Forum of Mathematics Pi最新文献_第7页

Point counting for foliations over number fields 数域上叶理的点计数

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-09-02 DOI: 10.1017/fmp.2021.20

Gal Binyamini

{"title":"Point counting for foliations over number fields","authors":"Gal Binyamini","doi":"10.1017/fmp.2021.20","DOIUrl":"https://doi.org/10.1017/fmp.2021.20","url":null,"abstract":"Abstract Let${mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${mathbb K}$. For an algebraic $Vsubset {mathbb M}$ over ${mathbb K}$, write $delta _{V}$ for the maximum of the degree and log-height of V. Write $Sigma _{V}$ for the points where the leaves intersect V improperly. Fix a compact subset ${mathcal B}$ of a leaf ${mathcal L}$. We prove effective bounds on the geometry of the intersection ${mathcal B}cap V$. In particular, when $operatorname {codim} V=dim {mathcal L}$ we prove that $#({mathcal B}cap V)$ is bounded by a polynomial in $delta _{V}$ and $log operatorname {dist}^{-1}({mathcal B},Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of ${mathcal B}cap V$ by an algebraic map $Phi $. For instance, under suitable conditions we show that $Phi ({mathcal B}cap V)$ contains at most $operatorname {poly}(g,h)$ algebraic points of log-height h and degree g. We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections $P,Q$ of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever $P,Q$ are simultaneously torsion their order of torsion is bounded effectively by a polynomial in $delta _{P},delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given $Vsubset {mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in $delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 10

Endoscopic decompositions and the Hausel–Thaddeus conjecture 内窥镜分解与Hauser–Thaddeus猜想

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-08-19 DOI: 10.1017/fmp.2021.7

D. Maulik, Junliang Shen

引用次数: 17

On locally analytic vectors of the completed cohomology of modular curves 模曲线完全上同调的局部解析向量

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-08-17 DOI: 10.1017/fmp.2022.1

Lue Pan

引用次数: 22

Quadratic Klein-Gordon equations with a potential in one dimension 一维势的二次Klein-Gordon方程

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-06-28 DOI: 10.1017/fmp.2022.9

P. Germain, F. Pusateri

{"title":"Quadratic Klein-Gordon equations with a potential in one dimension","authors":"P. Germain, F. Pusateri","doi":"10.1017/fmp.2022.9","DOIUrl":"https://doi.org/10.1017/fmp.2022.9","url":null,"abstract":"Abstract This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions. More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions. These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the \u0000$phi ^4$\u0000 problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2020-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46588163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 29

The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials 阿贝尔微分模空间的Chern类和欧拉特征

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-06-23 DOI: 10.1017/fmp.2022.10

Matteo Costantini, Martin Möller, Jonathan Zachhuber

引用次数: 22

Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers 具有不相交源和接收器的非线性双曲型方程的反问题

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-06-22 DOI: 10.1017/fmp.2021.11

A. Feizmohammadi, M. Lassas, L. Oksanen

{"title":"Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers","authors":"A. Feizmohammadi, M. Lassas, L. Oksanen","doi":"10.1017/fmp.2021.11","DOIUrl":"https://doi.org/10.1017/fmp.2021.11","url":null,"abstract":"Abstract The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension \u0000$n+1geqslant 3$\u0000 and with partial data. We consider the case when the set \u0000$Omega _{mathrm{in}}$\u0000 , where the sources are supported, and the set \u0000$Omega _{mathrm{out}}$\u0000 , where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point \u0000$p_{in}in Omega _{mathrm{in}}$\u0000 and the past of the point \u0000$p_{out}in Omega _{mathrm{out}}$\u0000 . In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in \u0000$Omega _{mathrm{in}}$\u0000 and observations in \u0000$Omega _{mathrm{out}}$\u0000 , determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2020-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45589284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 10

Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on $mathbb {T}^{4}$ $mathbb｛T｝^｛4｝上能量临界非线性Schrödinger方程的无条件唯一性$

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-06-10 DOI: 10.1017/fmp.2021.16

Xuwen Chen, J. Holmer

引用次数: 4

Resonance-based schemes for dispersive equations via decorated trees 通过装饰树的色散方程的基于共振的格式

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-05-04 DOI: 10.1017/fmp.2021.13

Y. Bruned, Katharina Schratz

{"title":"Resonance-based schemes for dispersive equations via decorated trees","authors":"Y. Bruned, Katharina Schratz","doi":"10.1017/fmp.2021.13","DOIUrl":"https://doi.org/10.1017/fmp.2021.13","url":null,"abstract":"Abstract We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48874055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 40

K-stability of Fano varieties via admissible flags 通过容许标志测定法诺品种的k稳定性

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-03-30 DOI: 10.1017/fmp.2022.11

Hamid Abban, Ziquan Zhuang

引用次数: 46

The Asymptotic Statistics of Random Covering Surfaces 随机覆盖曲面的渐近统计量

IF 2.3 1区数学

Forum of Mathematics Pi Pub Date : 2020-03-12 DOI: 10.1017/fmp.2023.13

Michael Magee, Doron Puder

{"title":"The Asymptotic Statistics of Random Covering Surfaces","authors":"Michael Magee, Doron Puder","doi":"10.1017/fmp.2023.13","DOIUrl":"https://doi.org/10.1017/fmp.2023.13","url":null,"abstract":"Abstract Let \u0000$Gamma _{g}$\u0000 be the fundamental group of a closed connected orientable surface of genus \u0000$ggeq 2$\u0000 . We develop a new method for integrating over the representation space \u0000$mathbb {X}_{g,n}=mathrm {Hom}(Gamma _{g},S_{n})$\u0000 , where \u0000$S_{n}$\u0000 is the symmetric group of permutations of \u0000${1,ldots ,n}$\u0000 . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g. Given \u0000$phi in mathbb {X}_{g,n}$\u0000 and \u0000$gamma in Gamma _{g}$\u0000 , we let \u0000$mathsf {fix}_{gamma }(phi )$\u0000 be the number of fixed points of the permutation \u0000$phi (gamma )$\u0000 . The function \u0000$mathsf {fix}_{gamma }$\u0000 is a special case of a natural family of functions on \u0000$mathbb {X}_{g,n}$\u0000 called Wilson loops. Our new methodology leads to an asymptotic formula, as \u0000$nto infty $\u0000 , for the expectation of \u0000$mathsf {fix}_{gamma }$\u0000 with respect to the uniform probability measure on \u0000$mathbb {X}_{g,n}$\u0000 , which is denoted by \u0000$mathbb {E}_{g,n}[mathsf {fix}_{gamma }]$\u0000 . We prove that if \u0000$gamma in Gamma _{g}$\u0000 is not the identity and q is maximal such that \u0000$gamma $\u0000 is a q th power in \u0000$Gamma _{g}$\u0000 , then \u0000$$begin{align*}mathbb{E}_{g,n}left[mathsf{fix}_{gamma}right]=d(q)+O(n^{-1}) end{align*}$$\u0000 as \u0000$nto infty $\u0000 , where \u0000$dleft (qright )$\u0000 is the number of divisors of q. Even the weaker corollary that \u0000$mathbb {E}_{g,n}[mathsf {fix}_{gamma }]=o(n)$\u0000 as \u0000$nto infty $\u0000 is a new result of this paper. We also prove that \u0000$mathbb {E}_{g,n}[mathsf {fix}_{gamma }]$\u0000 can be approximated to any order \u0000$O(n^{-M})$\u0000 by a polynomial in \u0000$n^{-1}$\u0000 .","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44854396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 18