Finitary Analysis in Homogeneous Dynamics and Applications

齐次动力学有限分析及其应用

基本信息

  • 批准号:
    2055122
  • 负责人:
  • 金额:
    $ 29.44万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2021
  • 资助国家:
    美国
  • 起止时间:
    2021-07-01 至 2024-06-30
  • 项目状态:
    已结题

项目摘要

Dynamical systems study rules which predict the long-term behavior of a point in a space. The origin of dynamical systems may be traced back to Newtonian mechanics. Examples of mathematical models of dynamical systems include the study of fluid dynamics, airflow dynamics, and many others. Consider, for instance, gas particles moving in a space according to a certain rule. One is interested in understanding various states of the system over long-time intervals. A curious phenomenon is the following: if one assumes that initially all gas particles are in one half of the space, then infinitely often in the future, the gas molecules will collect in the initial portion of the space. This conclusion, which is predicted by mathematical aspects of the system, seems paradoxical. However, the mystery may be revealed if one observes that the number of the degree of freedom in this system is very large, thus in order to return to the original state, one needs to observe the system for very long (practically impossible) time intervals. This proposal aims at the study of finitary aspects of dynamical systems where one is interested in approximating various states of the system within an error. This project also include the training of a graduate student.The Principal Investigator seeks extensions and strengthening of certain rigidity results in dynamics and their applications in number theory and geometry. Special attention will be given to finitary and effective aspects of the analysis. The following will be the main objectives: (i) Dynamical systems have become a major player in modern mathematics. However, arguments relying on techniques from ergodic theory and dynamics are often non-quantitative. Providing finitary versions of these arguments are challenging and much sought after, especially in view of various applications. The principal investigator seeks results in this direction with two main goals in mind: provide finitary arguments which yield polynomial rates for strong rigidity results in homogeneous dynamics in some specific examples with interesting applications to number theory and geometry; provide finitary versions of these celebrated results in great generality, even though one may not obtain a polynomial rate in general. (ii) Geodesic planes in hyperbolic 3-manifolds have been studied from different angles. These investigations have made it clear that the behavior of geodesic planes is intimately related to the geometric, topological, and arithmetic properties of the ambient manifold. This proposal seeks to refine and strengthen results in this direction using tools from homogeneous dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
动力系统研究的是预测空间中某点长期行为的规则。动力系统的起源可以追溯到牛顿力学。动力系统的数学模型的例子包括流体动力学,气流动力学和许多其他的研究。例如,考虑气体粒子按照一定的规则在空间中运动。人们对了解系统在长时间间隔内的各种状态感兴趣。下面是一个奇怪的现象:如果假设最初所有的气体粒子都在空间的一半,那么在未来,气体分子将无限地聚集在空间的初始部分。这一结论,这是预测的数学方面的系统,似乎自相矛盾。然而,如果我们观察到这个系统的自由度非常大,那么这个奥秘就可能被揭示出来,因此为了回到原始状态,我们需要观察系统很长时间(实际上是不可能的)。这个建议的目的是在有限方面的动力系统的研究,其中一个有兴趣在一个错误内近似系统的各种状态。该项目还包括一名研究生的培训。主要研究者寻求在动力学中某些刚性结果的扩展和加强及其在数论和几何中的应用。将特别注意分析的有限性和有效性方面。以下将是主要目标:(一)动力系统已成为现代数学的主要参与者。然而,依赖于遍历理论和动力学技术的论点往往是非定量的。提供这些论点的有限版本具有挑战性,并且备受追捧,特别是考虑到各种应用。主要研究人员寻求结果在这个方向上有两个主要目标:提供有限的论点,产生多项式率的强刚性结果在齐次动力学在一些具体的例子与有趣的应用数论和几何;提供有限的版本,这些著名的结果在很大的一般性,即使一个可能不会获得多项式率一般。(ii)从不同的角度研究了双曲三维流形中的测地平面。这些研究表明,测地平面的行为与周围流形的几何、拓扑和算术性质密切相关。该奖项反映了NSF的法定使命,并通过使用基金会的知识价值和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Polynomial effective density in quotients of $${\mathbb {H}}^3$$ and $${\mathbb {H}}^2\times {\mathbb {H}}^2$$
多项式有效密度,以 $${mathbb {H}}^3$$ 和 $${mathbb {H}}^2 imes {mathbb {H}}^2$$ 的商表示
  • DOI:
    10.1007/s00222-022-01162-5
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    3.1
  • 作者:
    Lindenstrauss, E.;Mohammadi, A.
  • 通讯作者:
    Mohammadi, A.
Isolations of geodesic planes in the frame bundle of a hyperbolic 3-manifold
双曲 3 流形的框架丛中测地平面的隔离
  • DOI:
    10.1112/s0010437x22007928
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    1.8
  • 作者:
    Mohammadi, Amir;Oh, Hee
  • 通讯作者:
    Oh, Hee
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Amir Mohammadi其他文献

