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  • TILTING THEORY FOR GORENSTEIN RINGS IN DIMENSION ONE
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-07-03
    RAGNAR-OLAF BUCHWEITZ; OSAMU IYAMA; KOTA YAMAURA

    In representation theory, commutative algebra and algebraic geometry, it is an important problem to understand when the triangulated category $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting (respectively, silting) object for a $\mathbb{Z}$ -graded commutative Gorenstein ring $R=\bigoplus _{i\geqslant 0}R_{i}$ . Here $\mathsf{D}_{\o

    更新日期:2020-07-03
  • GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-06-30
    ANDREA LUCCHINI; CLAUDE MARION; GARETH TRACEY

    For a finite group $G$ , let $d(G)$ denote the minimal number of elements required to generate $G$ . In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method

    更新日期:2020-06-30
  • SOLVING DIFFERENCE EQUATIONS IN SEQUENCES: UNIVERSALITY AND UNDECIDABILITY
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-06-30
    GLEB POGUDIN; THOMAS SCANLON; MICHAEL WIBMER

    We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz

    更新日期:2020-06-30
  • CATEGORICAL COMPLEXITY
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-06-30
    SAUGATA BASU; UMUT ISIK

    We introduce a notion of complexity of diagrams (and, in particular, of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest such as finite sets, Boolean functions, topological spaces, vector spaces, semilinear and

    更新日期:2020-06-30
  • WAVE FRONT HOLONOMICITY OF $\text{C}^{\text{exp}}$ -CLASS DISTRIBUTIONS ON NON-ARCHIMEDEAN LOCAL FIELDS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-06-30
    AVRAHAM AIZENBUD; RAF CLUCKERS

    Many phenomena in geometry and analysis can be explained via the theory of $D$ -modules, but this theory explains close to nothing in the non-archimedean case, by the absence of integration by parts. Hence there is a need to look for alternatives. A central example of a notion based on the theory of $D$ -modules is the notion of holonomic distributions. We study two recent alternatives of this notion

    更新日期:2020-06-30
  • A CLASS OF NONHOLOMORPHIC MODULAR FORMS II: EQUIVARIANT ITERATED EISENSTEIN INTEGRALS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-05-28
    FRANCIS BROWN

    We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular

    更新日期:2020-05-28
  • COARSE AND FINE GEOMETRY OF THE THURSTON METRIC
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-05-26
    DAVID DUMAS; ANNA LENZHEN; KASRA RAFI; JING TAO

    We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $S$ . Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and

    更新日期:2020-05-26
  • INNER AMENABLE GROUPOIDS AND CENTRAL SEQUENCES
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-05-26
    YOSHIKATA KIDA; ROBIN TUCKER-DROB

    We introduce inner amenability for discrete probability-measure-preserving (p.m.p.) groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives

    更新日期:2020-05-26
  • RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-05-26
    MAIKE GRUCHOT; ALASTAIR LITTERICK; GERHARD RÖHRLE

    We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely

    更新日期:2020-05-26
  • A LIPSCHITZ METRIC FOR THE CAMASSA–HOLM EQUATION
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-05-21
    JOSÉ A. CARRILLO; KATRIN GRUNERT; HELGE HOLDEN

    We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown

    更新日期:2020-05-21
  • PERIOD IDENTITIES OF CM FORMS ON QUATERNION ALGEBRAS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-05-20
    CHARLOTTE CHAN

    Waldspurger’s formula gives an identity between the norm of a torus period and an $L$ -function of the twist of an automorphic representation on GL(2). For any two Hecke characters of a fixed quadratic extension, one can consider the two torus periods coming from integrating one character against the automorphic induction of the other. Because the corresponding $L$ -functions agree, (the norms of)

    更新日期:2020-05-20
  • APPROXIMATING SMOOTH, MULTIVARIATE FUNCTIONS ON IRREGULAR DOMAINS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-05-20
    BEN ADCOCK; DAAN HUYBRECHS

    In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain

    更新日期:2020-05-20
  • FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-05-08
    NATHAN CHEN; DAVID STAPLETON

    We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index  $e$ , then the degree of irrationality of a very general complex Fano hypersurface of index  $e$ and dimension n is bounded from below by a constant times  $\sqrt{n}$ . To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality

    更新日期:2020-05-08
  • ON THE IRREDUCIBLE COMPONENTS OF SOME CRYSTALLINE DEFORMATION RINGS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-04-24
    ROBIN BARTLETT

