Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): James C. Robinson, Aníbal Rodríguez-BernalAbstractWe analyse the behaviour of solutions of the linear heat equation in Rd for initial data in the classes Mε(Rd) of Radon measures with ∫Rde−ε x 2d u0 0Mε(Rd) consists of those initial data for which a solution of the heat equation can be given for all time using the heat kernel representation formula. We prove existence, uniqueness, and regularity results for such initial data, which can grow rapidly at infinity, and then show that they give rise to properties associated more often with nonlinear models. We demonstrate the finite-time blowup of solutions, showing that the set of blowup points is the complement of a convex set, and that given any closed convex set there is an initial condition whose solutions remain bounded precisely on this set at the ‘blowup time’. We also show that wild oscillations are possible from non-negative initial data as t→∞ and that one can prescribe the behaviour of u(0,t) to be any real-analytic function γ(t) on [0,∞).

Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): Nima Amini, Petter BrändénAbstractThe generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. Hyperbolic polynomials give rise to a class of (hyperbolic) matroids which properly contains the class of matroids representable over the complex numbers. This connection was used by the second author to construct counterexamples to algebraic (stronger) versions of the generalized Lax conjecture by considering a non-representable hyperbolic matroid. The Vámos matroid and a generalization of it are, prior to this work, the only known instances of non-representable hyperbolic matroids.We prove that the Non-Pappus and Non-Desargues matroids are non-representable hyperbolic matroids by exploiting a connection between Euclidean Jordan algebras and projective geometries. We further identify a large class of hyperbolic matroids which contains the Vámos matroid and the generalized Vámos matroids recently studied by Burton, Vinzant and Youm. This proves a conjecture of Burton et al. We also prove that many of the matroids considered here are non-representable. The proof of hyperbolicity for the matroids in the class depends on proving nonnegativity of certain symmetric polynomials. In particular we generalize and strengthen several inequalities in the literature, such as the Laguerre–Turán inequality and an inequality due to Jensen. Finally we explore consequences to algebraic versions of the generalized Lax conjecture.

Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): Alexandros Eskenazis, Piotr Nayar, Tomasz TkoczAbstractThis article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors uniformly distributed on the unit ball of the space ℓqn for q∈(2,∞), complementing past works that treated q∈(0,2]∪{∞}. As a byproduct of this result, we prove an extremal property for weighted sums of symmetric uniform distributions among all symmetric unimodal distributions. In the second part we provide a one-to-one correspondence between vectors of moments of symmetric log-concave functions and two simple classes of piecewise log-affine functions. These functions are shown to be the unique extremisers of the p-th moment functional, under the constraint of a finite number of other moments being fixed, which is a refinement of the description of extremisers provided by the generalised localisation theorem of Fradelizi and Guédon (2006) [7].

Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): C. Brech, J. Lopez-Abad, S. TodorcevicAbstractWe study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal κ smaller than the first Mahlo cardinal there is a reflexive Banach space of density κ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic structure of the spaces. The collections of families are of independent interest and their existence is proved inductively. The fundamental stepping up argument is the analysis of such collections of families on trees.

Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): Nicola Soave, Susanna TerraciniAbstractWe are concerned with the nodal set of solutions to equations of the form−Δu=λ+(u+)q−1−λ−(u−)q−1in B1 where λ+,λ−>0, q∈[1,2), B1=B1(0) is the unit ball in RN, N≥2, and u+:=max{u,0}, u−:=max{−u,0} are the positive and the negative part of u, respectively. This class includes, the unstable two-phase membrane problem (q=1), as well as sublinear equations for 1

Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): J. Ederson M. Braga, Diego MoreiraAbstractIn this paper, we construct new barriers for the Pucci extremal operators with unbounded RHS. The geometry of these barriers is given by a Harnack inequality up to the boundary type estimate. Under the possession of these barriers, we prove a new quantitative version of the Hopf–Oleĭnik Lemma for quasilinear elliptic equations with g-Laplace type growth. Finally, we prove (sharp) regularity for ω-semiconvex supersolutions for some nonlinear PDEs. These results are new even for second order linear elliptic equations in nondivergence form. Moreover, these estimates extend and improve a classical a priori estimate proven by L. Caffarelli, J.J. Kohn, J. Spruck and L. Nirenberg in [13] in 1985 as well as a more recent result on the C1,1 regularity for convex supersolutions obtained by C. Imbert in [33] in 2006.

Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): Cristian Lenart, Arthur LubovskyAbstractKirillov–Reshetikhin (KR) crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor product of column shape Kirillov–Reshetikhin crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the so-called quantum alcove model. We enhance this model by using it to give a uniform realization of the corresponding combinatorial R-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of column shape KR crystals. In other words, we are generalizing to all Lie types Schützenberger's sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial R-matrix in type A. Our construction is in terms of certain combinatorial moves, called quantum Yang–Baxter moves, which are explicitly described by reduction to the rank 2 root systems. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves joining the fundamental one to a translation of it.

Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): D. KaledinAbstractFollowing an old suggestion of M. Kontsevich, and inspired by recent work of A. Beilinson and B. Bhatt, we introduce a new version of periodic cyclic homology for DG algebras and DG categories. We call it co-periodic cyclic homology. It is always torsion, so that it vanishes in char 0. However, we show that co-periodic cyclic homology is derived-Morita invariant, and that it coincides with the usual periodic cyclic homology for smooth cohomologically bounded DG algebras over a torsion ring. For DG categories over a field of odd positive characteristic, we also establish a non-commutative generalization of the conjugate spectral sequence converging to our co-periodic cyclic homology groups.

Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): Cesar CuencaAbstractThe (BC type) z-measures are a family of four parameter z,z′,a,b probability measures on the path space of the nonnegative Gelfand–Tsetlin graph with Jacobi-edge multiplicities. We can interpret the z-measures as random point processes Pz,z′,a,b on the punctured positive real line X=R>0∖{1}. Our main result is that these random processes are determinantal and moreover we compute their correlation kernels explicitly in terms of hypergeometric functions.For very special values of the parameters z,z′, the processes Pz,z′,a,b on X are essentially scaling limits of Racah orthogonal polynomial ensembles and their correlation kernels can be computed simply from some limits of the Racah polynomials. Thus, in the language of random matrices, we study certain analytic continuations of processes that are limits of Racah ensembles, and such that they retain the determinantal structure. Another interpretation of our results, and the main motivation of this paper, is the representation theory of big groups. In representation-theoretic terms, this paper solves a natural problem of harmonic analysis for several infinite-dimensional symmetric spaces.

