3,296 results
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2. Derived length of zero entropy groups acting on projective varieties in arbitrary characteristic — A remark to a paper of Dinh-Oguiso-Zhang
- Author
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Sichen Li
- Subjects
Automorphism group ,Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Zhàng ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,01 natural sciences ,0103 physical sciences ,Computer Science::General Literature ,Entropy (information theory) ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Projective test ,Projective variety ,Mathematics - Abstract
Let [Formula: see text] be a projective variety of dimension [Formula: see text] over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of [Formula: see text]. Let [Formula: see text] be a group of zero entropy automorphisms of [Formula: see text] and [Formula: see text] the set of elements in [Formula: see text] which are isotopic to the identity. We show that after replacing [Formula: see text] by a suitable finite-index subgroup, [Formula: see text] is a unipotent group of the derived length at most [Formula: see text]. This result was first proved by Dinh et al. for compact Kähler manifolds.
- Published
- 2020
3. Special Ulrich bundles on regular Weierstrass fibrations
- Author
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Joan Pons-Llopis and Rosa M. Miró-Roig
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Short paper ,Elliptic surfaces ,Ulrich bundles ,01 natural sciences ,Mathematics::Algebraic Geometry ,Simple (abstract algebra) ,0103 physical sciences ,Weierstrass fibrations ,Rank (graph theory) ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations \(\pi : S\rightarrow \mathbb {P}^1\). Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations \(S\rightarrow C_0\) and we deal with the (non-)existence of rank one Ulrich bundles on them.
- Published
- 2019
4. Derived Non-archimedean analytic Hilbert space
- Author
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Mauro Porta, Jorge António, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Fiber (mathematics) ,General Mathematics ,010102 general mathematics ,Short paper ,Formal scheme ,Hilbert space ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Mathematics - Algebraic Geometry ,Mathematics::Category Theory ,0103 physical sciences ,Localization theorem ,FOS: Mathematics ,symbols ,010307 mathematical physics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics - Abstract
In this short paper we combine the representability theorem introduced in [17, 18] with the theory of derived formal models introduced in [2] to prove the existence representability of the derived Hilbert space RHilb(X) for a separated k-analytic space X. Such representability results relies on a localization theorem stating that if X is a quasi-compact and quasi-separated formal scheme, then the \infty-category Coh^+(X^rig) of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the \infty-category Coh^+(X). Along the way, we prove several results concerning the the \infty-categories of formal models for almost perfect modules on derived k-analytic spaces., 28 pages
- Published
- 2019
5. On Beilinson’s equivalence for p-adic cohomology
- Author
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Daniel Caro, Tomoyuki Abe, Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo (UTokyo), Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)
- Subjects
Pure mathematics ,Derived category ,Functor ,Holonomic ,General Mathematics ,010102 general mathematics ,Short paper ,General Physics and Astronomy ,Unipotent ,01 natural sciences ,Cohomology ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,010307 mathematical physics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,Equivalence (formal languages) ,Mathematics::Representation Theory ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
In this short paper, we construct a unipotent nearby cycle functor and show a p-adic analogue of Beilinson’s equivalence comparing two derived categories: the derived category of holonomic arithmetic $${\mathcal {D}}$$ -modules and the derived category of arithmetic $${\mathcal {D}}$$ -modules whose cohomologies are holonomic.
- Published
- 2018
6. Iterates of Generic Polynomials and Generic Rational Functions
- Author
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Jamie Juul
- Subjects
Pure mathematics ,Degree (graph theory) ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,Galois group ,37P05, 11G50, 14G25 ,Rational function ,01 natural sciences ,Unpublished paper ,Generic polynomial ,Number theory ,Symmetric group ,Iterated function ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In 1985, Odoni showed that in characteristic 0 0 the Galois group of the n n -th iterate of the generic polynomial with degree d d is as large as possible. That is, he showed that this Galois group is the n n -th wreath power of the symmetric group S d S_d . We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.
- Published
- 2014
7. Non-negative Ricci curvature on closed manifolds under Ricci flow
- Author
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Davi Maximo
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Short paper ,Ricci flow ,01 natural sciences ,Mathematics::Geometric Topology ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Bounded curvature ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Mathematics::Differential Geometry ,0101 mathematics ,10. No inequality ,Mathematics::Symplectic Geometry ,Ricci curvature ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result B\"ohm and Wilking have for dimensions twelve and above, \cite{BW}. Moreover, the manifolds constructed here are \Kahler manifolds and relate to a question raised by Xiuxiong Chen in \cite{XC}, \cite{XCL}., Comment: New version with added references and corrected typos
- Published
- 2009
- Full Text
- View/download PDF
8. Order 3 symplectic automorphisms on K3 surfaces
- Author
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Alice Garbagnati and Yulieth Prieto Montañez
- Subjects
Pure mathematics ,Endomorphism ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Order (ring theory) ,Automorphism ,01 natural sciences ,Cohomology ,14J28, 14J50 ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics ,Symplectic geometry - Abstract
The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms., Comment: 28 pages. Version 2: this is the published version of the paper. The last section of the previous version (v1) was erased (the results are only stated) and it is now contained in arXiv:2209.10141
- Published
- 2021
9. On Nilpotent Extensions of ∞-Categories and the Cyclotomic Trace
- Author
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Elden Elmanto and Vladimir Sosnilo
- Subjects
Trace (semiology) ,Pure mathematics ,Nilpotent ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty $-categories) for additive $\infty $-categories, (2) define the notion of nilpotent extensions for suitable $\infty $-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas–Goodwillie–McCarthy theorem for stable $\infty $-categories that are not monogenically generated (such as the stable $\infty $-category of Voevodsky’s motives or the stable $\infty $-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarko’s notion of weight structures, which provides a “ring-with-many-objects” analog of a connective $\mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work by Hoyois–Krishna for homotopy $K$-theory and establish new cases of Blanc’s lattice conjecture.