    We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_{K}$ for a finite extension $K/\mathbb{Q}_{p}$ . This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the unrestricted deformation ring. For $K$ unramified over $\mathbb{Q}_{p}$ and Hodge–Tate weights in $[0,p]$ , we study the geometry of this space

    更新日期:2020-04-24
  • A MODULI STACK OF TROPICAL CURVES
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-04-24
    RENZO CAVALIERI; MELODY CHAN; MARTIN ULIRSCH; JONATHAN WISE

    We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces

    更新日期:2020-04-24
  • THE LIPMAN–ZARISKI CONJECTURE IN GENUS ONE HIGHER
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-04-23
    HANNAH BERGNER; PATRICK GRAF

    We prove the Lipman–Zariski conjecture for complex surface singularities with $p_{g}-g-b\leqslant 2$ . Here $p_{g}$ is the geometric genus, $g$ is the sum of the genera of exceptional curves and $b$ is the first Betti number of the dual graph. This improves on a previous result of the second author. As an application, we show that a compact complex surface with a locally free tangent sheaf is smooth

    更新日期:2020-04-23
  • THE FREE ENERGY OF THE TWO-DIMENSIONAL DILUTE BOSE GAS. I. LOWER BOUND
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-04-21
    ANDREAS DEUCHERT; SIMON MAYER; ROBERT SEIRINGER

    We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density $\unicode[STIX]{x1D70C}$ and inverse temperature $\unicode[STIX]{x1D6FD}$ differs from the one of the noninteracting system by the correction term $4\unicode[STIX]{x1D70B}\unicode[STIX]{x1D70C}^{2}|\ln \,a^{2}\unicode[STIX]{x1D70C}|^{-1}(2

    更新日期:2020-04-21
  • HODGE IDEALS FOR $\mathbb{Q}$ -DIVISORS, $V$ -FILTRATION, AND MINIMAL EXPONENT
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-04-17
    MIRCEA MUSTAŢĂ; MIHNEA POPA

    We compute the Hodge ideals of $\mathbb{Q}$ -divisors in terms of the $V$ -filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein–Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log

    更新日期:2020-04-17
  • STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-03-30
    MARCUS DE CHIFFRE; LEV GLEBSKY; ALEXANDER LUBOTZKY; ANDREAS THOM

    Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\text{Sym}(n)$ (in the sofic case) or the finite-dimensional unitary groups $\text{U}(n)$ (in the hyperlinear case)? In the case of $\text{U}(n)$ , the question can be asked with respect to different metrics and norms

    更新日期:2020-03-30
  • FRIEZE PATTERNS WITH COEFFICIENTS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-03-26
    MICHAEL CUNTZ; THORSTEN HOLM; PETER JØRGENSEN

    Frieze patterns, as introduced by Coxeter in the 1970s, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp. In this paper, we study these frieze patterns with coefficients systematically and prove various fundamental results, generalizing

    更新日期:2020-03-26
  • MULTIPLICATIVE PARAMETRIZED HOMOTOPY THEORY VIA SYMMETRIC SPECTRA IN RETRACTIVE SPACES
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-03-19
    FABIAN HEBESTREIT; STEFFEN SAGAVE; CHRISTIAN SCHLICHTKRULL

    In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set-level framework for parametrized homotopy theory. We provide a convolution smash product that descends to the corresponding $\infty$ -categorical product and allows for convenient constructions of commutative parametrized ring spectra. As an immediate application, we compare various models for generalized

    更新日期:2020-03-19
  • PERIODIC TWISTS OF $\operatorname{GL}_{3}$ -AUTOMORPHIC FORMS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-03-12
    EMMANUEL KOWALSKI; YONGXIAO LIN; PHILIPPE MICHEL; WILL SAWIN

    We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on  $\operatorname{SL}_{3}(\mathbf{Z})$ do not correlate with $q$ -periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky–Nelson, corresponding to multiplicative Dirichlet characters, and applies, in particular, to trace functions of small conductor modulo

    更新日期:2020-03-12
  • ERRATUM TO APPENDIX TO ‘2-ADIC INTEGRAL CANONICAL MODELS’
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-03-11
    KEERTHI MADAPUSI PERA

    We use Lau’s classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial.

