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Forum of Mathematics, Sigma
Number of Followers: 1  Open Access journalISSN (Online) 2050-5094 Published by Cambridge University Press  [386 journals]
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- STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS
Authors: MARCUS DE CHIFFRE; LEV GLEBSKY, ALEXANDER LUBOTZKY, ANDREAS THOM
Abstract: 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. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by 
$\text{U}(n)$ with respect to the Frobenius norm 
$\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$ . Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple 
$p$ -adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.
PubDate: 2020-03-30T00:00:00.000Z
DOI: 10.1017/fms.2020.5
Issue No: Vol. 8 (2020)
- FRIEZE PATTERNS WITH COEFFICIENTS
Authors: MICHAEL CUNTZ; THORSTEN HOLM, PETER JØRGENSEN
Abstract: 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generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in an unpublished manuscript by Propp.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 classic results for frieze patterns. As a consequence, we see how frieze patterns with coefficients can be obtained from classic frieze patterns by cutting out subpolygons from the triangulated polygons associated with classic Conway–Coxeter frieze patterns. We address the question of which frieze patterns with coefficients can be obtained in this way and solve this problem completely for triangles. Finally, we prove a finiteness result for frieze patterns with coefficients by showing that for a given boundary sequence there are only finitely many (nonzero) frieze patterns with coefficients with entries in a subset of the complex numbers without an accumulation point.
PubDate: 2020-03-26T00:00:00.000Z
DOI: 10.1017/fms.2020.13
Issue No: Vol. 8 (2020)
- MULTIPLICATIVE PARAMETRIZED HOMOTOPY THEORY VIA SYMMETRIC SPECTRA IN
RETRACTIVE SPACES
Authors: FABIAN HEBESTREIT; STEFFEN SAGAVE, CHRISTIAN SCHLICHTKRULL
Abstract: 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 Thom spectra. In a companion paper, this approach is used to compare homotopical and operator algebraic models for twisted 
$K$ -theory.
PubDate: 2020-03-19T00:00:00.000Z
DOI: 10.1017/fms.2020.11
Issue No: Vol. 8 (2020)
Authors: EMMANUEL KOWALSKI; YONGXIAO LIN, PHILIPPE MICHEL, WILL SAWIN
Abstract: 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 primes.
PubDate: 2020-03-12T00:00:00.000Z
DOI: 10.1017/fms.2020.7
Issue No: Vol. 8 (2020)
- ERRATUM TO APPENDIX TO ‘2-ADIC INTEGRAL CANONICAL MODELS’
Authors: KEERTHI MADAPUSI PERA
PubDate: 2020-03-11T00:00:00.000Z
DOI: 10.1017/fms.2020.2
Issue No: Vol. 8 (2020)
Authors: JOACKIM BERNIER; ERWAN FAOU, BENOÎT GRÉBERT
Abstract: 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 into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on 
$\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when 
$d=1$ while our strategy applies in any dimension.
PubDate: 2020-03-06T00:00:00.000Z
DOI: 10.1017/fms.2020.8
Issue No: Vol. 8 (2020)
Authors: SOPHIE MORIER-GENOUD; VALENTIN OVSIENKO
Abstract: 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 the 
$q$ -rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, 
$q$ -deformation of the Farey graph, matrix presentations and 
$q$ -continuants are given, as well as a relation to the Jones polynomial of rational knots.
PubDate: 2020-03-06T00:00:00.000Z
DOI: 10.1017/fms.2020.9
Issue No: Vol. 8 (2020)
- WEIGHTED BESOV AND TRIEBEL–LIZORKIN SPACES ASSOCIATED WITH OPERATORS
AND APPLICATIONS
Authors: HUY-QUI BUI; THE ANH BUI, XUAN THINH DUONG
Abstract: 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 
$0
PubDate: 2020-02-26T00:00:00.000Z
DOI: 10.1017/fms.2020.6
Issue No: Vol. 8 (2020)
- MEASURABLE REALIZATIONS OF ABSTRACT SYSTEMS OF CONGRUENCES
Authors: CLINTON T. CONLEY; ANDREW S. MARKS, SPENCER T. UNGER
Abstract: 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 completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the 
$2$ -sphere. This answers a question by Wagon. We also construct Borel realizations of abstract systems of congruences for the action of 
$\mathsf{PSL}_{2}(\mathbb{Z})$ on 
$\mathsf{P}^{1}(\mathbb{R})$ . The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.