- Published
- 2021
10. An index theorem for higher orbital integrals
- Author
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Xiang Tang, Peter Hochs, and Yanli Song
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Index (economics) ,General Mathematics ,01 natural sciences ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,Algebra over a field ,Operator Algebras (math.OA) ,Mathematics ,Group (mathematics) ,010102 general mathematics ,Mathematics - Operator Algebras ,Lie group ,K-Theory and Homology (math.KT) ,Elliptic operator ,Differential Geometry (math.DG) ,Mathematics - K-Theory and Homology ,Equivariant map ,010307 mathematical physics ,Atiyah–Singer index theorem ,Mathematics - Representation Theory - Abstract
Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions., 40 pages; updates based on referee comments; expanded proof of Proposition 3.3
- Published
- 2021
11. Correction to: Seifert fibrations of lens spaces
- Author
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Christian Lange and Hansjörg Geiges
- Subjects
Lemma (mathematics) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Lens (geology) ,Astrophysics::Cosmology and Extragalactic Astrophysics ,Base (topology) ,Mathematics::Geometric Topology ,01 natural sciences ,Number theory ,Differential geometry ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Mathematics::Symplectic Geometry ,Orbifold ,Mathematics - Abstract
We classify the Seifert fibrations of lens spaces where the base orbifold is non-orientable. This is an addendum to our earlier paper ‘Seifert fibrations of lens spaces’. We correct Lemma 4.1 of that paper and fill the gap in the classification that resulted from the erroneous lemma.
- Published
- 2021
12. Wild Cantor actions
- Author
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Ramón Barral Lijó, Hiraku Nozawa, Jesús A. Álvarez López, and Olga Lukina
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Closure (topology) ,Mathematics::General Topology ,Dynamical Systems (math.DS) ,16. Peace & justice ,Equicontinuity ,01 natural sciences ,Centralizer and normalizer ,Cantor set ,Group action ,Wreath product ,0103 physical sciences ,FOS: Mathematics ,Countable set ,2020: 37B05, 37E25, 20E08, 20E15, 20E18, 20E22, 22F05, 22F50 (Primary), 20F22, 57R30, 57R50 (Secondary) ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations., 20 pages, 1 figure. The condition of finite generation in Thm 1.9 was replaced by countability. The proof of Thm 1.9 has been simplified. The notation used in 5 has been modified. Several minor corrections across the paper
- Published
- 2022
13. Graded Bourbaki ideals of graded modules
- Author
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Jürgen Herzog, Dumitru I. Stamate, and Shinya Kumashiro
- Subjects
Noetherian ,Pure mathematics ,Sequence ,Class (set theory) ,Ideal (set theory) ,Mathematics::Commutative Algebra ,Mathematics::General Mathematics ,General Mathematics ,Mathematics::History and Overview ,010102 general mathematics ,Structure (category theory) ,Mathematics::General Topology ,Field (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematik ,0103 physical sciences ,FOS: Mathematics ,Homomorphism ,13A02, 13A30, 13D02, 13H10 ,010307 mathematical physics ,0101 mathematics ,Rees algebra ,Mathematics - Abstract
In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal., Comment: 29 pages
- Published
- 2021
14. The factorisation property ofl∞(Xk)
- Author
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Paul F. X. Müller, Thomas Schlumprecht, Pavlos Motakis, and Richard Lechner
- Subjects
Pure mathematics ,Property (philosophy) ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Diagonal ,Banach space ,01 natural sciences ,Identity (music) ,Bounded operator ,Factorization ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we consider the following problem: letXk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by${(e_{(k,j)}^*)_j}$, for$k\in\N$, let$Z=\ell^\infty(X_k:k\kin\N)$be theirl∞-sum, and let$T:Z\to Z$be a bounded linear operator with a large diagonal,i.e.,$$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$Under which condition does the identity onZfactor throughT? The purpose of this paper is to formulate general conditions for which the answer is positive.
- Published
- 2020
15. Hyperbolicity and Uniformity of Varieties of Log General type
- Author
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Amos Turchet, Kristin DeVleming, Kenneth Ascher, Ascher, Kenneth, Devleming, Kristin, and Turchet, Amos
- Subjects
Pure mathematics ,Conjecture ,Mathematics - Number Theory ,Generalization ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Sheaf ,Trigonometric functions ,Uniform boundedness ,Cotangent bundle ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Variety (universal algebra) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded., v5: exposition greatly improved. Previous section on function fields removed, to be expanded upon in a future paper. To appear in IMRN
- Published
- 2020
16. Sobolev regular solutions for the incompressible Navier–Stokes equations in higher dimensions: asymptotics and representation formulae
- Author
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Weiping Yan and Vicenţiu D. Rădulescu
- Subjects
Pure mathematics ,Regular polyhedron ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,01 natural sciences ,Domain (mathematical analysis) ,Sobolev space ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Asymptotic expansion ,Representation (mathematics) ,Navier–Stokes equations ,Mathematics - Abstract
In this paper, we consider the steady incompressible Navier–Stokes equations in a smooth bounded domain $$\Omega \subset \mathbb R^n$$ Ω ⊂ R n with the dimension $$n\ge 3$$ n ≥ 3 . We first establish asymptotic expansion formulae of Sobolev regular finite energy solutions in $$\Omega$$ Ω . In the second part of this paper, explicit representation formulae of Sobolev regular solutions are showed in the regular polyhedron $$\Omega :=[0,T]^n$$ Ω : = [ 0 , T ] n .