Abstract: Publication date: 20 August 2018Source: Advances in Mathematics, Volume 334Author(s): Maria Carvalho, Fagner B. Rodrigues, Paulo VarandasAbstractIn this paper we introduce a notion of measure theoretical entropy for a finitely generated free semigroup action and establish a variational principle when the semigroup is generated by continuous self maps on a compact metric space and has finite topological entropy. In the case of semigroups generated by Ruelle-expanding maps we prove the existence of equilibrium states and describe some of their properties. Of independent interest are the different ways we will present to compute the metric entropy and a characterization of the stationary measures.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Imma Gálvez-Carrillo, Joachim Kock, Andrew TonksAbstractThis is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in ∞-groupoids. A decomposition space is a simplicial ∞-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition.In this paper, we introduce various technical conditions on decomposition spaces. The first is a completeness condition (weaker than Rezk completeness), needed to control simplicial nondegeneracy. For complete decomposition spaces we establish a general Möbius inversion principle, expressed as an explicit equivalence of ∞-groupoids. Next we analyse two finiteness conditions on decomposition spaces. The first, that of locally finite length, guarantees the existence of the important length filtration for the associated incidence coalgebra. We show that a decomposition space of locally finite length is actually the left Kan extension of a semi-simplicial space. The second finiteness condition, local finiteness, ensures we can take homotopy cardinality to pass from the level of ∞-groupoids to the level of Q-vector spaces.These three conditions — completeness, locally finite length, and local finiteness — together define our notion of Möbius decomposition space, which extends Leroux's notion of Möbius category (in turn a common generalisation of the locally finite posets of Rota et al. and of the finite decomposition monoids of Cartier–Foata), but which also covers many coalgebra constructions which do not arise from Möbius categories, such as the Faà di Bruno and Connes–Kreimer bialgebras.Note: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov [6] who call them unital 2-Segal spaces.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Alexey PokrovskiyAbstractAharoni and Berger conjectured that in every proper edge-colouring of a bipartite multigraph by n colours with at least n+1 edges of each colour there is a rainbow matching using every colour. This conjecture generalizes a longstanding problem of Brualdi and Stein about transversals in Latin squares. Here an approximate version of the Aharoni–Berger Conjecture is proved—it is shown that if there are at least n+o(n) edges of each colour in a proper n-edge-colouring of a bipartite multigraph then there is a rainbow matching using every colour.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Yu-Shen LinAbstractIn the paper, we study the wall-crossing phenomenon of reduced open Gromov–Witten invariants on K3 surfaces with rigid special Lagrangian boundary condition. As a corollary, we derived the multiple cover formula for the reduced open Gromov–Witten invariants, which is an open analog of the Gopakumar–Vafa conjecture of genus zero.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): David Ayala, John Francis, Nick RozenblyumAbstractWe construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata. Our main result constructs labeling systems on disk-stratified vari-framed n-manifolds from (∞,n)-categories. These (∞,n)-categories, in contrast with the literature to date, are not required to have adjoints. This allows the following conceptual definition: the factorization homology∫MC of a framed n-manifold M with coefficients in an (∞,n)-category C is the classifying space of C-labeled disk-stratifications over M. The core calculation underlying our main result is the following: for any disk-stratified manifold, the space of conically smooth diffeomorphisms which preserve a vari-framing is discrete.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Eli Glasner, Yonatan Gutman, XiangDong YeAbstractWe introduce higher order regionally proximal relations suitable for an arbitrary acting group. For minimal abelian group actions, these relations coincide with the ones introduced by Host, Kra and Maass. Our main result is that these relations are equivalence relations whenever the action is minimal. This was known for abelian actions by a result of Shao and Ye. We also show that these relations lift through extensions between minimal systems. Answering a question by Tao, given a minimal system, we prove that the regionally proximal equivalence relation of order d corresponds to the maximal dynamical Antolín Camarena–Szegedy nilspace factor of order at most d. In particular the regionally proximal equivalence relation of order one corresponds to the maximal abelian group factor. Finally by using a result of Gutman, Manners and Varjú under some restrictions on the acting group, it follows that the regionally proximal equivalence relation of order d corresponds to the maximal pronilfactor of order at most d (a factor which is an inverse limit of nilsystems of order at most d).

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): James East, James D. Mitchell, Nik Ruškuc, Michael TorpeyAbstractWe give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley–Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I→M onto the minimal ideal, a congruence on M, and a normal subgroup of a maximal subgroup outside I.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): David Bate, Sean LiAbstractWe demonstrate the necessity of a Poincaré type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon–Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect nearby points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger–Kleiner [12], we show that our conditions are also sufficient.We also develop another characterization of RNP Lipschitz differentiability spaces by connecting points by curves that form a rich structure of partial derivatives that were first discussed in [5].

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Peter Albers, Serge TabachnikovAbstractIn this article we introduce a simple dynamical system called symplectic billiards. As opposed to usual/Birkhoff billiards, where length is the generating function, for symplectic billiards symplectic area is the generating function. We explore basic properties and exhibit several similarities, but also differences of symplectic billiards to Birkhoff billiards.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Tom Koornwinder, Aleksey Kostenko, Gerald TeschlAbstractThe present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrödinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrödinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Amin Coja-Oghlan, Florent Krzakala, Will Perkins, Lenka ZdeborováAbstractVindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models. We work out several concrete applications of these general results. For instance, we pinpoint the exact condensation phase transition in the Potts antiferromagnet on the random graph, thereby improving prior approximate results (Contucci et al., 2013) [34]. Further, we prove the conjecture from Krzakala et al. (2007) [55] about the condensation phase transition in the random graph coloring problem for any number q≥3 of colors. Moreover, we prove the conjecture on the information-theoretic threshold in the disassortative stochastic block model (Decelle et al., 2011) [35]. Additionally, our general result implies the conjectured formula for the mutual information in Low-Density Generator Matrix codes (Montanari, 2005) [73].