Taylor wavelets collocation technique for solving fractional nonlinear singular PDEs
  • DOI:
    10.1007/s40096-022-00483-z
  • 发表时间:
    2022-07-12
  • 期刊:
  • 影响因子:
    2.300
  • 作者:
    Nasser Aghazadeh;Amir Mohammadi;Gamze Tanoglu
  • 通讯作者:
    Gamze Tanoglu
Exploring two-phase bubble dynamics and convective mass transfer in thermochemical Cu–Cl cycle for hydrogen production
探究热化学铜 - 氯循环制氢中的两相气泡动力学和对流传质
  • DOI:
    10.1016/j.ces.2025.121476
  • 发表时间:
    2025-05-01
  • 期刊:
  • 影响因子:
    4.300
  • 作者:
    Amir Mohammadi;Ofelia A. Jianu;Canan Acar
  • 通讯作者:
    Canan Acar
Property (τ) in positive characteristic
  • DOI:
    10.1007/s11856-024-2660-7
  • 发表时间:
    2024-09-03
  • 期刊:
  • 影响因子:
    0.800
  • 作者:
    Amir Mohammadi;Nattalie Tamam
  • 通讯作者:
    Nattalie Tamam
Association between air pollution and sudden sensorineural hearing loss (SSHL): A systematic review and meta-analysis
空气污染与突发性感觉神经性听力损失(SSHL)之间的关联:系统综述和荟萃分析
  • DOI:
    10.1016/j.envres.2023.117392
  • 发表时间:
    2023-12-15
  • 期刊:
  • 影响因子:
    7.700
  • 作者:
    Fatemeh Ranjdoost;Mohammad-Ebrahim Ghaffari;Faramarz Azimi;Amir Mohammadi;Reza Fouladi-Fard;Maria Fiore
  • 通讯作者:
    Maria Fiore
Comprehensive analysis of hospital solid waste levels and HSE risks using FMEA technique: A case study in northwest Iran
  • DOI:
    10.1016/j.cscee.2024.100646
  • 发表时间:
    2024-06-01
  • 期刊:
  • 影响因子:
  • 作者:
    Saeed Hosseinpoor;Towhid Dadashi;Amir Mohammadi
  • 通讯作者:
    Amir Mohammadi

Amir Mohammadi的其他文献

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{{ truncateString('Amir Mohammadi', 18)}}的其他基金

Homogeneous and Teichmuller Dynamics: A Quantitative Viewpoint
齐次和 Teichmuller 动力学:定量观点
  • 批准号:
    1764246
  • 财政年份:
    2018
  • 资助金额:
    $ 29.44万
  • 项目类别:
    Continuing Grant
Dynamics on homogeneous spaces and Moduli spaces
齐次空间和模空间上的动力学
  • 批准号:
    1724316
  • 财政年份:
    2017
  • 资助金额:
    $ 29.44万
  • 项目类别:
    Continuing Grant
Dynamics on homogeneous spaces and Moduli spaces
齐次空间和模空间上的动力学
  • 批准号:
    1500677
  • 财政年份:
    2015
  • 资助金额:
    $ 29.44万
  • 项目类别:
    Continuing Grant
Homogeneous Dynamics and Number Theory
齐次动力学和数论
  • 批准号:
    1200388
  • 财政年份:
    2012
  • 资助金额:
    $ 29.44万
  • 项目类别:
    Continuing Grant

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