- Published
- 2020
17. Vector-valued q-variational inequalities for averaging operators and the Hilbert transform
- Author
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Tao Ma, Wei Liu, and Guixiang Hong
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Variational inequality ,symbols ,010307 mathematical physics ,Hilbert transform ,0101 mathematics ,Martingale (probability theory) ,Mathematics - Abstract
Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.
- Published
- 2020
18. On Counting Certain Abelian Varieties Over Finite Fields
- Author
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Chia-Fu Yu and Jiangwei Xue
- Subjects
Isogeny ,Pure mathematics ,Class (set theory) ,Current (mathematics) ,Mathematics - Number Theory ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Connection (mathematics) ,Finite field ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Mathematics - Abstract
This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number $\sqrt{q}$. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and ideal complexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of Yu on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigation., Comment: 23 pages. Section 5.4 corrected
- Published
- 2020
19. Remarks on the geodesic-Einstein metrics of a relative ample line bundle
- Author
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Xueyuan Wan and Xu Wang
- Subjects
Ample line bundle ,Pure mathematics ,Geodesic ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Fibration ,Type (model theory) ,01 natural sciences ,Mathematics::Algebraic Geometry ,Flow (mathematics) ,Bounded function ,Bundle ,0103 physical sciences ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space $$\mathcal {X}$$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $$(\mathcal {X}, L)$$ is nonlinear semistable if the associated Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time existence property. We also define the associated S-classes and C-classes for $$(\mathcal {X}, L)$$ and obtain two inequalities between them when L admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relative ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian–Einstein.
- Published
- 2020
20. SNC Log Symplectic Structures on Fano Products
- Author
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Katsuhiko Okumura
- Subjects
Pure mathematics ,Mathematics::Algebraic Geometry ,General Mathematics ,Poisson manifold ,010102 general mathematics ,0103 physical sciences ,Projective space ,010307 mathematical physics ,Fano plane ,0101 mathematics ,01 natural sciences ,Symplectic geometry ,Mathematics - Abstract
This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.
- Published
- 2020
21. On the local density formula and the Gross–Keating invariant with an Appendix ‘The local density of a binary quadratic form’ by T. Ikeda and H. Katsurada
- Author
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Cho Sungmun
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Local factor ,01 natural sciences ,Quadratic form ,0103 physical sciences ,FOS: Mathematics ,11E08, 11E95, 14L15, 20G25 ,Binary quadratic form ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Local field ,Fourier series ,Mathematics - Abstract
T. Ikeda and H. Katsurada have developed the theory of the Gross-Keating invariant of a quadratic form in their recent papers [IK1] and [IK2]. In particular, they prove that the local factor of the Fourier coefficients of the Siegel-Eisenstein series is completely determined by the Gross-Keating invariant with extra datum, called the extended GK datum, in [IK2]. On the other hand, such local factor is a special case of the local densities for a pair of two quadratic forms. Thus we propose a general question if the local density can be determined by certain series of the Gross-Keating invariants and the extended GK datums. In this paper, we prove that the answer to this question is affirmative, for the local density of a single quadratic form defined over an unramified finite extension of $\mathbb{Z}_2$. In the appendix, T. Ikeda and H. Katsurada compute the local density formula of a single binary quadratic form defined over any finite extension of $\mathbb{Z}_2$., 32 pages
- Published
- 2020
22. Zeros of Holomorphic One-Forms and Topology of Kähler Manifolds
- Author
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Stefan Schreieder
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Holomorphic function ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Topology (chemistry) ,Mathematics - Abstract
A conjecture of Kotschick predicts that a compact Kähler manifold X fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao [10], we use our approach to prove Kotschick’s conjecture for smooth projective three-folds.
