Geometric Topology and Manifolds
Geometric Topology and Manifolds
批准号:
1615056
负责人:
James Davis
金额:
$22.37万
依托单位:
依托单位国家:
美国
项目类别:
Standard Grant
财政年份:
2016
资助国家:
美国
项目状态:
已结题
起止时间:
2016-09-01 至 2021-08-31
中文摘要
奖项:DMS 1615056,首席研究员:James F. davis理论数学的主要分支包括拓扑,代数,几何,分析和组合。这个项目将这些不同的线索编织在一起。主要的焦点是流形的研究,流形是在欧几里德空间上局部建模的点的集合。二维流形的样例由平面、环面和球面给出。然而,流形存在于所有维度中,对它们的考虑导致了各种各样的例子,并使用了各种各样的工具。这个项目的重点是流形的分析简单,但拓扑复杂。上面提到的不同领域之间的联系意味着人们可以将拓扑用作预言器,在理论数学的其他领域产生有趣的问题、示例和研究。流形理论与数学的大多数领域,以及物理现象,如宇宙学、弦理论、经典力学和量子力学联系在一起。首席研究员提出了六个项目。第一个是l2 -无环流形的本体。这与低维拓扑(结调和)、代数(Hilberts第17问题和函数场的Witt群)、高维拓扑(外科理论的一种新形式)和分析(可服从群和L2-betti数)有关。第二部分给出了拓扑等变刚性的系统方法,使用来自外科理论、分层空间、代数k和l理论以及法雷尔-琼斯猜想的工具。三是研究和应用基于定向拟阵的组合特征类。第四个是尼尔森实现问题,由于法雷尔-琼斯猜想和对其基本群具有无限挠性的流形的理解,现在可能取得进展。第五部分是代数l论的基础工作,特别是n变量整系数多项式环的l论计算和群的自由积群环的l论计算。六是研究由代数几何产生的布里斯科恩流形,进而用代数几何来研究这些流形。这些问题都是相互关联的;基本的主题是流形的分类,他们的束,和他们的对称性。
英文摘要
Award: DMS 1615056, Principal Investigator: James F. DavisMajor branches of theoretical mathematics include topology, algebra, geometry, analysis, and combinatorics. This project weaves these various threads together. The major focus is the study of manifolds, which are sets of points locally modeled on Euclidean space. Sample manifolds in dimension 2 are given by a plane, the surface of a torus, and the surface of a sphere. However manifolds exist in all dimensions, and their consideration leads to a rich variety of examples and uses a rich variety of tools. This project focuses on manifolds which are analytically simple but topologically complex. The connection between the disparate fields mentioned above means that one can use topology as a oracle, producing interesting questions, examples, and research in other areas of theoretical mathematics. Manifold theory connects with most areas of mathematics, as well as physical phenomena such as cosmology, string theory, and classical and quantum mechanics.The principal investigator proposes six projects. The first is the bordism of L2-acyclic manifolds. This connects with low-dimensional topology (knot concordance), with algebra (Hilberts 17th problem and the Witt group of function fields), with high-dimensional topology (a new form of surgery theory), and analysis (amenable groups and L2-betti numbers). The second gives a systematic approach to topological equivariant rigidity, using tools from surgery theory, stratified spaces, algebraic K-and L-theory, and the Farrell-Jones Conjecture. The third is to study and apply combinatorial characteristic classes based on oriented matroids. The fourth is the Nielsen Realization question, where progress in now possible due to the Farrell-Jones conjecture and the understanding of manifolds whose fundamental groups are infinite with torsion. The fifth is foundational work in algebraic L-theory, in particular the computation of the L-theory of the polynomial ring in n variables with integral coefficients and the group ring of a free product of groups. The sixth is to study Brieskorn manifolds produced by algebraic geometry, and, in turn, to use algebraic geometry to study these manifolds. These problems are all interrelated with each other; the basic theme being classification of manifolds, their bundles, and their symmetries.
期刊论文(2)
专著(0)
科研奖励(0)
会议论文
DOI:
10.4310/jdg/1463404119
发表时间:
2015-01
期刊:
arXiv: Differential Geometry
影响因子:
--
作者:
[James F. Davis;F. Fang]
通讯作者:
James F. Davis;F. Fang
Any finite group acts freely and homologically trivially on a product of spheres
任何有限群都可以自由且同调地作用于球体的乘积
DOI:
10.1090/proc/12435
发表时间:
2016
期刊:
Proceedings of the American Mathematical Society
影响因子:
1
作者:
[Davis, James F.]
通讯作者:
Davis, James F.