PubDate: 2020-02-24T00:00:00.000Z
DOI: 10.1017/fms.2020.4
Issue No: Vol. 8 (2020)
- SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE
CLOSED GEODESICS ON HYPERBOLIC SURFACES
Authors: FRANCISCO ARANA-HERRERA; JAYADEV S. ATHREYA
Abstract: Given integers
$g,n\geqslant 0$ satisfying 
$2-2g-n
PubDate: 2020-02-04T00:00:00.000Z
DOI: 10.1017/fms.2019.49
Issue No: Vol. 8 (2020)
- THE INTERSECTION MOTIVE OF THE MODULI STACK OF SHTUKAS
Authors: TIMO RICHARZ; JAKOB SCHOLBACH
Abstract: 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 correspondence for function fields.
PubDate: 2020-02-03T00:00:00.000Z
DOI: 10.1017/fms.2019.32
Issue No: Vol. 8 (2020)
- ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL
DATA
Authors: THIERRY DAUDÉ; NIKY KAMRAN, FRANÇOIS NICOLEAU
Abstract: 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 of order
$\unicode[STIX]{x1D70C}
PubDate: 2020-01-28T00:00:00.000Z
DOI: 10.1017/fms.2020.1
Issue No: Vol. 8 (2020)
- SPLITTING LOOPS AND NECKLACES: VARIANTS OF THE SQUARE PEG PROBLEM
Authors: JAI ASLAM; SHUJIAN CHEN, FLORIAN FRICK, SAM SALOFF-COSTE, LINUS SETIABRATA, HUGH THOMAS
Abstract: 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 the loop into four pieces which can be rearranged to form two loops of equal length. (The previous two results are independently due to Schwartz.) A rectifiable loop in 
$d$ -space can be cut into 
$(r-1)(d+1)+1$ pieces that can be rearranged by translations to form 
$r$ loops of equal length. We relate our results to fair divisions of necklaces in the sense of Alon and to Tverberg-type results. This provides a new approach and a common framework to obtain inscribability results for the class of all continuous curves.
PubDate: 2020-01-20T00:00:00.000Z
DOI: 10.1017/fms.2019.51
Issue No: Vol. 8 (2020)
- OPTIMAL LINE PACKINGS FROM FINITE GROUP ACTIONS
Authors: JOSEPH W. IVERSON; JOHN JASPER, DUSTIN G. MIXON
Abstract: 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 variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.
PubDate: 2020-01-20T00:00:00.000Z
DOI: 10.1017/fms.2019.48
Issue No: Vol. 8 (2020)
Authors: KATHRIN BRINGMANN; BEN KANE
Abstract: 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 
$\mathfrak{z}$ . Moreover, define 
$\unicode[STIX]{x1D6E5}(z):=q\prod _{n\geqslant 1}(1-q^{n})^{24}$ , where 
$q:=e^{2\unicode[STIX]{x1D70B}iz}$ , and set 
$\unicode[STIX]{x1D55B}(z):=\frac{1}{6}\log (y^{6}|\unicode[STIX]{x1D6E5}(z)|)+1$
PubDate: 2020-01-15T00:00:00.000Z
DOI: 10.1017/fms.2019.46
Issue No: Vol. 8 (2020)
Authors: ANTHONY BAK; ANURADHA S. GARGE
Abstract: 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, due to the first author, starts by recalling the notion of a global action and describes in detail the global actions attached to the general linear, elementary, and Steinberg groups. With these examples in mind, we describe the elementary homotopy theory of arbitrary global actions, construct their homotopy groups, and revisit their covering theory. We then equip the set
$Um_{n}(R)$ of all unimodular row vectors of length 
$n$ over a ring 
$R$ with a global action. Its homotopy groups 
$\unicode[STIX]{x1D70B}_{i}(Um_{n}(R)),i\geqslant 0$ are christened the vector 
$K$ -theory groups 
$K_{i+1}(Um_{n}(R)),i\geqslant 0$ of 
$Um_{n}(R)$ . It is known that the homotopy groups 
$\unicode[STIX]{x1D70B}_{i}(\text{GL}_{n}(R))$ of the general linear group 
$\text{GL}_{n}(R)$ viewed as a global action are the Volodin 
...
PubDate: 2020-01-15T00:00:00.000Z
DOI: 10.1017/fms.2019.30
Issue No: Vol. 8 (2020)
REDUCTIVE GROUPS
Authors: FLORIAN HERZIG; KAROL KOZIOŁ, MARIE-FRANCE VIGNÉRAS
Abstract: 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 
$C$ .
PubDate: 2020-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.50
Issue No: Vol. 8 (2020)
- EXISTENCE AND COMPACTNESS THEORY FOR ALE SCALAR-FLAT KÄHLER SURFACES
Authors: JIYUAN HAN; JEFF A. VIACLOVSKY
Abstract: 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 metrics for several infinite families of Kähler ALE spaces.
PubDate: 2020-01-01T00:00:00.000Z
DOI: 10.1017/fms.2019.42
Issue No: Vol. 8 (2020)