- Published
- 2020
23. On the polar Orlicz-Minkowski problems and the p-capacitary Orlicz-Petty bodies
- Author
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Xiaokang Luo, Deping Ye, and Baocheng Zhu
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Metric Geometry (math.MG) ,Type (model theory) ,01 natural sciences ,Measure (mathematics) ,52A20, 52A38, 52A39, 52A40, 53A15 ,Mathematics - Metric Geometry ,0103 physical sciences ,Minkowski space ,FOS: Mathematics ,Mathematics::Metric Geometry ,Polar ,010307 mathematical physics ,Orthogonal matrix ,0101 mathematics ,Isoperimetric inequality ,Mathematics - Abstract
In this paper, we propose and study the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure $\mu$ and a continuous function $\varphi:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K\in\mathcal{K}_0$ such that $K$ is an optimizer of the following optimization problems: \begin{equation*} \inf/\sup \bigg\{\int_{S^{n-1}}\varphi\big( h_L \big) \,d \mu: L \in \mathcal{K}_{0} \ \text{and}\ |L^\circ|=\omega_{n}\bigg\}. \end{equation*} The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $\varphi,$ the existence of a solution is proved for a nonzero finite measure $\mu$ on $S^{n-1}$ which is not concentrated on any hemisphere of $S^{n-1}.$ Another part of this paper deals with the $p$-capacitary Orlicz-Petty bodies. In particular, the existence of the $p$-capacitary Orlicz-Petty bodies is established and the continuity of the $p$-capacitary Orlicz-Petty bodies is proved., Comment: This paper has been accepted by Indiana University Mathematics Journal
- Published
- 2020
24. Archimedean non-vanishing, cohomological test vectors, and standard L-functions of $${\mathrm {GL}}_{2n}$$: real case
- Author
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Cheng Chen, Fangyang Tian, Dihua Jiang, and Bingchen Lin
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Linear model ,Structure (category theory) ,22E45 (Primary), 11F67 (Secondary) ,Type (model theory) ,Lambda ,Infinity ,01 natural sciences ,Invariant theory ,Linear form ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics ,media_common - Abstract
The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $\Lambda_{s,\chi}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,\pi\otimes\chi)$ as a meromorphic function of $s\in \mathbb{C}$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector, and hence recovers a non-vanishing result of Binyong Sun via a completely different method. Our main result indicates a complete solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun and Fangyang Tian with full details and with applications to the global period relations for the twisted standard $L$-functions at critical places., Comment: 39 pages. The current version of this paper is significantly shorter than the previous one, as the first author pointed out a conceptual intepretation of construction of cohomological test vector in the old version of this paper. Section 4 is completely rewritten. Also fix some inaccuracies
- Published
- 2019
25. A sparse approach to mixed weak type inequalities
- Author
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Marcela Caldarelli and Israel P. Rivera-Ríos
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Novelty ,Singular integral ,Weak type ,01 natural sciences ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,GEOM ,media_common ,Mathematics - Abstract
In this paper we provide some quantitative mixed weak-type estimates assuming conditions that imply that $$uv\in A_{\infty }$$ for Calderon–Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in Domingo-Salazar et al. (Bull Lond Math Soc 48(1):63–73, 2016) and extended in Lerner et al. (Adv Math 319:153–181, 2017) and Li et al. (J Geom Anal, 2018).
- Published
- 2019
26. Masur–Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves
- Author
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Vincent Delecroix, Elise Goujard, Peter Zograf, Anton Zorich, Groupe Sociétés, Religions, Laïcités (GSRL), Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), École Pratique des Hautes Études (EPHE), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and ANR-19-CE40-0021,Phymath,physique mathématique(2019)
- Subjects
Teichmüller space ,Surface (mathematics) ,Pure mathematics ,Geodesic ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Dynamical Systems (math.DS) ,Algebraic geometry ,01 natural sciences ,Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,Mathematics - Dynamical Systems ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quadratic differential ,Mathematics ,Meromorphic function ,010102 general mathematics ,Geometric Topology (math.GT) ,Mathematics::Geometric Topology ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Mapping class group ,Moduli space ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Combinatorics (math.CO) ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,010307 mathematical physics - Abstract
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent., Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042
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- 2021
27. Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials
- Author
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Vidya Venkateswaran, Jasper V. Stokman, Siddhartha Sahi, Algebra, Geometry & Mathematical Physics (KDV, FNWI), Quantum Matter and Quantum Information, KdV Other Research (FNWI), Faculty of Science, and KDV (FNWI)
- Subjects
Weyl group ,Polynomial ,Pure mathematics ,Algebraic combinatorics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,20C08 (Primary), 11F68, 22E50 (Secondary) ,Rational function ,01 natural sciences ,symbols.namesake ,Macdonald polynomials ,Gauss sum ,0103 physical sciences ,symbols ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Dirichlet series ,Mathematics - Representation Theory ,Mathematics - Abstract
Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group multiple Dirichlet series associated to metaplectic (n-fold) covers of algebraic groups. In subsequent joint work with Puskas, they extended this action to a "metaplectic" representation of the equal parameter affine Hecke algebra, which allowed them to obtain explicit formulas for the p-parts of these Dirichlet series. They have also verified by a computer check the remarkable fact that their formulas continue to define a group action for general (unspecialized) parameters. In the first part of paper we give a conceptual explanation of this fact, by giving a uniform and elementary construction of the "metaplectic" representation for generic Hecke algebras as a suitable quotient of a parabolically induced affine Hecke algebra module, from which the associated Chinta-Gunnells Weyl group action follows through localization. In the second part of the paper we extend the metaplectic representation to the double affine Hecke algebra, which provides a generalization of Cherednik's basic representation. This allows us to introduce a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials. In this paper, we provide the details of the construction of metaplectic polynomials in type A; the general case will be handled in the sequel to this paper., 39 pages. Version 2 is a significant revision. Added second part introducing a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials and metaplectic Iwahori-Whittaker functions. Title has been changed and the introduction has been expanded
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- 2021
28. Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities
- Author
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Lu Zhang, Guozhen Lu, and Nguyen Lam
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Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Function (mathematics) ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Infimum and supremum ,Symmetry (physics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics ,media_common - Abstract
Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space R n , including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in [25] , combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.
