Analytic and geometric aspects of convexity theory with applications

凸性理论的解析和几何方面及其应用

基本信息

  • 批准号:
    RGPIN-2018-05159
  • 负责人:
  • 金额:
    $ 1.68万
  • 依托单位:
  • 依托单位国家:
    加拿大
  • 项目类别:
    Discovery Grants Program - Individual
  • 财政年份:
    2022
  • 资助国家:
    加拿大
  • 起止时间:
    2022-01-01 至 2023-12-31
  • 项目状态:
    已结题

项目摘要

The omnipresent convexity appears naturally when describing objects of interest in many mathematical related sciences. For instance, the set of quantum states in finite dimensional quantum systems and its subset containing all separable quantum states (i.e., not entangled quantum states) are convex compact sets. Hence, understanding the analytic and/or geometric aspects of convexity theory is in great demand and is prerequisite to understanding convex objects of interest. To this end, my proposed program of research aims to study properties of convexity theory, and to apply tools from convexity theory to attack problems arising in other areas such as mathematical physics, partial differential equations, probability theory and (quantum) information theory. One part of the proposed program of research deals with the modern geometric extensions of the Brunn-Minkowski theory and its dual. The emphasis is on understanding the properties of affine invariants (e.g., affine and geominimal surface areas), establishing new affine isoperimetric and isocapacitary inequalities, and solving Minkowski type problems (e.g., the Orlicz-Minkowski problem as well as its dual and/or polar analogues). Several projects are proposed to further explore the connections of the Brunn-Minkowski theory of convex bodies and its dual with partial differential equations, with particular attention paid to geometric inequalities, (polar or dual) Minkowski type problems, and the development of a dual Brunn-Minkowski theory for various variational functionals. Another part of the proposed program of research lies in the areas of geometrization of log-concave measures (or functions) and the information theory. The geometrization of log-concave measures can be viewed as the functional analogue of the Brunn-Minkowski theory. I aim to build a framework of the functional Lp and/or Orlicz Brunn-Minkowski theories for log-concave or quasi-concave functions, extend the entropy power inequality to their Lp and/or Orlicz analogues, and discover new geometric inequalities for quantum states. It is expected that these projects help further advance the connections between information theory and the Brunn-Minkowski theory, with particular attention paid to geometric inequalities for quantum states, and the generalizations of the entropy power inequality and Fisher information (in both classical and quantum settings). I will continue my commitment to the training of (undergraduate and graduate) students and postdocs. This program of research includes multiple diverse and interdisciplinary research topics, which makes it easier to attract Highly Qualified Personnel (HQP) and helps produce knowledgeable mathematicians of next generation.

项目成果

期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)

暂无数据

数据更新时间:2024-06-01

Ye, Deping其他文献

On the Bures volume of separable quantum states
Phase transitions for random states and a semicircle law for the partial transpose
  • DOI:
    10.1103/physreva.85.030302
    10.1103/physreva.85.030302
  • 发表时间:
    2012-03-12
    2012-03-12
  • 期刊:
  • 影响因子:
    2.9
  • 作者:
    Aubrun, Guillaume;Szarek, Stanislaw J.;Ye, Deping
    Aubrun, Guillaume;Szarek, Stanislaw J.;Ye, Deping
  • 通讯作者:
    Ye, Deping
    Ye, Deping
共 2 条
  • 1
前往

