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  • A class of continuous non-associative algebras arising from algebraic groups including
    Forum Math. Sigma (IF 1.464) Pub Date : 2021-01-14
    Maurice Chayet; Skip Garibaldi

    We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type $E_8$, the algebra A is obtained by adjoining a unit to the 3875-dimensional representation; and (2) it is effective, in that the product operation on A can be implemented

    更新日期:2021-01-14
  • Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates
    Forum Math. Sigma (IF 1.464) Pub Date : 2021-01-11
    Neal Bez; Sanghyuk Lee; Shohei Nakamura

    We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein–Gordon and fractional Schrödinger equations. Our estimates extend those of Frank–Sabin in the case of the wave and Klein–Gordon equations, and generalize work of Frank et al. and Frank–Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger

    更新日期:2021-01-11
  • On exceptional Lie geometries
    Forum Math. Sigma (IF 1.464) Pub Date : 2021-01-11
    Anneleen De Schepper; Jeroen Schillewaert; Hendrik Van Maldeghem; Magali Victoor

    Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected

    更新日期:2021-01-11
  • Log -modules and index theorems
    Forum Math. Sigma (IF 1.464) Pub Date : 2021-01-11
    Lei Wu; Peng Zhou

    We study log $\mathscr {D}$-modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah’s generalized b-functions. As applications, we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective

    更新日期:2021-01-11
  • Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings
    Forum Math. Sigma (IF 1.464) Pub Date : 2021-01-11
    Alex Chirvasitu; Ryo Kanda; S. Paul Smith

    The elliptic algebras in the title are connected graded $\mathbb {C}$-algebras, denoted $Q_{n,k}(E,\tau )$, depending on a pair of relatively prime integers $n>k\ge 1$, an elliptic curve E and a point $\tau \in E$. This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety

    更新日期:2021-01-11
  • Verlinde formulae on complex surfaces: K-theoretic invariants
    Forum Math. Sigma (IF 1.464) Pub Date : 2021-01-11
    L. Göttsche; M. Kool; R. A. Williams

    We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool

    更新日期:2021-01-11
  • Localizing virtual structure sheaves for almost perfect obstruction theories
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-12-07
    Young-Hoon Kiem; Michail Savvas

    Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin

    更新日期:2020-12-18
  • Dynamics of plane partitions: Proof of the Cameron–Fon-Der-Flaass conjecture
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-12-07
    Rebecca Patrias; Oliver Pechenik

    One of the oldest outstanding problems in dynamical algebraic combinatorics is the following conjecture of P. Cameron and D. Fon-Der-Flaass (1995): consider a plane partition P in an $a \times b \times c$ box ${\sf B}$ . Let $\Psi (P)$ denote the smallest plane partition containing the minimal elements of ${\sf B} - P$ . Then if $p= a+b+c-1$ is prime, Cameron and Fon-Der-Flaass conjectured that the

    更新日期:2020-12-18
  • One-sided reflected Brownian motions and the KPZ fixed point
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-12-09
    Mihai Nica; Jeremy Quastel; Daniel Remenik

    We consider the system of one-sided reflected Brownian motions that is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling-invariant Markov process defined in

    更新日期:2020-12-18
  • The cohomology of Torelli groups is algebraic
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-12-16
    Alexander Kupers; Oscar Randal-Williams

    The Torelli group of $W_g = \#^g S^n \times S^n$ is the group of diffeomorphisms of $W_g$ fixing a disc that act trivially on $H_n(W_g;\mathbb{Z} )$ . The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $\text{Sp}_{2g}(\mathbb{Z} )$ or $\text{O}_{g,g}(\mathbb{Z} )$ . In this article we prove that for $2n \geq 6$ and $g \geq 2$ , they are in fact algebraic

    更新日期:2020-12-18
  • Corrigendum: Abelian n-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-30
    Anthony Várilly-Alvarado; Bianca Viray

    There is an error in the statement and proof of [VAV17, Proposition 5.1] that affects the statements of [VAV17, Corollaries 5.2 and 5.3]. In this note, we correct the statement of [VAV17, Proposition 5.1] and explain how to rectify subsequent statements. In brief, for a statement about abelian Galois representations of a fixed level, ‘abelian’ should be replaced with ‘liftable abelian’ (Definition

    更新日期:2020-12-01
  • Noncommutative strong maximals and almost uniform convergence in several directions
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-20
    José M. Conde-Alonso; Adrián M. González-Pérez; Javier Parcet

    Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$ -norm of the $\limsup $ of a sequence

    更新日期:2020-11-21
  • Tropically constructed Lagrangians in mirror quintic threefolds
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-20
    Cheuk Yu Mak; Helge Ruddat

    We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special

    更新日期:2020-11-21
  • Solving the 4NLS with white noise initial data
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-18
    Tadahiro Oh; Nikolay Tzvetkov; Yuzhao Wang

    We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type

    更新日期:2020-11-18
  • Non-existence of bi-infinite geodesics in the exponential corner growth model
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-16
    Márton Balázs; Ofer Busani; Timo Seppäläinen

    This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process.

    更新日期:2020-11-17
  • RC-positive metrics on rationally connected manifolds
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-16
    Xiaokui Yang

    In this paper, we prove that if a compact Kähler manifold X has a smooth Hermitian metric $\omega $ such that $(T_X,\omega )$ is uniformly RC-positive, then X is projective and rationally connected. Conversely, we show that, if a projective manifold X is rationally connected, then there exists a uniformly RC-positive complex Finsler metric on $T_X$ .

