Desingularization and applications. Analysis on and Geometry of singular spaces
去奇异化和应用。
基本信息
- 批准号:RGPIN-2018-04445
- 负责人:
- 金额:$ 1.46万
- 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:2020
- 资助国家:加拿大
- 起止时间:2020-01-01 至 2021-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Singularities express irregularities of form in many branches of mathematics and its applications.
The important features of forms are often concentrated at singularities. Desingularization and
its applications are central to my work.
In the past years my research led me to discovery of fundamental links between algebraic, analytic
and geometric aspects of singularities: solutions of the long-standing problems posed by Whitney,
Thom and Hironaka; to an extension to a singular setting of the classical Bernstein-Markov and
Gagliardo-Nirenberg type inequalities of analysis; to sharp new bounds on the heat kernel via
desingularizing weights, to tangential Markov inequalities on singular varieties, to a Chow type
theorem for ideals and by means of the latter and desingularization to a simple construction of a
Poincare type metric off singularities.
Within the last 6 years I discovered a Bertini-type theorem crucial for my characterization of
Universal Stratifications satisfying Thom and Whitney-a conditions, established complexity bounds
for classical desingularization (turned out to be very high) and low complexity bounds (polynomial)
for Normalized Nash Desingularization in essential dimension 2 , proved Geometric Auslander
criteria for openness and for flatness of algebraic morphisms and also advanced my 15 years old
`geometric minimal models' program to a classification of `minimal singularities' in 4 and 3 variables
(i.e. a minimal list of singularities besides normal crossings with existence of desingularizations
isomorphic off these singularities, e.g. just the Whitney Umbrella for surfaces). Recently I also
established desingularization of the cotangent bundle of singular threefolds (and, in any dimension,
its equivalence to a desingularization of the induced metric) and extended my arcanalyticity and
Malgrange type division by analytic functions results to the quasi-analytic classes. I plan to work on natural extensions of these results.
The main objective of the proposed research is to find a closer link between desingularization and the
information that may be encoded in the geometry of and analysis on singular spaces. My longer term objective would be a reconstruction of the entire process of resolution of singularities in terms of geometry and/or analysis. I also would like to show that complex analytic stable leaves of a `rigid' Morse-Smale complexes of projective algebraic manifolds extend to algebraic subvarieties of the same dimension, and that function on a closed set which is separately a restriction of a semi-algebraic function and of a k-times
differentiable function is a restriction of a k-times differentiable semi-algebraic function, etc.
Finally, my attempts to clarify diverse problems in algebra, geometry and analysis proved to be a
fertile ground for raising excellent mathematicians from graduate students. My research plans
aim to repeat this experience.
在数学及其应用的许多分支中,奇点表示形式的不规则性。
形式的重要特征往往集中在奇点上。去单一化和
它的应用是我工作的核心。
在过去的几年里,我的研究使我发现了代数、分析
奇点的几何方面:惠特尼提出的长期问题的解决方案,
Thom和Hironaka;推广到经典Bernstein-Markov型和
分析的Gagliardo-Nirenberg型不等式;通过
将奇异簇上的切向马氏不等式的权重去单一化为Chow型
关于理想的一个定理,并借助于后者和去奇异化为一个简单的构造
Poincare类型度规的奇点。
在过去的6年里,我发现了一个贝尔蒂尼型定理,这对我刻画
满足Thom和Whitney-a条件的普适分层,建立了复杂性界
对于经典去奇异化(结果是非常高的)和低复杂性界(多项式)
对于本质维为2的归一化Nash去单值化,证明了几何Auslander
代数态射的开性和平坦性的判据,也比我15岁早
“几何极小模型”程序对4个和3个变量的“最小奇点”的分类
(即,除正常交叉点外存在去单角化的奇点的最小列表
与这些奇点同构,例如仅曲面的惠特尼伞)。最近我也
奇数三重余切丛的已建立的去单角化(并且,在任何维度中,
它等价于诱导度量的去奇异),并扩展了我的弧解析性和
用解析函数进行Malgrange型除法得到拟解析类。我计划致力于这些结果的自然延伸。
这项研究的主要目的是在去单语化和语言学习之间找到一个更紧密的联系
可以在奇异空间的几何和分析中编码的信息。我的长期目标是从几何和/或分析的角度重建奇点分解的整个过程。我还想证明,射影代数流形的“刚性”Morse-Smer复形的复解析稳定叶扩张到同维的代数子簇,以及闭集上的函数分别是半代数函数和k次函数的限制
可微函数是k次可微半代数函数的限制等。
最后,我试图澄清代数、几何和分析中的各种问题,结果证明是一种
从研究生中培养优秀数学家的沃土。我的研究计划
目的是重演这段经历。
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)
数据更新时间:{{ journalArticles.updateTime }}
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
数据更新时间:{{ journalArticles.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ monograph.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ sciAawards.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ conferencePapers.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ patent.updateTime }}
Milman, Pierre其他文献
Milman, Pierre的其他文献
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
{{ truncateString('Milman, Pierre', 18)}}的其他基金
Desingularization and applications. Analysis on and Geometry of singular spaces
去奇异化和应用。
- 批准号:
RGPIN-2018-04445 - 财政年份:2022
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
Desingularization and applications. Analysis on and Geometry of singular spaces
去奇异化和应用。
- 批准号:
RGPIN-2018-04445 - 财政年份:2021
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
Desingularization and applications. Analysis on and Geometry of singular spaces
去奇异化和应用。
- 批准号:
RGPIN-2018-04445 - 财政年份:2019
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
Desingularization and applications. Analysis on and Geometry of singular spaces
去奇异化和应用。
- 批准号:
RGPIN-2018-04445 - 财政年份:2018
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
Resolution of Singularities and its applications. Analysis on and geometry of singular spaces.