- Published
- 2019
29. Homological behavior of idempotent subalgebras and Ext algebras
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Charles Paquette and Colin Ingalls
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Ring (mathematics) ,Pure mathematics ,Noetherian ring ,Conjecture ,Reduction (recursion theory) ,Mathematics::Commutative Algebra ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,01 natural sciences ,Global dimension ,16E10, 16G10 ,0103 physical sciences ,Idempotence ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\Gamma$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $\Gamma$. We also investigate the situation where both $A,\Gamma$ have finite global dimension. When $A$ is Koszul and finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determiant of artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $\Gamma$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture., Comment: 14 pages
- Published
- 2019
30. Normal crossings singularities for symplectic topology
- Author
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Mark McLean, Aleksey Zinger, and Mohammad Farajzadeh Tehrani
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Pure mathematics ,Logarithm ,Divisor ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics - Symplectic Geometry ,0103 physical sciences ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,53D05, 53D45, 14N35 ,Gravitational singularity ,010307 mathematical physics ,0101 mathematics ,Equivalence (formal languages) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Symplectic sum ,Symplectic geometry ,Mathematics - Abstract
We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors., Comment: 65 pages, 4 figures; a number of typos fixed; the exposition has been significantly revised, fixing a technical error in the non-compact case in the process; this paper is now restricted to the simple normal crossings case; the arbitrary normal crossings case will be detailed in a followup paper
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- 2018
31. Signature characters of invariant Hermitian forms on irreducible Verma modules and Hall–Littlewood polynomials
- Author
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Wai Ling Yee
- Subjects
Pure mathematics ,Verma module ,General Mathematics ,010102 general mathematics ,Positive-definite matrix ,01 natural sciences ,Unitary state ,Hermitian matrix ,Hall–Littlewood polynomials ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics::Representation Theory ,Alcove ,Affine Hecke algebra ,Mathematics - Abstract
The Unitary Dual Problem is one of mathematics’ most important open problems: classify the irreducible unitary representations of a group. The general approach has been to classify all representations admitting non-degenerate invariant Hermitian forms, compute the signatures of those forms, and then determine which forms are positive definite. Signature character algorithms and formulas arising from deforming representations and analysing changes at reducibility points, as in Adams et al. (Unitary representations of real reductive groups (ArXiv e-prints), 2012) and Yee (Represent Theory 9:638–677, 2005), produce very complicated formulas or algorithms from the resulting recursion. This paper shows that in the case of irreducible Verma modules all of the complexity can be encapsulated by the affine Hecke algebra: for compact real forms and for alcoves corresponding to translations of the fundamental alcove by a regular weight, signature characters of irreducible Verma modules are in fact “negatives” of Hall–Littlewood polynomial summands evaluated at $$q=-\,1$$ times a version of the Weyl denominator, establishing a simple signature character formula and drawing an important connection between signature characters and the affine Hecke algebra. Signature characters of irreducible highest weight modules are shown to be related to Kazhdan-Lusztig basis elements. This paper also handles noncompact real forms. The current state of the art for the unitary dual is a computer algorithm for determining if a given representation is unitary. These results suggest the potential to move the state of the art to a closed form classification for the entire unitary dual.
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- 2018
32. An Application of the S-Functional Calculus to Fractional Diffusion Processes
- Author
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Jonathan Gantner and Fabrizio Colombo
- Subjects
Pure mathematics ,Spectral theory ,Vector operator ,General Mathematics ,01 natural sciences ,Functional calculus ,Mathematics - Spectral Theory ,Operator (computer programming) ,Unit vector ,0103 physical sciences ,FOS: Mathematics ,Mathematics (all) ,0101 mathematics ,Spectral Theory (math.SP) ,Commutative property ,Mathematics ,fractional diffusion and fractional evolution processes ,S-spectrum ,010102 general mathematics ,Operator theory ,Quaternionic analysis ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,H∞ functional calculus for quaternionic operators ,010307 mathematical physics ,fractional powers of vector operators - Abstract
In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the $${H^\infty}$$ functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form $$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$ where $${e_{\ell}, {\ell} = 1, 2, 3}$$ are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables $${x = (x_{1}, x_{2}, x_{3})}$$ and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version $${T^{\alpha}, {\rm for} \alpha \in (0, 1)}$$ , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.
- Published
- 2018
33. Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials
- Author
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Daniel Erman, Steven V Sam, and Andrew Snowden
- Subjects
Pure mathematics ,General Mathematics ,media_common.quotation_subject ,MathematicsofComputing_GENERAL ,Hilbert's basis theorem ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematics - Algebraic Geometry ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Ideal (ring theory) ,0101 mathematics ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics ,media_common ,Conjecture ,Hilbert's syzygy theorem ,Mathematics::Commutative Algebra ,Degree (graph theory) ,Applied Mathematics ,010102 general mathematics ,13A02, 13D02 ,Mathematics - Commutative Algebra ,Infinity ,Bounded function ,symbols ,010307 mathematical physics - Abstract
Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin
- Published
- 2018
34. Quasi-elliptic cohomology I
- Author
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Zhen Huan
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Elliptic cohomology ,16. Peace & justice ,Space (mathematics) ,Mathematics::Algebraic Topology ,01 natural sciences ,Cohomology ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Equivariant map ,Mathematics - Algebraic Topology ,010307 mathematical physics ,55N34, 55P35 ,0101 mathematics ,Tate curve ,Constant (mathematics) ,Computer Science::Databases ,Quotient ,Orbifold ,Mathematics - Abstract
Quasi-elliptic cohomology is a variant of elliptic cohomology theories. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. Thus, the constructions on it can be made in a neat way. This theory reflects the geometric nature of the Tate curve. In this paper we provide a systematic introduction of its construction and definition., Comment: Final Version. 26 pages. To appear in Advances in Mathematics. In this paper we generalize the construction in arXiv:1612.00930. The subtle point of this generalization is explained in Section 2
- Published
- 2018
35. Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality
- Author
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Aparajita Dasgupta and Michael Ruzhansky
- Subjects
Pure mathematics ,CONVOLUTION ,General Mathematics ,Structure (category theory) ,Boundary (topology) ,Type (model theory) ,Universality ,01 natural sciences ,Mathematics - Spectral Theory ,symbols.namesake ,Mathematics - Analysis of PDEs ,Primary 46F05 ,Tensor (intrinsic definition) ,0103 physical sciences ,FOS: Mathematics ,DISTRIBUTIONS ,Secondary 22E30 ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematics ,Hilbert spaces ,Sequence ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,Universality (philosophy) ,Eigenfunction ,Sequence spaces ,Smooth functions ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Mathematics and Statistics ,Physics and Astronomy ,Komatsu classes ,symbols ,Tensor representations ,010307 mathematical physics ,Primary 46F05, Secondary 22E30 ,Analysis ,Analysis of PDEs (math.AP) - Abstract
In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary., 23 pages
- Published
- 2021
36. Decompositions of principal series representations of Iwahori-Hecke algebras for Kac-Moody groups over local fields
- Author
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Auguste Hébert, Université Jean Monnet [Saint-Étienne] (UJM), and Université de Lyon
- Subjects
Pure mathematics ,Series (mathematics) ,[MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT] ,General Mathematics ,010102 general mathematics ,Principal (computer security) ,01 natural sciences ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,Irreducible representation ,Mathematics::Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Irreducibility ,010307 mathematical physics ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
International audience; Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we introduced principal series representations for these algebras and partially generalized Kato's irreducibility criterion. In this paper, we study how some of these representations decompose when they are reducible and deduce information on the irreducible representations of these algebras.