Workshops on Smart Manufacturing with Open and Scaled Data Sharing in Semiconductor and Microelectronics Manufacturing; Virtual and In-Person; Washington, DC; October/November 2023
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批准号:2334590
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项目类别:Standard Grant
-
资助金额:$10.0万
-
财政年份:2023
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负责人:James Davis
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依托单位:
MICA: Stomasense: A New Route to the Proactive Detection and Management of Leaks within Ostomy Pouches
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批准号:MR/W029561/1
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项目类别:Research Grant
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资助金额:$34.28万
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财政年份:2023
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负责人:James Davis
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依托单位:
Collaborative Research: SaTC: CORE: Small: Improving Sanitization and Avoiding Denial of Service Through Correct and Safe Regexes
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批准号:2135156
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项目类别:Standard Grant
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资助金额:$27.4万
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财政年份:2022
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负责人:James Davis
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依托单位:
Symposium on the Strategy for Resilient Manufacturing Ecosystems through AI
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批准号:2132067
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项目类别:Standard Grant
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资助金额:$8.94万
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财政年份:2021
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负责人:James Davis
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依托单位:
CAS: Collaborative Research: Boronium Ionic Liquids - Impact of Structure on Chemistry, Electrochemical Stability, Ion Dynamics, and Charge Transport
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批准号:2102978
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项目类别:Standard Grant
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资助金额:$51.02万
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财政年份:2021
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负责人:James Davis
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依托单位:
Workshop: Aligning AI and U.S. Advanced Manufacturing Competitiveness
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批准号:2049670
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项目类别:Standard Grant
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资助金额:$9.83万
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财政年份:2020
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负责人:James Davis
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依托单位:
Finite Fields and their Applications at Simon Fraser University
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批准号:1905024
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项目类别:Standard Grant
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资助金额:$2.5万
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财政年份:2019
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负责人:James Davis
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依托单位:
Topology of Manifolds: Interactions between High and Low Dimensions
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批准号:1850620
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项目类别:Standard Grant
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资助金额:$3.0万
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财政年份:2019
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负责人:James Davis
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依托单位:
Ionic and Molecular Materials of High Thermal Stability: Design, Structure, and Function
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批准号:1800122
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项目类别:Standard Grant
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资助金额:$33.0万
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财政年份:2018
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负责人:James Davis
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依托单位:
Summer School on Surgery and the Classification of Manifolds
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批准号:1638464
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项目类别:Standard Grant
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资助金额:$2.14万
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财政年份:2016
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负责人:James Davis
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依托单位:
Workshop to Expand the SMLC Roadmap and Action Plan on Integrated Sensor, Control and Platform Modeling for Smart and Digital Manufacturing
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批准号:1523237
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项目类别:Standard Grant
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资助金额:$4.0万
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财政年份:2015
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负责人:James Davis
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依托单位:
Ionic Liquids of Improved Thermal Stability
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批准号:1464740
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项目类别:Standard Grant
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资助金额:$28.5万
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财政年份:2015
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负责人:James Davis
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依托单位:
I-Corps: Sensors For Relightable Images
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批准号:1449695
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项目类别:Standard Grant
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资助金额:$5.0万
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财政年份:2014
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负责人:James Davis
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依托单位:
Smart Manufacturing Advancement
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批准号:1321470
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项目类别:Standard Grant
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资助金额:$3.0万
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财政年份:2013
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负责人:James Davis
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依托单位:
Collaborative Research: The impact of time-dependent mantle rheology and 3-D structure on models and observations of Glacial Isostatic Adjustment
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批准号:1315254
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项目类别:Continuing Grant
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资助金额:$23.46万
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财政年份:2013
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负责人:James Davis
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依托单位:
Geometric and Algebraic Topology
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批准号:1210991
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项目类别:Standard Grant
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资助金额:$18.36万
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财政年份:2012
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负责人:James Davis
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依托单位:
EAGER: Bridging Campus Resources via GENI and OpenFlow
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批准号:1247357
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项目类别:Standard Grant
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资助金额:$29.68万
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财政年份:2012
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负责人:James Davis
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依托单位:
STEMing the Tide: Exploring Factors Related to Minority Males Interest, Engagement and Achievement in Mathematics and Science
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批准号:1240757
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项目类别:Standard Grant
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资助金额:$20.0万
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财政年份:2012
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负责人:James Davis
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依托单位:
Collaborative Research: Dynamics of Crust-Mantle Coupling through Combined Analysis and Modeling of EarthScope Seismic, Geodetic, and Geologic Data
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批准号:1053292
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项目类别:Continuing Grant
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资助金额:$21.84万
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财政年份:2011
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负责人:James Davis
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依托单位:
Smart Composites for Minimising Bacterial Biofilm Formation
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批准号:EP/I01179X/1
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项目类别:Research Grant
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资助金额:$24.21万
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财政年份:2011
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负责人:James Davis
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依托单位:
海外基金