    更新日期:2020-11-17
  • A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-16
    David Favero; Daniel Kaplan; Tyler L. Kelly

    We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category that does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.

    更新日期:2020-11-17
  • Product formalisms for measures on spaces with binary tree structures: representation, visualization, and multiscale noise
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-13
    Devasis Bassu; Peter W. Jones; Linda Ness; David Shallcross

    In this paper, we present a theoretical foundation for a representation of a data set as a measure in a very large hierarchically parametrized family of positive measures, whose parameters can be computed explicitly (rather than estimated by optimization), and illustrate its applicability to a wide range of data types. The preprocessing step then consists of representing data sets as simple measures

    更新日期:2020-11-13
  • Lattice isomorphisms between projection lattices of von Neumann algebras
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-13
    Michiya Mori

    Generalizing von Neumann’s result on type II $_1$ von Neumann algebras, I characterise lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, I give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type

    更新日期:2020-11-13
  • Generic Newton points and the Newton poset in Iwahori-double cosets
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-13
    Elizabeth Milićević; Eva Viehmann

    We consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck’s conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori-double

    更新日期:2020-11-13
  • The Brouwer invariance theorems in reverse mathematics
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-13
    Takayuki Kihara

    In [12], John Stillwell wrote, ‘finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics.’ In this article, we solve Stillwell’s problem by showing that (some forms of) the Brouwer invariance theorems are equivalent to the weak König’s lemma over the base system ${\sf RCA}_0$ . In particular, there exists an explicit

    更新日期:2020-11-13
  • Motivic Steenrod operations in characteristic p
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-13
    Eric Primozic

    For a prime p and a field k of characteristic $p,$ we define Steenrod operations $P^{n}_{k}$ on motivic cohomology with $\mathbb {F}_{p}$ -coefficients of smooth varieties defined over the base field $k.$ We show that $P^{n}_{k}$ is the pth power on $H^{2n,n}(-,\mathbb {F}_{p}) \cong CH^{n}(-)/p$ and prove an instability result for the operations. Restricted to mod p Chow groups, we show that the operations

    更新日期:2020-11-13
  • Profinite invariants of arithmetic groups
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-13
    Holger Kammeyer; Steffen Kionke; Jean Raimbault; Roman Sauer

    We prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results

    更新日期:2020-11-13
  • Boolean lattices in finite alternating and symmetric groups
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-13
    Andrea Lucchini; Mariapia Moscatiello; Sebastien Palcoux; Pablo Spiga

    Given a group G and a subgroup H, we let $\mathcal {O}_G(H)$ denote the lattice of subgroups of G containing H. This article provides a classification of the subgroups H of G such that $\mathcal {O}_{G}(H)$ is Boolean of rank at least $3$ when G is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type

    更新日期:2020-11-13
  • Semisimplification for subgroups of reductive algebraic groups
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-09
    Michael Bate; Benjamin Martin; Gerhard Röhrle

    Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If $G= {GL }_n$ , then there is a degeneration process for obtaining from H a completely reducible subgroup $H'$ of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results

    更新日期:2020-11-09
  • Bounds for twisted symmetric square L-functions via half-integral weight periods
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-09
    Paul D. Nelson

    We establish the first moment bound $$\begin{align*}\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_\varepsilon p^{5/4+\varepsilon} \end{align*}$$ for triple product L-functions, where $\Psi $ is a fixed Hecke–Maass form on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and $\varphi $ runs over the Hecke–Maass newforms on $\Gamma _0(p)$ of bounded eigenvalue. The proof is via

    更新日期:2020-11-09
  • The construction problem for Hodge numbers modulo an integer in positive characteristic
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-09
    Remy van Dobben de Bruyn; Matthias Paulsen

    Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

    更新日期:2020-11-09
  • The integral cohomology of the Hilbert scheme of points on a surface
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-04
    Burt Totaro

    We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

    更新日期:2020-11-04
  • A dichotomy of sets via typical differentiability
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-04
    Michael Dymond; Olga Maleva

    We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function: namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has a zero-length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical

    更新日期:2020-11-04
  • The bandwidth theorem for locally dense graphs
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-04
    Katherine Staden; Andrew Treglown

    The bandwidth theorem of Böttcher, Schacht, and Taraz [Proof of the bandwidth conjecture of Bollobás and Komlós, Mathematische Annalen, 2009] gives a condition on the minimum degree of an n-vertex graph G that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth $o(n)$ , thereby proving a conjecture of Bollobás and Komlós [The Blow-up Lemma, Combinatorics, Probability

    更新日期:2020-11-04
  • Vanishing theorems for the mod p cohomology of some simple Shimura varieties
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-03
    Teruhisa Koshikawa

    We show that the mod p cohomology of a simple Shimura variety treated in Harris-Taylor’s book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree under a mild additional assumption.

    更新日期:2020-11-03
  • Almost all Steiner triple systems are almost resolvable
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-11-03
    Asaf Ferber; Matthew Kwan

    We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).

    更新日期:2020-11-03
  • A rainbow blow-up lemma for almost optimally bounded edge-colourings
    Forum Math. Sigma (IF 1.464) Pub Date : 2020-10-30
    Stefan Ehard; Stefan Glock; Felix Joos

    A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills

    更新日期:2020-10-30
  • 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 -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 -DIVISORS, -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
  • -DEFORMED RATIONALS AND -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
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