- Published
- 2021
37. Functional relations of solutions of $q$-difference equations
- Author
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Thomas Dreyfus, Charlotte Hardouin, Julien Roques, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Combinatoire, théorie des nombres (CTN), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), ANR-19-CE40-0018,DeRerumNatura,Décider l'irrationalité et la transcendance(2019), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Series (mathematics) ,General Mathematics ,[MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC] ,010102 general mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,39A06, 12H10 ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Operator (computer programming) ,Algebraic relations ,0103 physical sciences ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,Algebraic independence ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,Mathematics - Abstract
In this paper, we study the algebraic relations satisfied by the solutions of q-difference equations and their transforms with respect to an auxiliary operator. Our main tools are the parametrized Galois theories developed in Hardouin and Singer (Math Ann 342(2):333–377, 2008) and Ovchinnikov and Wibmer (Int Math Res Not 12:3962–4018, 2015). The first part of this paper is concerned with the case where the auxiliary operator is a derivation, whereas the second part deals with a $$\mathbf {q}$$ -difference operator. In both cases, we give criteria to guarantee the algebraic independence of a series, solution of a q-difference equation, with either its successive derivatives or its $$\mathbf {q}$$ -transforms. We apply our results to q-hypergeometric series.
- Published
- 2021
38. Generalized absolute values, ideals and homomorphisms in mixed lattice groups
- Author
-
Lassi Paunonen, Jani Jokela, Sirkka-Liisa Eriksson, Tampere University, Computing Sciences, and Department of Mathematics and Statistics
- Subjects
Pure mathematics ,General Mathematics ,High Energy Physics::Lattice ,Lattice (group) ,Structure (category theory) ,Riesz space ,01 natural sciences ,Theoretical Computer Science ,0103 physical sciences ,111 Mathematics ,Absolute value ,Ideal (order theory) ,Mixed lattice ,0101 mathematics ,Lattice ordered group ,Mathematics ,Mixed lattice group ,Group (mathematics) ,010102 general mathematics ,Mixed lattice semigroup ,Operator theory ,Ideal ,010307 mathematical physics ,Quotient group ,Analysis ,Vector space - Abstract
A mixed lattice group is a generalization of a lattice ordered group. The theory of mixed lattice semigroups dates back to the 1970s, but the corresponding theory for groups and vector spaces has been relatively unexplored. In this paper we investigate the basic structure of mixed lattice groups, and study how some of the fundamental concepts in Riesz spaces and lattice ordered groups, such as the absolute value and other related ideas, can be extended to mixed lattice groups and mixed lattice vector spaces. We also investigate ideals and study the properties of mixed lattice group homomorphisms and quotient groups. Most of the results in this paper have their analogues in the theory of Riesz spaces.
- Published
- 2021
39. Fourier multipliers on graded lie groups
- Author
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Michael Ruzhansky and Veronique Fischer
- Subjects
Pure mathematics ,Mathematics(all) ,General Mathematics ,Graded nilpotent Lie groups ,Type (model theory) ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Representation Theory (math.RT) ,Analysis on Lie groups ,Mathematics ,Group (mathematics) ,010102 general mathematics ,Lie group ,Dual (category theory) ,Functional Analysis (math.FA) ,Sobolev space ,Mathematics - Functional Analysis ,Nilpotent ,Fourier transform ,symbols ,010307 mathematical physics ,Fourier multipliers ,Mathematics - Representation Theory ,Primary: 43A22, Secondary: 43A15, 22E30 - Abstract
In this paper we study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that H\"ormander type conditions on the Fourier multipliers imply $L^p$-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper., Comment: 23 pages
- Published
- 2020
40. The fully marked surface theorem
- Author
-
Mehdi Yazdi and David Gabai
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,Taut foliation ,Homology (mathematics) ,01 natural sciences ,symbols.namesake ,Mathematics - Geometric Topology ,Euler characteristic ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,16. Peace & justice ,Surface (topology) ,Mathematics::Geometric Topology ,Cohomology ,Manifold ,57R30, 57K32, 57M50 ,symbols ,Foliation (geology) ,010307 mathematical physics ,Mathematics::Differential Geometry ,Euler class - Abstract
In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F'}$ is homotopic to that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $\mathcal{F'}$ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper., Comment: 36 pages, 16 figures. Portions of this work previously appeared as an appendix to arXiv:1603.03822, but has evolved into its own work and has been accepted for publication separately. Final version to appear in Acta Mathematica
- Published
- 2020
41. Singularities of Hermitian–Yang–Mills connections and Harder–Narasimhan–Seshadri filtrations
- Author
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Song Sun and Xuemiao Chen
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,Vector bundle ,Algebraic geometry ,01 natural sciences ,Harder–Narasimhan–Seshadri filtrations ,Mathematics::Algebraic Geometry ,Singularity ,Mathematics::K-Theory and Homology ,32G13 ,0103 physical sciences ,FOS: Mathematics ,Projective space ,70S15 ,32Q15 ,0101 mathematics ,Holomorphic vector bundle ,Mathematics ,Hermitian–Yang–Mills connections ,010102 general mathematics ,Tangent cone ,Reflexive sheaf ,53C07 ,Differential Geometry (math.DG) ,reflexive sheaves ,Sheaf ,010307 mathematical physics ,instantons ,singularities - Abstract
This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable., Final version
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- 2020
42. Homological dimension of elementary amenable groups
- Author
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Conchita Martínez-Pérez and Peter H. Kropholler
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics ,Global dimension - Abstract
In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.
