Song, Xiaowei Wang We give criterions for the existence of toric conical Kähler–Einstein and Kähler–Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry–Emery–Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical Kähler–Einstein metrics in the Gromov–Hausdorff topology.

Authors:Eric M. Friedlander Pages: 84 - 113 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Eric M. Friedlander We investigate rational G-modules M for a linear algebraic group G over an algebraically closed field k of characteristic p > 0 using filtrations by sub-coalgebras of the coordinate algebra k [ G ] of G. Even in the special case of the additive group G a , interesting structures and examples are revealed. The “degree” filtration we consider for unipotent algebraic groups leads to a “filtration by exponential degree” applicable to rational G modules for any linear algebraic group G of exponential type; this filtration is defined in terms of 1-parameter subgroups and is related to support varieties introduced recently by the author for such rational G-modules. We formulate in terms of this filtration a necessary and sufficient condition for rational injectivity for rational G-modules. Our investigation leads to the consideration of two new classes of rational G-modules: those that are “mock injective” and those that are “mock trivial”.

Authors:Martin Henk; Hannes Pollehn Pages: 114 - 141 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Martin Henk, Hannes Pollehn We prove tight subspace concentration inequalities for the dual curvature measures C ˜ q ( K , ⋅ ) of an n-dimensional origin-symmetric convex body for q ≥ n + 1 . This supplements former results obtained in the range q ≤ n .

Authors:Jaiung Jun Pages: 142 - 192 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Jaiung Jun We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is ‘multi-valued’. This paper largely consists of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme ( X , O X ) and prove that Γ ( X , O X ) ≃ R for any integral affine hyperring scheme X = Spec R .

Authors:Joachim Toft; Elmira Nabizadeh Pages: 193 - 225 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Joachim Toft, Elmira Nabizadeh We characterize periodic elements in Gevrey classes, Gelfand–Shilov distribution spaces and modulation spaces, in terms of estimates of involved Fourier coefficients, and by estimates of their short-time Fourier transforms. If q ∈ [ 1 , ∞ ) , ω is a suitable weight and ( E 0 E ) ′ is the set of all E-periodic elements, then we prove that the dual of M ( ω ) ∞ , q ∩ ( E 0 E ) ′ equals M ( 1 / ω ) ∞ , q ′ ∩ ( E 0 E ) ′ by suitable extensions of Bessel's identity.

Authors:Agnieszka Bodzenta; Alexey Bondal Pages: 226 - 278 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Agnieszka Bodzenta, Alexey Bondal Given a relatively projective birational morphism f : X → Y of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over Y) generators T X , f and S X , f in D b ( X ) . We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that D b ( X ) has such a filtration L where the lattice is the set of all birational decompositions f : X → g Z → h Y with smooth Z. The t-structures related to T X , f and S X , f are proved to be glued via filtrations left and right dual to L . We realise all such Z as the fine moduli spaces of simple quotients of O X in the heart of the t-structure for which S X , g is a relative projective generator over Y. This implements the program of interpreting relevant smooth contractions of X in terms of a suitable system of t-structures on D b ( X ) .

Authors:Kyungkeun Kang; Hideyuki Miura; Tai-Peng Tsai Pages: 326 - 366 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Kyungkeun Kang, Hideyuki Miura, Tai-Peng Tsai We derive refined estimates of the Green tensor of the stationary Stokes system in the half space. We then investigate the spatial asymptotics of stationary solutions of the incompressible Navier–Stokes equations in the half space. We also discuss the asymptotics of fast decaying flows in the whole space and exterior domains. In the Appendix we consider axisymmetric self-similar solutions.