奇点的解决及其应用。
- 批准号:
8949-2013 - 财政年份:2017
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
Resolution of Singularities and its applications. Analysis on and geometry of singular spaces.
奇点的解决及其应用。
- 批准号:
8949-2013 - 财政年份:2016
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
Resolution of Singularities and its applications. Analysis on and geometry of singular spaces.
奇点的解决及其应用。
- 批准号:
8949-2013 - 财政年份:2015
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
Resolution of Singularities and its applications. Analysis on and geometry of singular spaces.
奇点的解决及其应用。
- 批准号:
8949-2013 - 财政年份:2014
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
Resolution of Singularities and its applications. Analysis on and geometry of singular spaces.
奇点的解决及其应用。
- 批准号:
8949-2013 - 财政年份:2013
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
Analysis on and geometry of singular spaces towards a geometric desingularization
奇异空间的分析和几何走向几何去奇异化
- 批准号:
8949-2008 - 财政年份:2012
- 资助金额:
$ 1.46万 - 项目类别:
Discovery Grants Program - Individual
相似海外基金
Biophilica - Analysis of bio-coatings as an alternative to PU-coatings for advanced product applications
Biophilica - 分析生物涂层作为先进产品应用的 PU 涂层的替代品
- 批准号:
10089592 - 财政年份:2024
- 资助金额:
$ 1.46万 - 项目类别:
Collaborative R&D
CAREER: Gauge-theoretic Floer invariants, C* algebras, and applications of analysis to topology
职业:规范理论 Floer 不变量、C* 代数以及拓扑分析应用
- 批准号:
2340465 - 财政年份:2024
- 资助金额:
$ 1.46万 - 项目类别:
Continuing Grant
REU Site: Graph Learning and Network Analysis: from Foundations to Applications (GraLNA)
REU 网站:图学习和网络分析:从基础到应用 (GraLNA)
- 批准号:
2349369 - 财政年份:2024
- 资助金额:
$ 1.46万 - 项目类别:
Standard Grant
Conference: Analysis on fractals and networks with applications, at Luminy
会议:分形和网络分析及其应用,在 Luminy 举行
- 批准号:
2334026 - 财政年份:2024
- 资助金额:
$ 1.46万 - 项目类别:
Standard Grant
ROBIN: Rotation-based Buckling Instability Analysis, and Applications to Creation of Novel Soft Mechanisms
ROBIN:基于旋转的屈曲不稳定性分析及其在新型软机构创建中的应用
- 批准号:
24K00847 - 财政年份:2024
- 资助金额:
$ 1.46万 - 项目类别:
Grant-in-Aid for Scientific Research (B)
Analysis of discrete dynamical systems described by max-plus equations and their applications
最大加方程描述的离散动力系统分析及其应用
- 批准号:
23K03238 - 财政年份:2023
- 资助金额:
$ 1.46万 - 项目类别:
Grant-in-Aid for Scientific Research (C)
Accelerating genomic analysis for time critical clinical applications
加速时间紧迫的临床应用的基因组分析
- 批准号:
10593480 - 财政年份:2023
- 资助金额:
$ 1.46万 - 项目类别:
CAREER: Temporal Network Analysis: Models, Algorithms, and Applications
职业:时态网络分析:模型、算法和应用
- 批准号:
2236789 - 财政年份:2023
- 资助金额:
$ 1.46万 - 项目类别:
Continuing Grant
Model Reduction Methods for Extended Quantum Systems: Analysis and Applications
扩展量子系统的模型简化方法:分析与应用
- 批准号:
2350325 - 财政年份:2023
- 资助金额:
$ 1.46万 - 项目类别:
Continuing Grant
Analysis of the paralinguistic production mechanism by Japanese learners and applications to pronunciation teaching
日语学习者副语言产生机制分析及其在发音教学中的应用
- 批准号:
22KF0429 - 财政年份:2023
- 资助金额:
$ 1.46万 - 项目类别:
Grant-in-Aid for JSPS Fellows