    更新日期:2020-03-11
  • LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE $d$ -DIMENSIONAL TORUS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-03-06
    JOACKIM BERNIER; ERWAN FAOU; BENOÎT GRÉBERT

    We consider the nonlinear wave equation (NLW) on the $d$ -dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics

    更新日期:2020-03-06
  • $q$ -DEFORMED RATIONALS AND $q$ -CONTINUED FRACTIONS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-03-06
    SOPHIE MORIER-GENOUD; VALENTIN OVSIENKO

    We introduce a notion of $q$ -deformed rational numbers and $q$ -deformed continued fractions. A $q$ -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$ -deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining

    更新日期:2020-03-06
  • WEIGHTED BESOV AND TRIEBEL–LIZORKIN SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-02-26
    HUY-QUI BUI; THE ANH BUI; XUAN THINH DUONG

    Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper, we develop the theory of weighted Besov spaces ${\dot{B}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ and weighted Triebel–Lizorkin spaces ${\dot{F}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ associated with the operator $L$ for the full range

    更新日期:2020-02-26
  • MEASURABLE REALIZATIONS OF ABSTRACT SYSTEMS OF CONGRUENCES
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-02-24
    CLINTON T. CONLEY; ANDREW S. MARKS; SPENCER T. UNGER

    An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$ -divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We

    更新日期:2020-02-24
  • SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-02-04
    FRANCISCO ARANA-HERRERA; JAYADEV S. ATHREYA

    Given integers $g,n\geqslant 0$ satisfying $2-2g-n<0$ , let ${\mathcal{M}}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$ which assigns to $X\in {\mathcal{M}}_{g,n}$ the Thurston measure of the set of measured

    更新日期:2020-02-04
  • THE INTERSECTION MOTIVE OF THE MODULI STACK OF SHTUKAS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-02-03
    TIMO RICHARZ; JAKOB SCHOLBACH

    For a split reductive group $G$ over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated $G$ -shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic $t$ -structures on triangulated categories of motives. This is in accordance with general expectations on the independence of $\ell$ in the Langlands

    更新日期:2020-02-03
  • ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL DATA
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-01-28
    THIERRY DAUDÉ; NIKY KAMRAN; FRANÇOIS NICOLEAU

    We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous

    更新日期:2020-01-28
  • SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-01-20
    JAI ASLAM; SHUJIAN CHEN; FLORIAN FRICK; SAM SALOFF-COSTE; LINUS SETIABRATA; HUGH THOMAS

    Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz’s square peg problem. We prove Hadwiger’s 1971 conjecture that any simple loop in $3$ -space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop. If the loop is rectifiable, there is a rectangle that cuts

    更新日期:2020-01-20
  • OPTIMAL LINE PACKINGS FROM FINITE GROUP ACTIONS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-01-20
    JOSEPH W. IVERSON; JOHN JASPER; DUSTIN G. MIXON

    We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a

    更新日期:2020-01-20
  • AN EXTENSION OF ROHRLICH’S THEOREM TO THE $j$ -FUNCTION
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-01-15
    KATHRIN BRINGMANN; BEN KANE

    We start by recalling the following theorem of Rohrlich [17]. To state it, let $\unicode[STIX]{x1D714}_{\mathfrak{z}}$ denote half of the size of the stabilizer $\unicode[STIX]{x1D6E4}_{\mathfrak{z}}$ of $\mathfrak{z}\in \mathbb{H}$ in $\text{SL}_{2}(\mathbb{Z})$ and for a meromorphic function $f:\mathbb{H}\rightarrow \mathbb{C}$ let $\text{ord}_{\mathfrak{z}}(f)$ be the order of vanishing of $f$ at 

    更新日期:2020-01-15
  • GLOBAL ACTIONS AND VECTOR $K$ -THEORY
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-01-15
    ANTHONY BAK; ANURADHA S. GARGE

    Purely algebraic objects like abstract groups, coset spaces, and G-modules do not have a notion of hole as do analytical and topological objects. However, equipping an algebraic object with a global action reveals holes in it and thanks to the homotopy theory of global actions, the holes can be described and quantified much as they are in the homotopy theory of topological spaces. Part I of this article

    更新日期:2020-01-15
  • ON THE EXISTENCE OF ADMISSIBLE SUPERSINGULAR REPRESENTATIONS OF $p$ -ADIC REDUCTIVE GROUPS
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-01-13
    FLORIAN HERZIG; KAROL KOZIOŁ; MARIE-FRANCE VIGNÉRAS

    Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_{p}$ and that $C$ is a field of characteristic  $p$ . We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over  $C$ .

    更新日期:2020-01-13
  • EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-01-10
    JIYUAN HAN; JEFF A. VIACLOVSKY

    Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat Kähler metrics on a minimal Kähler surface whose Kähler classes stay in a compact subset of the interior of the Kähler cone must have a convergent subsequence. As an application, we prove the existence of global moduli spaces of scalar-flat Kähler ALE

    更新日期:2020-01-10
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