Authors:Chris Hall; Doron Puder; William F. Sawin Pages: 367 - 410 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Chris Hall, Doron Puder, William F. Sawin Let G be a finite connected graph, and let ρ be the spectral radius of its universal cover. For example, if G is k-regular then ρ = 2 k − 1 . We show that for every r, there is an r-covering (a.k.a. an r-lift) of G where all the new eigenvalues are bounded from above by ρ. It follows that a bipartite Ramanujan graph has a Ramanujan r-covering for every r. This generalizes the r = 2 case due to Marcus, Spielman and Srivastava [26]. Every r-covering of G corresponds to a labeling of the edges of G by elements of the symmetric group S r . We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by [26], a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from [27]. Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the r-th matching polynomial of G to be the average matching polynomial of all r-coverings of G. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside [ − ρ , ρ ] .

Authors:Liang Kong; Hao Zheng Pages: 411 - 426 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Liang Kong, Hao Zheng We define the Drinfeld center of a monoidal category enriched over a braided monoidal category, and show that every modular tensor category can be realized in a canonical way as the Drinfeld center of a self-enriched monoidal category. We also give a generalization of this result for important applications in physics.

Authors:A.B. Dieker; F.V. Saliola Pages: 427 - 485 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): A.B. Dieker, F.V. Saliola We compute the eigenvalues and eigenspaces of random-to-random Markov chains. We use a family of maps which reveal a remarkable recursive structure of the eigenspaces, yielding an explicit and effective construction of all eigenbases starting from bases of the kernels.

Authors:Nicolai Reshetikhin; Jasper Stokman; Bart Vlaar Pages: 486 - 528 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Nicolai Reshetikhin, Jasper Stokman, Bart Vlaar We construct integral representations of solutions to the boundary quantum Knizhnik–Zamolodchikov equations. These are difference equations taking values in tensor products of Verma modules of quantum affine sl 2 , with the K-operators acting diagonally. The integrands in question are products of scalar-valued elliptic weight functions with vector-valued trigonometric weight functions (boundary Bethe vectors). These integrals give rise to a basis of solutions of the boundary qKZ equations over the field of quasi-constant meromorphic functions in weight subspaces of the tensor product.

Authors:Y. Angelopoulos; S. Aretakis; D. Gajic Pages: 529 - 621 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Y. Angelopoulos, S. Aretakis, D. Gajic We derive precise late-time asymptotics for solutions to the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes including as special cases the Schwarzschild and Reissner–Nordström families of black holes. We also obtain late-time asymptotics for the time derivatives of all orders and for the radiation field along null infinity. We show that the leading-order term in the asymptotic expansion is related to the existence of the conserved Newman–Penrose quantities on null infinity. As a corollary we obtain a characterization of all solutions which satisfy Price's polynomial law τ − 3 as a lower bound. Our analysis relies on physical space techniques and uses the vector field approach for almost-sharp decay estimates introduced in our companion paper. In the black hole case, our estimates hold in the domain of outer communications up to and including the event horizon. Our work is motivated by the stability problem for black hole exteriors and strong cosmic censorship for black hole interiors.

Authors:Rohini Ramadas Pages: 622 - 667 Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Rohini Ramadas Hurwitz correspondences are certain multi-valued self-maps of the moduli space M 0 , N . They arise in the study of Thurston's topological characterization of rational functions. We compare the dynamics of Hurwitz correspondence H on two different compactifications of M 0 , N : the Deligne-Mumford compactification M ‾ 0 , N , as well as a Hassett space of weighted stable curves. We use this comparison to show that the k-th dynamical degree of H is the absolute value of the dominant eigenvalue of the pushforward induced by H on a natural quotient of H 2 k ( M ‾ 0 , N ) .

Authors:Pascale Roesch; Xiaoguang Wang; Yongcheng Yin Pages: 1 - 59 Abstract: Publication date: 15 December 2017 Source:Advances in Mathematics, Volume 322 Author(s): Pascale Roesch, Xiaoguang Wang, Yongcheng Yin In this article, we study the topology and bifurcations of the moduli space M 3 of cubic Newton maps. It's a subspace of the moduli space of cubic rational maps, carrying the Riemann orbifold structure ( C ˆ , ( 2 , 3 , ∞ ) ) . We prove two results: • The boundary of the unique unbounded hyperbolic component is a Jordan arc and the boundaries of all other hyperbolic components are Jordan curves; • The Head's angle map is surjective and monotone. The fibers of this map are characterized completely. The first result is a moduli space analogue of the first author's dynamical regularity theorem [37]. The second result confirms a conjecture of Tan Lei.

