Discrete Convexity, Curvature, and Implications
离散凸性、曲率和含义
基本信息
- 批准号:1811935
- 负责人:
- 金额:$ 19万
- 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:2018-08-01 至 2021-07-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Convexity in continuous domains has been known to be a powerful property and an important tool for more than a century. Thanks to several independent recent developments in the topics of optimal transport, partial differential equations, geometric and functional analysis, the interplay of convexity and curvature has further enriched the understanding of several topics in probability and analysis. While convexity has been developed through the important notion of submodularity in discrete settings, curvature remains elusive. In the past five years, the investigator and various collaborators have introduced discrete notions of the so-called Ricci curvature in Markov chains and graphs and derived several interesting consequences. However, much remains to be understood and developed in terms of convexity and curvature in discrete spaces. Identifying suitable applications in mathematics and computer science first will be an important guiding principle in developing any new theory. In discrete probability, the Ahlswede-Daykin Four Functions Theorem has had numerous applications (by way of the FKG inequality and other consequences) and similarly the Prekopa-Leindler inequality has been instrumental in deepening the understanding of the geometric and functional inequalities, while expanding the applications of the (equivalent) Brunn-Minkowski inequality to branches of probability, analysis and PDEs. A main technical objective of this project is to relate these two seminal results, using optimal transport-based couplings, and derive refined versions of each inequality. Other aspects include deriving higher-order Buser inequalities relating the k-th eigenvalue of the Laplacian to the higher-order (multipartite) Cheeger isoperimetric constant, under additional assumptions on volume growth and/or discrete curvature. The project also involves several specific problems for investigation with postdoctoral fellows and students.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.
一个多世纪以来,连续域上的凸性一直被认为是一个强大的性质和重要的工具。由于在最优传输、偏微分方程、几何和泛函分析等主题上的几个独立的最新发展,凸性和曲率的相互作用进一步丰富了对概率和分析中的几个主题的理解。虽然凸性是通过离散环境中的子模性这一重要概念发展起来的,但曲率仍然难以捉摸。在过去的五年里,这位研究人员和不同的合作者在马尔可夫链和图中引入了所谓的Ricci曲率的离散概念,并得出了几个有趣的结果。然而,在离散空间的凸性和曲率方面,仍有许多需要理解和发展的地方。首先确定在数学和计算机科学中的适当应用将是发展任何新理论的重要指导原则。在离散概率中,AhlSwede-Daykin四函数定理有许多应用(通过FKG不等式和其他结果),类似地,Prekopa-Leindler不等式有助于加深对几何和函数不等式的理解,同时将(等价的)Brunn-Minkowski不等式的应用扩展到概率论、分析和偏微分方程的分支。该项目的一个主要技术目标是将这两个开创性的结果联系起来,使用最优的基于交通的耦合,并得出每个不平等的改进版本。其他方面包括在关于体积增长和/或离散曲率的附加假设下,推导出将拉普拉斯的第k个特征值与更高阶(多部分)Cheeger等周常数联系起来的高阶Buser不等式。该项目还涉及与博士后研究员和学生一起调查的几个具体问题。该奖项反映了NSF的法定使命,并通过使用基金会的智力优势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(2)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Volume Growth, Curvature, and Buser-Type Inequalities in Graphs
- DOI:10.1093/imrn/rnz305
- 发表时间:2018-02
- 期刊:
- 影响因子:1
- 作者:B. Benson;P. Ralli;P. Tetali
- 通讯作者:B. Benson;P. Ralli;P. Tetali
Finding cliques using few probes
使用少量探针发现派系
- DOI:10.1002/rsa.20896
- 发表时间:2019
- 期刊:
- 影响因子:1
- 作者:Feige, Uriel;Gamarnik, David;Neeman, Joe;Rácz, Miklós Z.;Tetali, Prasad
- 通讯作者:Tetali, Prasad
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Prasad Tetali其他文献
The Number of Linear Extensions of the Boolean Lattice
- DOI:
10.1023/b:orde.0000034596.50352.f7 - 发表时间:
2003-11-01 - 期刊:
- 影响因子:0.300
- 作者:
Graham R. Brightwell;Prasad Tetali - 通讯作者:
Prasad Tetali
Prasad Tetali的其他文献
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{{ truncateString('Prasad Tetali', 18)}}的其他基金
Conference: 2024 19th Annual Graduate Students Combinatorics Conference
会议:2024年第19届研究生组合学年会
- 批准号:
2334815 - 财政年份:2024
- 资助金额:
$ 19万 - 项目类别:
Standard Grant
New Approaches to Questions in Sampling, Counting, and Optimization
解决采样、计数和优化问题的新方法
- 批准号:
2151283 - 财政年份:2021
- 资助金额:
$ 19万 - 项目类别:
Standard Grant
New Approaches to Questions in Sampling, Counting, and Optimization
解决采样、计数和优化问题的新方法
- 批准号:
2055022 - 财政年份:2021
- 资助金额:
$ 19万 - 项目类别:
Standard Grant
Graph Structure, the Four Color Theorem, and Generalizations
图结构、四色定理和概括
- 批准号:
1700157 - 财政年份:2017
- 资助金额:
$ 19万 - 项目类别:
Continuing Grant
EAGER: Physical Flow and other Industrial Challenges
EAGER:物理流动和其他工业挑战
- 批准号:
1415496 - 财政年份:2014
- 资助金额:
$ 19万 - 项目类别:
Standard Grant
Displacement Convexity, Curvature and Concentration in Discrete Settings
离散设置中的位移凸度、曲率和浓度
- 批准号:
1407657 - 财政年份:2014
- 资助金额:
$ 19万 - 项目类别:
Continuing Grant
Random graph interpolation, Sumset inequalities and Submodular problems
随机图插值、和集不等式和子模问题
- 批准号:
1101447 - 财政年份:2011
- 资助金额:
$ 19万 - 项目类别:
Standard Grant
Extremal Problems in Combinatorics and Their Applications
组合学中的极值问题及其应用
- 批准号:
0901355 - 财政年份:2009
- 资助金额:
$ 19万 - 项目类别:
Standard Grant
Information Inequalities and Combinatorial Applications
信息不等式和组合应用
- 批准号:
0701043 - 财政年份:2007
- 资助金额:
$ 19万 - 项目类别:
Continuing Grant
Graph Homomorphisms, Stochastic Networks, Discrete Mass Transport
图同态、随机网络、离散质量传输
- 批准号:
0401239 - 财政年份:2004
- 资助金额:
$ 19万 - 项目类别:
Standard Grant
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