- Published
- 2020
43. Finitary birepresentations of finitary bicategories
- Author
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Vanessa Miemietz, Marco Mackaay, Daniel Tubbenhauer, Volodymyr Mazorchuk, and Xiaoting Zhang
- Subjects
Pure mathematics ,Reduction (recursion theory) ,Generalization ,General Mathematics ,Coalgebra ,01 natural sciences ,Simple (abstract algebra) ,Computer Science::Logic in Computer Science ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Finitary ,Category Theory (math.CT) ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics ,Transitive relation ,Applied Mathematics ,010102 general mathematics ,Mathematics - Category Theory ,16. Peace & justice ,Mathematics::Logic ,Double centralizer theorem ,010307 mathematical physics ,Bijection, injection and surjection ,Mathematics - Representation Theory - Abstract
In this paper, we discuss the generalization of finitary $2$-representation theory of finitary $2$-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive $2$-representations of a given $2$-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of $2$-representations. In this paper, we generalize them to biequivalences between certain $2$-categories of birepresentations. Furthermore, we prove an analog of the double centralizer theorem in finitary birepresentation theory., Significant revision of the original version
- Published
- 2020
44. The KK-theory of amalgamated free products
- Author
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Emmanuel Germain, Pierre Fima, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)
- Subjects
Vertex (graph theory) ,Exact sequence ,Pure mathematics ,Mathematics::Operator Algebras ,Direct sum ,General Mathematics ,Unital ,010102 general mathematics ,[MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA] ,Mathematics - Operator Algebras ,KK-theory ,K-Theory and Homology (math.KT) ,Conditional expectation ,01 natural sciences ,Free product ,0103 physical sciences ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,[MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT] ,010307 mathematical physics ,0101 mathematics ,Operator Algebras (math.OA) ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen., V.3, the paper has been splitted into two papers, this is the first part on amalgamated free products
- Published
- 2020
45. Affine category O, Koszul duality and Zuckerman functors
- Author
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Ruslan Maksimau, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), and Université de Montpellier (UM)
- Subjects
Pure mathematics ,Koszul duality ,General Mathematics ,Structure (category theory) ,Zuckerman functor ,Category O ,Mathematics::Algebraic Topology ,01 natural sciences ,Fock space ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics ,Subcategory ,Functor ,[MATH.MATH-RT]Mathematics [math]/Representation Theory [math.RT] ,010102 general mathematics ,17B10 ,16. Peace & justice ,Dual (category theory) ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf{A}$ at level $-N-e$ can be decomposed in terms of the components of the functor $F$ for the category $\mathbf{A}$ at level $-N-e-1$. To prove this, we use the following fact: a category with an action of $\widetilde{\mathfrak sl}_{e+1}$ contains a (canonically defined) subcategory with an action of $\widetilde{\mathfrak sl}_{e}$. We also prove a general statement that says that in some general situation a functor that satisfies a list of axioms is automatically Koszul dual to some sort of Zuckerman functor., 71 pages. This represents a portion of arXiv:1512.04878 which was split into two parts, this is the second part. This paper is rewritten (compared to arXiv:1512.04878) in a way that we never use KLR algebras explicitly. This makes the paper more independent from the first part
- Published
- 2020
46. On Some Families of Certain Divisible Polynomials and Their Zeta Functions
- Author
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Koji Chinen
- Subjects
12D10 ,Pure mathematics ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,13A50 ,Differential operator ,01 natural sciences ,Riemann hypothesis ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,11T71 ,Mathematics - Abstract
The formal weight enumerators were first introduced by M. Ozeki, and it was shown in the author's previous paper that there are various families of similar divisible polynomials. Among them, three families are dealt with in this paper and their properties are investigated: they are analogues of the Mallows--Sloane bound, the extremal property, the Riemann hypothesis analogue, etc. In the course of the investigation, some generalizations of the theory of invariant differential operators developed by I. Duursma and T. Okuda are deduced.