Authors:Nitu Kitchloo; Vitaly Lorman; W. Stephen Wilson Pages: 60 - 82 Abstract: Publication date: 15 December 2017 Source:Advances in Mathematics, Volume 322 Author(s): Nitu Kitchloo, Vitaly Lorman, W. Stephen Wilson We take advantage of the internal algebraic structure of the Bockstein spectral sequence converging to E R ( n ) ⁎ ( p t ) to prove that for spaces Z that are part of Landweber flat real pairs with respect to E ( n ) (see Definition 2.9), the cohomology ring E R ( n ) ⁎ ( Z ) can be obtained from E ( n ) ⁎ ( Z ) by base change. In particular, our results allow us to compute the Real Johnson–Wilson cohomology of the Eilenberg–MacLane spaces Z = K ( Z , 2 m + 1 ) , K ( Z / 2 q , 2 m ) , K ( Z / 2 , m ) for any natural numbers m and q, as well as connective covers of BO: BO , BSO , BSpin and BO 〈 8 〉 .

Authors:Alice Rizzardo Pages: 83 - 96 Abstract: Publication date: 15 December 2017 Source:Advances in Mathematics, Volume 322 Author(s): Alice Rizzardo Given a Fourier–Mukai functor Φ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to Φ, and also give explicit formulas for them. These formulas are simple and natural, and recover the usual formulas when the Fourier–Mukai kernel is a perfect complex. This extends previous work of [1,12,13] and has applications to the twist autoequivalences of [9].

Authors:Alex Massarenti Pages: 97 - 131 Abstract: Publication date: 15 December 2017 Source:Advances in Mathematics, Volume 322 Author(s): Alex Massarenti Let X [ n ] be the Fulton–MacPherson compactification of the configuration space of n ordered points on a smooth projective variety X. We prove that if either n ≠ 2 or dim ( X ) ≥ 2 , then the connected component of the identity of Aut ( X [ n ] ) is isomorphic to the connected component of the identity of Aut ( X ) . When X = C is a curve of genus g ( C ) ≠ 1 we classify the dominant morphisms C [ n ] → C [ r ] , and thanks to this we manage to compute the whole automorphism group of C [ n ] , namely Aut ( C [ n ] ) ≅ S n × Aut ( C ) for any n ≠ 2 , while Aut ( C [ 2 ] ) ≅ S 2 ⋉ ( Aut ( C ) × Aut ( C ) ) . Furthermore, we extend these results on the automorphisms to the case where X = C 1 × . . . × C r is a product of curves of genus g ( C i ) ≥ 2 . Finally, using the techniques developed to deal with Fulton–MacPherson spaces, we study the automorphism groups of some Kontsevich moduli spaces M ‾ 0 , n ( P N , d ) .

Authors:Chongying Dong; Li Ren; Feng Xu Pages: 1 - 30 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Chongying Dong, Li Ren, Feng Xu Let V be a simple vertex operator algebra and G a finite automorphism group of V such that V G is regular. It is proved that every irreducible V G -module occurs in an irreducible g-twisted V-module for some g ∈ G . Moreover, the quantum dimensions of irreducible V G -modules are determined and a global dimension formula for V in terms of twisted modules is obtained. In particular, the orbifold theory conjecture is completely solved if G is solvable.

