Convexity and Applications
凸性及其应用
基本信息
- 批准号:1811146
- 负责人:
- 金额:$ 23.5万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2018
- 资助国家:美国
- 起止时间:2018-06-01 至 2021-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The principal investigator's research is in asymptotic geometric analysis and affine convex geometry. One main emphasis of her research is on high-dimensional objects and phenomena. This leads to applications of her research in areas as diverse as physics, biology and medicine, computer science, optimization and economics and material science: Indeed, a mathematical description of a scientific or engineering question often requires lots of independent numbers, leading to a geometric space of high dimension. For example, specifying the location, direction and speed of one gas molecule in a room six separate numbers in all. If you want to track 100 distinct molecules of the air in the room, then you will need 600 independent numerical coordinates to collect all of the relevant measurements. As these dimensions increase then the difficulty of sampling and computation go up rapidly, a phenomenon data scientists sometimes call "the curse of dimensionality." However, there are also patterns that emerge as dimension increases which are not visible in low dimensions. We can exploit those patterns, thus converting the "curse of dimensionality" into "blessing of dimensionality". It is one purpose of this award to study such high-dimensional phenomena. Important features of this project are the study of high-dimensional objects and phenomena and their links with other areas of mathematics and mathematical sciences, such as probability, statistics and information theory. Of particular interest are the affine invariant functionals on convex bodies in high dimensions. Among the most important such functionals are affine surface area and the p-affine surface area (a family of functionals parametrized by a real number p). Their corresponding affine isoperimetric inequalities, established by the PI and collaborators for all p, are stronger than their Euclidean counterparts and related to the famous Mahler conjecture which is still open in dimensions four and higher. It was shown by the principal investigator that p-affine surface areas are directly related to entropies of cone measures of convex bodies which establishes a link between convex geometry and information theory. This link will be further explored, also in the context of log concave functions which are a natural extension of convex bodies in the realm of functions. Moreover, affine surface area appears naturally in questions on approximation of convex bodies by polytopes, a further main topic of study. The goal is to establish optimal dependence on all the relevant parameters involved in the approximation, for example the dimension and the number of vertices of the approximating polytopes. The principal investigator and her collaborators also extended the notions of affine surface area recently to a functional setting and to spherical and hyperbolic space. To establish the corresponding inequalities in those settings is a further topic of study.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.
主要研究方向是渐近几何分析和仿射凸几何。她的研究重点之一是高维物体和现象。这导致她的研究在物理学,生物学和医学,计算机科学,优化和经济学以及材料科学等不同领域的应用:事实上,科学或工程问题的数学描述通常需要大量的独立数字,从而导致高维几何空间。 例如,指定房间中一个气体分子的位置,方向和速度总共六个独立的数字。 如果你想跟踪房间里100个不同的空气分子,那么你需要600个独立的数字坐标来收集所有相关的测量值。 随着这些维度的增加,采样和计算的难度迅速上升,数据科学家有时称之为“维度灾难”。“然而,随着维度的增加,也会出现一些模式,这些模式在低维度中是不可见的。 我们可以利用这些模式,从而将“维度灾难”转化为“维度祝福”。 这个奖项的目的之一就是研究这种高维现象。该项目的重要特点是研究高维对象和现象及其与数学和数学科学的其他领域,如概率,统计和信息论的联系。 特别感兴趣的是在高维凸体上的仿射不变泛函。其中最重要的是仿射表面积和p-仿射表面积(一个由真实的数p参数化的泛函族)。他们相应的仿射等周不等式,由PI和合作者建立的所有p,比他们的欧几里德同行更强,并与著名的马勒猜想有关,该猜想在四维和更高维中仍然是开放的。 主要研究者表明,p-仿射表面积与凸体的锥测度的熵直接相关,这建立了凸几何和信息论之间的联系。这种联系将进一步探讨,也在日志凹函数的情况下,这是一个自然的凸体在功能领域的延伸。 此外,仿射表面积自然出现在凸体的多面体逼近问题,进一步的主要研究课题。 我们的目标是建立最佳的依赖于所有相关参数的近似,例如尺寸和近似多面体的顶点数。首席研究员和她的合作者最近还将仿射表面积的概念扩展到函数设置以及球面和双曲空间。在这些环境中建立相应的不平等是进一步研究的主题。该奖项反映了NSF的法定使命,并被认为值得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来支持。
项目成果
期刊论文数量(10)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
The Löwner Function of a Log-Concave Function
对数凹函数的 Löwner 函数
- DOI:10.1007/s12220-019-00270-8
- 发表时间:2021
- 期刊:
- 影响因子:0
- 作者:Li, Ben;Schütt, Carsten;Werner, Elisabeth M.