- Published
- 2020
47. Explicit equations of a fake projective plane
- Author
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Lev A. Borisov and JongHae Keum
- Subjects
Surface (mathematics) ,fake projective planes ,Pure mathematics ,Betti number ,General Mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,ball quotient ,equations ,elliptic surfaces ,0103 physical sciences ,FOS: Mathematics ,Ball (mathematics) ,0101 mathematics ,14J29 ,Algebraic Geometry (math.AG) ,32N15 ,Quotient ,Mathematics ,Complex conjugate ,14J29, 14F05, 32Q40, 32N15 ,Fake projective plane ,14F05 ,010102 general mathematics ,Automorphism ,32Q40 ,bicanonical embedding ,010307 mathematical physics ,Projective plane - Abstract
Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism., Comment: This is a full version of "Research announcement: equations of a fake projective plane", arXiv:1710.04501. Key tables and some M2 and Magma code from the paper are included in separate files for convenience
- Published
- 2020
48. Generic conformally flat hypersurfaces and surfaces in 3-sphere
- Author
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Suyama Yoshihiko
- Subjects
Surface (mathematics) ,Mathematics - Differential Geometry ,Pure mathematics ,Gauss map ,General Mathematics ,010102 general mathematics ,Conformal map ,Space (mathematics) ,01 natural sciences ,3-sphere ,symbols.namesake ,Hypersurface ,Differential Geometry (math.DG) ,Primary 53B25, Secondary 53E40 ,0103 physical sciences ,Euclidean geometry ,Gaussian curvature ,symbols ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere. The conformal structure of generic conformally flat (local-)hypersurfaces is characterized as conformally flat (local-)3-metrics with the Guichard condition. Then, there is a certain class of orthogonal analytic (local-)Riemannian 2-metrics with constant Gauss curvature -1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition. In this paper, we firstly relate 2-metrics of the class to surfaces in the 3-sphere: for a 2-metric of the class, a 5-dimensional set of (non-isometric) analytic surfaces in the 3-sphere is determined such that any surface of the set gives rise to an evolution of surfaces in the 3-sphere issuing from the surface and the evolution is the Gauss map of a generic conformally flat hypersurface in the Euclidean 4-space. Secondly, we characterize analytic surfaces in the 3-sphere which give rise to generic conformally flat hypersurfaces., 39 pages
- Published
- 2020
49. The structure theory of nilspaces I
- Author
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Yonatan Gutman, Freddie Manners, Péter P. Varjú, and Apollo - University of Cambridge Repository
- Subjects
medicine.medical_specialty ,Class (set theory) ,Pure mathematics ,General Mathematics ,Topological dynamics ,Dynamical Systems (math.DS) ,01 natural sciences ,Morphism ,math.GN ,0103 physical sciences ,medicine ,FOS: Mathematics ,Mathematics - Combinatorics ,math.CO ,Mathematics - Dynamical Systems ,0101 mathematics ,Abelian group ,Axiom ,Mathematics - General Topology ,Mathematics ,010102 general mathematics ,General Topology (math.GN) ,Pure Mathematics ,Compact space ,010307 mathematical physics ,Inverse limit ,Combinatorics (math.CO) ,math.DS ,Analysis ,Structured program theorem - Abstract
This paper forms the first part of a series by the authors [GMV2,GMV3] concerning the structure theory of nilspaces of Antol\'in Camarena and Szegedy. A nilspace is a compact space $X$ together with closed collections of cubes $C^n(X)\subseteq X^{2^n}$, $n=1,2,\ldots$ satisfying some natural axioms. Antol\'in Camarena and Szegedy proved that from these axioms it follows that (certain) nilspaces are isomorphic (in a strong sense) to an inverse limit of nilmanifolds. The aim of our project is to provide a new self-contained treatment of this theory and give new applications to topological dynamics. This paper provides an introduction to the project from the point of view of applications to higher order Fourier analysis. We define and explain the basic definitions and constructions related to cubespaces and nilspaces and develop the weak structure theory, which is the first stage of the proof of the main structure theorem for nilspaces. Vaguely speaking, this asserts that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group. We also make some modest innovations and extensions to this theory. In particular, we consider a class of maps that we term fibrations, which are essentially equivalent to what are termed fiber-surjective morphisms by Anatol\'in Camarena and Szegedy, and we formulate and prove a relative analogue of the weak structure theory alluded to above for these maps. These results find applications elsewhere in the project., Comment: 64 pages, minor revision based on referee's report, results and proofs have not been changed, this version is accepted for publication in J. Anal. Math
- Published
- 2020
50. Locally constrained curvature flows and geometric inequalities in hyperbolic space
- Author
-
Yong Wei, Haizhong Li, and Yingxiang Hu
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mean curvature ,General Mathematics ,Hyperbolic space ,010102 general mathematics ,Order (ring theory) ,Type (model theory) ,Curvature ,01 natural sciences ,53C44, 52A39 ,Hypersurface ,Differential Geometry (math.DG) ,Flow (mathematics) ,Principal curvature ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics - Abstract
In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the $m$th quermassintegral and decreases $(m+1)$th quermassintegral, so the convergence of the flow yields sharp Alexandrov-Fenchel type inequalities in hyperbolic space. Some special cases have been studied in [7]. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of $k$th Gauss-Bonnet curvature of a smooth h-convex hypersurface to its $m$th quermassintegral (for $0\leq m\leq 2k+1\leq n$), and comparing the weighted integral of $k$th mean curvature to its $m$th quermassintegral (for $0\leq m\leq k\leq n$). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu in 2015. In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of $k$th shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang's [8] inequality., 38 pages, accepted version for Mathematische Annalen, add Corollary 1.10 to describe the application of the new locally constrained flow (1.11)
- Published
- 2020
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