Authors:Huyi Hu; Yongxia Hua; Weisheng Wu Pages: 31 - 68 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Huyi Hu, Yongxia Hua, Weisheng Wu We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphisms, which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier–Young, though we do not use increasing partitions. The unstable topological entropy is defined equivalently via separated sets, spanning sets and open covers along a piece of unstable leaf, and it coincides with the unstable volume growth along unstable foliation. We also obtain some properties for the unstable metric entropy such as affineness, upper semi-continuity and a version of Shannon–McMillan–Breiman theorem.

Authors:Mikhail Belolipetsky; Benjamin Linowitz Pages: 69 - 79 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Mikhail Belolipetsky, Benjamin Linowitz Given a simple Lie group H of real rank at least 2 we show that the maximum cardinality of a set of isospectral non-isometric H-locally symmetric spaces of volume at most x grows at least as fast as x c log x / ( log log x ) 2 where c = c ( H ) is a positive constant. In contrast with the real rank 1 case, this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of Belolipetsky and Lubotzky [2]. Our proof uses Sunada's method, results of [2], and some deep results from number theory. We also discuss an open number-theoretical problem which would imply an even faster growth estimate.

Authors:Joseph Chuang; Hyohe Miyachi; Kai Meng Tan Pages: 80 - 159 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Joseph Chuang, Hyohe Miyachi, Kai Meng Tan We provide closed formulas for a large subset of the canonical basis vectors of the Fock space representation of U q ( sl ˆ e ) . These formulas arise from parallelotopes which assemble to form polytopal complexes. The subgraphs of the Ext 1 -quivers of v-Schur algebras at complex e-th roots of unity generated by simple modules corresponding to these canonical basis vectors are given by the 1-skeletons of the polytopal complexes.

Authors:Sylvie Corteel; Olya Mandelshtam; Lauren Williams Pages: 160 - 204 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Sylvie Corteel, Olya Mandelshtam, Lauren Williams In previous work [12–14], the first and third authors introduced staircase tableaux, which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process (ASEP) and for the moments of the Askey–Wilson weight function. The fact that the ASEP and Askey–Wilson moments are related at all is unexpected, and is due to [45]. The ASEP is a model of particles hopping on a one-dimensional lattice of N sites with open boundaries; particles can enter and exit at both left and right borders. It was introduced around 1970 [34,43] and is cited as a model for both traffic flow and translation in protein synthesis. Meanwhile, the Askey–Wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials. So from this previous work, we have the relationship ASEP −− staircase tableaux −− Askey–Wilson moments. The Askey–Wilson polynomials can be viewed as the one-variable case of the multivariate Koornwinder polynomials, which are also known as the Macdonald polynomials attached to the non-reduced affine root system ( C n ∨ , C n ). It is natural then to ask whether one can generalize the relationships among the ASEP, Askey–Wilson moments, and staircase tableaux, in such a way that Koornwinder moments replace Askey–Wilson moments. In [15], we made a precise link between Koornwinder moments and the two-species ASEP, a generalization of the ASEP which has two species of particles with different “weights.” In this article we introduce rhombic staircase tableaux, and show that we have the relationship 2-species ASEP −− rhombic staircase tableaux −− Koornwinder moments. In particular, we give formulas for the stationary distribution of the two-species ASEP and for Koornwinder moments, in terms of rhombic staircase tableaux.

Authors:Keomkyo Seo Pages: 205 - 220 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Keomkyo Seo In this paper, we derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L 2 norm of the mean curvature vector in a Riemannian manifold.

Authors:Jesse Leo Kass; Nicola Pagani Pages: 221 - 268 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Jesse Leo Kass, Nicola Pagani The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth pointed curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable pointed curves using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with this parameter. We use this result to analyze divisors on the moduli space of smooth pointed curves that have recently been studied by Grushevsky–Zakharov, Hain and Müller. Finally, we compute the pullback of the theta divisor studied in Alexeev's work on stable semiabelic varieties and in Caporaso's work on theta divisors of compactified Jacobians.