- 通讯作者:Werner, Elisabeth M.
A Steiner formula in the $L_p$ Brunn Minkowski theory
$L_p$ Brunn Minkowski 理论中的 Steiner 公式
- DOI:
- 发表时间:2019
- 期刊:
- 影响因子:1.7
- 作者:Tatarko, Kateryna;Werner, Elisabeth M
- 通讯作者:Werner, Elisabeth M
Blaschke-Santalo inequality for many functions and geodesic barycenters of measures
许多函数的 Blaschke-Santalo 不等式和测度测地重心
- DOI:
- 发表时间:2022
- 期刊:
- 影响因子:1.7
- 作者:Kolesnikov, Alexander V;Werner, Elisabeth M
- 通讯作者:Werner, Elisabeth M
Surface area deviation between smooth convex bodies and polytopes
光滑凸体与多面体之间的表面积偏差
- DOI:10.1016/j.aam.2021.102218
- 发表时间:2021
- 期刊:
- 影响因子:0
- 作者:J. Grote;C. Thale;E. Werner
- 通讯作者:E. Werner
Constrained convex bodies with extremal affine surface areas
具有极值仿射表面积的约束凸体
- DOI:
- 发表时间:2020
- 期刊:
- 影响因子:1.7
- 作者:Giladi, O;Huang, H;Schuett, C;Werner, E
- 通讯作者:Werner, E
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Elisabeth Werner其他文献
Reaktionen mit phosphororganischen Verbindungen, 10. Mitt.: [Oxydation mit Pb(OAc)4, (C6H5COO)2, C6H5J(OAc)2 und PbO2] Zur Darstellung von α-Ketosäuremethylestern und α-Ketosäurethiophenylestern
- DOI:
10.1007/bf00901458 - 发表时间:
1966-01-01 - 期刊:
- 影响因子:1.900
- 作者:
E. Zbiral;Elisabeth Werner - 通讯作者:
Elisabeth Werner
The convex floating body of almost polygonal bodies
- DOI:
10.1007/bf00182948 - 发表时间:
1992-11-01 - 期刊:
- 影响因子:0.500
- 作者:
Cartest Schütt;Elisabeth Werner - 通讯作者:
Elisabeth Werner
Quasi-Banach spaces which are unique predual
- DOI:
10.1007/bf01450076 - 发表时间:
1988-12-01 - 期刊:
- 影响因子:1.400
- 作者:
Elisabeth Werner - 通讯作者:
Elisabeth Werner
A comprehensive study of fully automated combined upstream and downstream processes for the production of pharmaceutical proteins
- DOI:
10.1016/j.jbiotec.2007.07.338 - 发表时间:
2007-09-01 - 期刊:
- 影响因子:
- 作者:
Reiner Luttmann;Ali Kazemi-Seresht;Matthias Eicke;Elisabeth Werner;Birger Hahn;Andree Ellert - 通讯作者:
Andree Ellert
Sequential injection analyzer for glycerol monitoring in yeast cultivation medium
- DOI:
10.1016/j.talanta.2006.05.072 - 发表时间:
2007-02-15 - 期刊:
- 影响因子:
- 作者:
Burkhard Horstkotte;Elisabeth Werner;Ali Kazemi Seresht;Gesine Cornelissen;Olaf Elsholz;Víctor Cerdà Martín;Reiner Luttmann - 通讯作者:
Reiner Luttmann
Elisabeth Werner的其他文献
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{{ truncateString('Elisabeth Werner', 18)}}的其他基金
Workshop on Asymptotic Geometry in Paris
巴黎渐近几何研讨会
- 批准号:
0535305 - 财政年份:2006
- 资助金额:
$ 23.5万 - 项目类别:
Standard Grant
Mathematical Sciences: Banach Space Theory and Convexity Theory
数学科学:巴拿赫空间理论和凸性理论
- 批准号:
9401784 - 财政年份:1994
- 资助金额:
$ 23.5万 - 项目类别:
Standard Grant
Mathematical Sciences: Banach Spaces and Convexity Theory
数学科学:巴拿赫空间和凸性理论
- 批准号:
8915893 - 财政年份:1989
- 资助金额:
$ 23.5万 - 项目类别:
Continuing Grant
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英文专著《FRACTIONAL INTEGRALS AND DERIVATIVES: Theory and Applications》的翻译
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- 项目类别:数学天元基金项目
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