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Ignacio BarrosAbstractWe address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to genus 3≤g≤6, we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most g−1, hence showing in addition that these strata are unirational.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Ingrid Bauer, Fabrizio CataneseAbstractThis article investigates the subject of rigid compact complex manifolds. First of all we investigate the different notions of rigidity (local rigidity, global rigidity, infinitesimal rigidity, etale rigidity and strong rigidity) and the relations among them. Only for curves these notions coincide and the only rigid curve is the projective line. For surfaces we prove that a rigid surface which is not minimal of general type is either a Del Pezzo surface of degree ≥5 or an Inoue surface. We give examples of rigid manifolds of dimension n≥3 and Kodaira dimensions 0, and 2≤k≤n. Our main theorem is that the Hirzebruch Kummer coverings of exponent n≥4 branched on a complete quadrangle are infinitesimally rigid. Moreover, we pose a number of questions.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Maki Nakasuji, Ouamporn Phuksuwan, Yoshinori YamasakiAbstractWe introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler–Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi–Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding to the Schur multiple zeta functions. We obtain the similar results as above for them and, furthermore, describe the images of them by the antipode of the Hopf algebra of quasi-symmetric functions explicitly. Finally, we establish iterated integral representations of the Schur multiple zeta values of ribbon type, which yield a duality for them in some cases.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Ya'acov Peterzil, Sergei StarchenkoAbstractWe consider the covering map π:Cn→T of a compact complex torus. Given an algebraic variety X⊆Cn we describe the topological closure of π(X) in T. We obtain a similar description when T is a real torus and X⊆Rn is a set definable in an o-minimal structure over the reals.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Huabin Ge, Bobo HuaAbstractIn this paper, we prove the long time existence of the combinatorial Calabi flow in hyperbolic background geometry and the convergence of the flow to a smooth hyperbolic surface under Thurston's combinatorial condition in full generality, which improves the result in [14]. This flow provides a natural algorithm to find Thurston's hyperbolic circle patterns.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Scott A. WolpertAbstractDeformations of compact Riemann surfaces are considered using a Čech cohomology sliding overlaps approach. Cocycles are calculated for conformal cutting and regluing deformations at zeros of Abelian differentials. Deformations fixing the periods of a differential and deformations splitting zeros are considered. A second order deformation expansion is presented for the Riemann period matrix. A complete deformation expansion is presented for Abelian differentials. Schiffer's kernel function approach for deformations of a Green's function is followed.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Paul SobajeAbstractLet G be a simple and simply connected algebraic group over an algebraically closed field k of characteristic p>0. Assume that p is good for the root system of G and that the covering map Gsc→G is separable. In previous work we proved the existence of a (not necessarily unique) Springer isomorphism for G that behaved like the exponential map on the restricted nullcone of G.In the present paper we give a formal definition of these maps, which we call ‘generalized exponential maps.’ We provide an explicit and uniform construction of such maps for all root systems, demonstrate their existence over Z(p), and give a complete parameterization of all such maps. One application is that this gives a uniform approach to dealing with the “saturation problem” for a unipotent element u in G, providing a new proof of the known result that u lies inside a subgroup of CG(u) that is isomorphic to a truncated Witt group. We also develop a number of other explicit and new computations for g and for G. This paper grew out of an attempt to answer a series of questions posed to us by P. Deligne, who also contributed several of the new ideas that appear here.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Alexey Ananyevskiy, Andrei DruzhininAbstractA rigidity property for the homotopy invariant stable linear framed presheaves is established. As a consequence a variant of Gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a ϕ-torsion spectrum with ϕ∈GW(k) of rank coprime to the (exponential) characteristic of the base field k. It is shown that the values of such cohomology theories at an essentially smooth Henselian ring and its residue field coincide. The result is applicable to cohomology theories representable by n-torsion spectra as well as to the ones representable by η-periodic spectra and spectra related to Witt groups.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Eugene Gorsky, András NémethiAbstractWe study the qualitative structure of the set LS of integral L-space surgery slopes for links with two components. It is known that the set of L-space surgery slopes for a nontrivial L-space knot is always a positive half-line. However, already for two-component torus links the set LS has a very complicated structure, e.g. in some cases it can be unbounded from below. In order to probe the geometry of this set, we ask if it is bounded from below for more general L-space links with two components.For algebraic two-component links we provide three complete characterizations for the boundedness from below: one in terms of the Alexander polynomial, one in terms of the embedded resolution graph, and one in terms of the so-called h-function introduced by the authors in [9]. It turns out that LS is bounded from below for most algebraic links. If it is unbounded from below, it must contain a negative half-line parallel to one of the axes. We also give a sufficient condition for boundedness for arbitrary L-space links.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Pavel SechinAbstractIn this paper we investigate the structure of algebraic cobordism of Levine–Morel as a module over the Lazard ring with the action of Landweber–Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration Ω(r)⁎(X) are unions of finitely presented L-modules of very specific structure. Namely, these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules L/I(p,n)x where degx≥pn−1p−1. As a corollary we prove the Syzygies Conjecture of Vishik on the existence of certain free L-resolutions of Ω⁎(X), and show that algebraic cobordism of a smooth surface can be described in terms of K0 together with a topological filtration.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): M.E. Descotte, E.J. Dubuc, M. SzyldAbstractIn this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A⟶FB is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similarly we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary Cat-valued 2-functor as a conical σ-bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula.As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a Cat-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctorA⟶Catis flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Dennis Gaitsgory, Alexander Yom DinAbstractDrinfeld suggested the definition of a certain endo-functor, called the pseudo-identity functor, on the category of D-modules on an algebraic stack. We extend this definition to an arbitrary DG category, and show that if certain finiteness conditions are satisfied, this functor is the inverse of the Serre functor. We apply this to the category of (g,K)-modules, and we stipulate that the pseudo-identity functor is the analog of the Deligne–Lusztig functor. In order to support this, we show that the pseudo-identity functor for (g,K)-modules is isomorphic to the composition of cohomological and contragredient dualities, which is parallel to the corresponding assertion for p-adic groups in [4].