Authors:John H. Johnson; Florian Karl Richter Pages: 269 - 286 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): John H. Johnson, Florian Karl Richter Answering a question posed by Bergelson and Leibman in [6], we establish a nilpotent version of the Polynomial Hales–Jewett Theorem that contains the main theorem in [6] as a special case. Important to the formulation and the proof of our main theorem is the notion of a relative syndetic set (relative with respect to a closed non-empty subsets of β G ) [25]. As a corollary of our main theorem we prove an extension of the restricted van der Waerden Theorem to nilpotent groups, which involves nilprogressions.

Authors:Jacob Fox; László Miklós Lovász Pages: 287 - 297 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Jacob Fox, László Miklós Lovász Let p be a fixed prime. A triangle in F p n is an ordered triple ( x , y , z ) of points satisfying x + y + z = 0 . Let N = p n = F p n . Green proved an arithmetic triangle removal lemma which says that for every ϵ > 0 and prime p, there is a δ > 0 such that if X , Y , Z ⊂ F p n and the number of triangles in X × Y × Z is at most δ N 2 , then we can delete ϵN elements from X, Y, and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1 / δ can be taken to be an exponential tower of twos of height logarithmic in 1 / ϵ . We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in F p n . We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is an explicit number C p such that we may take δ = ( ϵ / 3 ) C p , and we must have δ ≤ ϵ C p − o ( 1 ) . In particular, C 2 = 1 + 1 / ( 5 / 3 − log 2 3 ) ≈ 13.239 , and C 3 = 1 + 1 / c 3 with c 3 = 1 − log b log 3 , b = a − 2 / 3 + a 1 / 3 + a 4 / 3 , and a = 33 − 1 8 , which gives C 3 ≈ 13.901 . The proof uses the essentially sharp bound on mu... PubDate: 2017-10-08T14:37:17Z DOI: 10.1016/j.aim.2017.09.037 Issue No:Vol. 321 (2017)

Authors:Josep Àlvarez Montaner; Craig Huneke; Luis Núñez-Betancourt Pages: 298 - 325 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Josep Àlvarez Montaner, Craig Huneke, Luis Núñez-Betancourt We study the structure of D-modules over a ring R which is a direct summand of a polynomial or a power series ring S with coefficients over a field. We relate properties of D-modules over R to D-modules over S. We show that the localization R f and the local cohomology module H I i ( R ) have finite length as D-modules over R. Furthermore, we show the existence of the Bernstein–Sato polynomial for elements in R. In positive characteristic, we use this relation between D-modules over R and S to show that the set of F-jumping numbers of an ideal I ⊆ R is contained in the set of F-jumping numbers of its extension in S. As a consequence, the F-jumping numbers of I in R form a discrete set of rational numbers. We also relate the Bernstein–Sato polynomial in R with the F-thresholds and the F-jumping numbers in R.

Authors:Maria Basterra; Irina Bobkova; Kate Ponto; Ulrike Tillmann; Sarah Yeakel Pages: 391 - 430 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Maria Basterra, Irina Bobkova, Kate Ponto, Ulrike Tillmann, Sarah Yeakel Motivated by the operad built from moduli spaces of Riemann surfaces, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions. We prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces. The recent, strong homological stability results of Galatius and Randal-Williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability. As a consequence diffeomorphism groups and mapping class groups are shown to give rise to infinite loop spaces. Furthermore, the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology is shown to be a map of infinite loop spaces.

Authors:Gus Schrader; Alexander Shapiro Pages: 431 - 474 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Gus Schrader, Alexander Shapiro We construct an algebra embedding of the quantum group U q ( g ) into a central extension of the quantum coordinate ring O q [ G w 0 , w 0 / H ] of the reduced big double Bruhat cell in G. This embedding factors through the Heisenberg double H q of the quantum Borel subalgebra U ≥ 0 , which we relate to O q [ G ] via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck–Springer resolution studied in [10], and the quantum Beilinson–Bernstein theorem investigated in [2] and [36].