Ye, Deping的其他基金

Analytic and geometric aspects of convexity theory with applications
凸性理论的解析和几何方面及其应用
  • 批准号:
    RGPIN-2018-05159
    RGPIN-2018-05159
  • 财政年份:
    2021
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
Analytic and geometric aspects of convexity theory with applications
凸性理论的解析和几何方面及其应用
  • 批准号:
    RGPIN-2018-05159
    RGPIN-2018-05159
  • 财政年份:
    2020
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
Analytic and geometric aspects of convexity theory with applications
凸性理论的解析和几何方面及其应用
  • 批准号:
    RGPIN-2018-05159
    RGPIN-2018-05159
  • 财政年份:
    2019
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
Analytic and geometric aspects of convexity theory with applications
凸性理论的解析和几何方面及其应用
  • 批准号:
    RGPIN-2018-05159
    RGPIN-2018-05159
  • 财政年份:
    2018
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
"Convex Geometric Analysis, Random Matrices, and Their Applications to Quantum Information Theory"
“凸几何分析、随机矩阵及其在量子信息论中的应用”
  • 批准号:
    418296-2012
    418296-2012
  • 财政年份:
    2017
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
"Convex Geometric Analysis, Random Matrices, and Their Applications to Quantum Information Theory"
“凸几何分析、随机矩阵及其在量子信息论中的应用”
  • 批准号:
    418296-2012
    418296-2012
  • 财政年份:
    2016
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
"Convex Geometric Analysis, Random Matrices, and Their Applications to Quantum Information Theory"
“凸几何分析、随机矩阵及其在量子信息论中的应用”
  • 批准号:
    418296-2012
    418296-2012
  • 财政年份:
    2015
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
"Convex Geometric Analysis, Random Matrices, and Their Applications to Quantum Information Theory"
“凸几何分析、随机矩阵及其在量子信息论中的应用”
  • 批准号:
    418296-2012
    418296-2012
  • 财政年份:
    2014
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
"Convex Geometric Analysis, Random Matrices, and Their Applications to Quantum Information Theory"
“凸几何分析、随机矩阵及其在量子信息论中的应用”
  • 批准号:
    418296-2012
    418296-2012
  • 财政年份:
    2013
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
"Convex Geometric Analysis, Random Matrices, and Their Applications to Quantum Information Theory"
“凸几何分析、随机矩阵及其在量子信息论中的应用”
  • 批准号:
    418296-2012
    418296-2012
  • 财政年份:
    2012
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual

相似国自然基金

强杂波下雷达弱小目标检测的矩阵信息几何方法
  • 批准号:
    62371458
  • 批准年份:
    2023
  • 资助金额:
    50 万元
  • 项目类别:
    面上项目
基于离散几何模型的高质量非结构曲面网格生成方法研究
  • 批准号:
    12301489
  • 批准年份:
    2023
  • 资助金额:
    30 万元
  • 项目类别:
    青年科学基金项目
拓扑棱态的微观几何性质及其在非线性光响应中的特征
  • 批准号:
    12374164
  • 批准年份:
    2023
  • 资助金额:
    52 万元
  • 项目类别:
    面上项目
流固复合膜的几何非线性弹性
  • 批准号:
    12374210
  • 批准年份:
    2023
  • 资助金额:
    53 万元
  • 项目类别:
    面上项目
离心叶轮冷热态双重不确定性几何变形的流动机理及鲁棒设计方法
  • 批准号:
    52376030
  • 批准年份:
    2023
  • 资助金额:
    51 万元
  • 项目类别:
    面上项目

相似海外基金

Geometric analysis of partial differential equations and inverse problems
偏微分方程和反问题的几何分析
  • 批准号:
    22K03381
    22K03381
  • 财政年份:
    2022
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
    Grant-in-Aid for Scientific Research (C)
Analytic and geometric aspects of convexity theory with applications
凸性理论的解析和几何方面及其应用
  • 批准号:
    RGPIN-2018-05159
    RGPIN-2018-05159
  • 财政年份:
    2021
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
Analytic, Geometric, and Probabilistic Aspects of High-Dimensional Phenomena
高维现象的分析、几何和概率方面
  • 批准号:
    1955175
    1955175
  • 财政年份:
    2020
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Standard Grant
    Standard Grant
Analytic and geometric aspects of convexity theory with applications
凸性理论的解析和几何方面及其应用
  • 批准号:
    RGPIN-2018-05159
    RGPIN-2018-05159
  • 财政年份:
    2020
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Discovery Grants Program - Individual
    Discovery Grants Program - Individual
Variational problems and geometric analysis for hypersurfaces with singular points, and novel development of discrete surface theory
奇点超曲面的变分问题和几何分析以及离散曲面理论的新发展
  • 批准号:
    20H01801
    20H01801
  • 财政年份:
    2020
  • 资助金额:
    $ 1.68万
    $ 1.68万
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
    Grant-in-Aid for Scientific Research (B)