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Éric RicardAbstractWe prove that the homogeneous functional calculus associated to x↦ x θ or x↦sgn(x) x θ for 0

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Xiao-Wu Chen, Yu YeAbstractWe introduce the notions of a D-standard abelian category and a K-standard additive category. We prove that for a finite dimensional algebra A, its module category is D-standard if and only if any derived autoequivalence on A is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. We prove that if the subcategory of projective A-modules is K-standard, then the module category is D-standard. We provide new examples of D-standard module categories.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Judit Abardia-Evéquoz, Andrea Colesanti, Eugenia Saorín GómezAbstractWe provide a description of the space of continuous and translation invariant Minkowski valuations Φ:Kn→Kn for which there is an upper and a lower bound for the volume of Φ(K) in terms of the volume of the convex body K itself. Although no invariance with respect to a group acting on the space of convex bodies is imposed, we prove that only two types of operators appear: a family of operators having only cylinders over (n−1)-dimensional convex bodies as images, and a second family consisting essentially of 1-homogeneous operators. Using this description, we give improvements of some known characterization results for the difference body.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Ulrich Bunke, Thomas Nikolaus, Georg TammeAbstractWe prove that the Beilinson regulator, which is a map from K-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E∞-ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R. The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K-theory spectrum is the source. To prove this result we compute the space of maps from the motivic K-theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E∞-refinement and prove a uniqueness result for these refinements.

Abstract: Publication date: 31 July 2018Source: Advances in Mathematics, Volume 333Author(s): Thomas Church, Jeremy Miller, Rohit Nagpal, Jens ReinholdAbstractWe prove two general results concerning spectral sequences of FI-modules. These results can be used to significantly improve stable ranges in a large portion of the stability theorems for FI-modules currently in the literature. We work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range. Previously, the best stable ranges known in these examples were exponential. Up to an additive constant, our work on congruence subgroups verifies a conjecture of Djament.