Authors:Ashutosh Kumar; Saharon Shelah Pages: 475 - 485 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Ashutosh Kumar, Saharon Shelah We show that for every partition of a set of reals into countable sets there is a transversal of the same outer measure.

Authors:Leandro Arosio; Filippo Bracci Pages: 486 - 512 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Leandro Arosio, Filippo Bracci We prove that a finite family of commuting holomorphic self-maps of the unit ball B q ⊂ C q admits a simultaneous holomorphic conjugacy to a family of commuting automorphisms of a possibly lower dimensional ball, and that such conjugacy satisfies a universal property. As an application we describe when a hyperbolic and a parabolic holomorphic self-map of B q can commute.

Authors:Xuhua He; Sian Nie Pages: 513 - 528 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Xuhua He, Sian Nie In this paper, we study the μ-ordinary locus of a Shimura variety with parahoric level structure. Under the axioms in [12], we show that μ-ordinary locus is a union of certain maximal Ekedahl–Kottwitz–Oort–Rapoport strata introduced in [12] and we give criteria on the density of the μ-ordinary locus.

Authors:Ariel Rapaport Pages: 529 - 546 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Ariel Rapaport We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than 1, such that there exist lines for which dimension conservation does not hold and the projection of the measure is singular. In fact, the set of such directions is residual and the typical slices of the measure, perpendicular to these directions, are discrete.

Authors:Luis Barreira; Davor Dragičević; Claudia Valls Pages: 547 - 591 Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Luis Barreira, Davor Dragičević, Claudia Valls For a sequence of bounded linear operators acting on a Banach space, we consider the notion of nonuniform spectrum. This is defined in terms of the existence of nonuniform exponential dichotomies with an arbitrarily small nonuniform part and can be seen as a nonuniform version of the spectrum introduced by Sacker and Sell in the case of a single trajectory, although now in the infinite-dimensional setting. We give a complete characterization of all possible forms of the nonuniform spectrum for sequences of compact linear operators and, more generally, for sequences of bounded linear operators satisfying a certain asymptotic compactness. Moreover, we provide explicit examples of sequences of compact linear operators acting on the l 2 space of sequences of real numbers for all the possible forms of the nonuniform spectrum. As nontrivial applications, we show that the nonuniform spectrum of a Lyapunov regular sequence is the set of Lyapunov exponents and that the asymptotic behavior persists under sufficiently small nonlinear perturbations, in the sense that the lower and upper Lyapunov exponents of the perturbed dynamics belong to a connected component of the nonuniform spectrum. Finally, we obtain appropriate versions of the results for nonuniformly hyperbolic cocycles.

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:Weiwei Abstract: Publication date: 7 January 2018 Source:Advances in Mathematics, Volume 323 Author(s): Weiwei Wu We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group G. Combining this with some generalizations of Seidel's algebraic frameworks from [35], we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call special isogenous tori. This extends the work of Abouzaid–Smith [2]. We also show that derived Fukaya categories are complete invariants of special isogenous tori.

Authors:Roman Bezrukavnikov; Alexander Braverman; Michael Finkelberg; Dennis Gaitsgory; Alexander Goncharov; Yakov Varshavsky Abstract: Publication date: Available online 8 November 2017 Source:Advances in Mathematics Author(s): Roman Bezrukavnikov, Alexander Braverman, Michael Finkelberg, Dennis Gaitsgory, Alexander Goncharov, Yakov Varshavsky

Authors:Geoffroy Horel Abstract: Publication date: 1 December 2017 Source:Advances in Mathematics, Volume 321 Author(s): Geoffroy Horel In this paper, we prove that the group of homotopy automorphisms of the profinite completion of the operad of little 2-disks is isomorphic to the profinite Grothendieck–Teichmüller group. In particular, the absolute Galois group of Q acts faithfully on the profinite completion of E 2 in the homotopy category of profinite weak operads.