Links between Algebraic Geometry and Complex Analysis
代数几何与复分析之间的联系
基本信息
- 批准号:EP/J002062/1
- 负责人:
- 金额:$ 88.39万
- 依托单位:
- 依托单位国家:英国
- 项目类别:Fellowship
- 财政年份:2012
- 资助国家:英国
- 起止时间:2012 至 无数据
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
An old idea, going back at least as far as Newton and probably much further, shows how it is possible to start with an polynomial equation that you wish to solve and end up with a certain polygon that captures important information of the original equation. Newton exploited this idea in his work on finding numerical solutions to polynomial equations, thereby allowing him to perform computations many centuries before any computers had been invented.One of the pieces of research in this proposal concerns a modern incarnation of this idea in the framework of algebraic geometry. Whereas Newton was considering a single polynomial equation, we now know how this works for several such equations simultaneously. An idea of Okounkov in the early 1980s showed how one can construct a certain solid in Euclidean space that similar to the Newton polygon but this time to associated an algebraic variety, and discovered that this shape captures some of the geometry of the original variety. One of the aims here is to study the geometry of this Okounkov body and to develop it as a tool connection algebraic and complex analysis.A second area of research in this proposal concerns a study of what is known as the Kahler-Einstein equations. These are some important differential equations whose solution should be thought of as giving the "best" shape of a space under consideration. These equations are analogous to the Einstein equations in general relativity, and have applications in various parts of pure mathematics and mathematical physics. One problem, however, is that the Kahler-Einstein equations are too complicated to be solved directly. In fact in many cases even knowing if there is a solution is beyond our current knowledge. However a deep and fascinating idea due to Yau-Tian-Donaldson states that it should be possible to detect the whether such a solution exists within algebraic geometry. In this proposal we aim to explore this circle of ideas, and to extend it to other frameworks and other kinds of differential equations.
一种古老的想法,至少可以追溯到牛顿,甚至更早,展示了如何从你想要求解的多项式方程开始,并以某个多边形来结束,该多边形捕捉了原始方程的重要信息。牛顿在寻找多项式方程的数值解的工作中利用了这一思想,从而使他能够在计算机发明之前的许多世纪进行计算。这项提议中的一项研究涉及在代数几何框架下这一思想的现代化身。虽然牛顿考虑的是一个多项式方程,但我们现在知道这是如何同时适用于几个这样的方程的。20世纪80年代初,奥孔科夫的一个想法展示了人们如何在欧几里得空间中构建类似于牛顿多边形的特定实体,但这一次是与代数变体相关联的,并发现这种形状捕捉了原始变体的一些几何形状。这里的目的之一是研究奥库科夫天体的几何学,并将其发展为一种连接代数和复数分析的工具。该提案中的第二个研究领域涉及对卡勒-爱因斯坦方程的研究。这些都是一些重要的微分方程,它们的解应该被认为是给出了所考虑的空间的“最佳”形状。这些方程类似于广义相对论中的爱因斯坦方程,在纯数学和数学物理的各个部分都有应用。然而,有一个问题是,卡勒-爱因斯坦方程太复杂了,无法直接求解。事实上,在许多情况下,即使知道是否有解决方案也超出了我们目前的知识范围。然而,Yau-Tian-Donaldson提出的一个深刻而有趣的想法是,应该有可能检测到在代数几何中是否存在这样的解。在这项提议中,我们的目标是探索这一思想圈,并将其扩展到其他框架和其他类型的微分方程。
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)
Semi-continuity of Stability for Sheaves and Variation of Gieseker Moduli Spaces
滑轮稳定性的半连续性和 Gieseker 模空间的变分
- DOI:
- 发表时间:2016
- 期刊:
- 影响因子:0
- 作者:Daniel Greb;Julius Ross;Matei Toma
- 通讯作者:Matei Toma
On cscK resolutions of conically singular cscK varieties
关于圆锥奇异 cscK 簇的 cscK 分辨率
- DOI:10.1016/j.jfa.2016.04.025
- 发表时间:2016
- 期刊:
- 影响因子:1.7
- 作者:Arezzo C
- 通讯作者:Arezzo C
Moduli of polarised manifolds via canonical Kähler metrics
通过规范 Kühler 度量的极化流形模
- DOI:10.48550/arxiv.1810.02576
- 发表时间:2018
- 期刊:
- 影响因子:0
- 作者:Dervan
- 通讯作者:Dervan
Hermitian Yang-Mills connections on blowups
赫米蒂安·杨-米尔斯在爆炸事件中的联系
- DOI:10.48550/arxiv.1707.07638
- 发表时间:2017
- 期刊:
- 影响因子:0
- 作者:Dervan
- 通讯作者:Dervan
Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds
- DOI:10.4310/pamq.2017.v13.n3.a5
- 发表时间:2017-04
- 期刊:
- 影响因子:0
- 作者:Cristiano Spotti;Song Sun
- 通讯作者:Cristiano Spotti;Song Sun
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Julius Ross其他文献
Interpolation, Prekopa and Brunn-Minkowski for emF/em-subharmonicity
插值、普雷科帕和布伦-闵可夫斯基在 emF/次调和性方面
- DOI:
10.1016/j.aim.2023.109405 - 发表时间:
2024-01-01 - 期刊:
- 影响因子:1.500
- 作者:
Julius Ross;David Witt Nyström - 通讯作者:
David Witt Nyström
Julius Ross的其他文献
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{{ truncateString('Julius Ross', 18)}}的其他基金
CAREER: Stability, Kahler Geometry, and the Hele-Shaw Flow
职业:稳定性、卡勒几何和赫勒肖流
- 批准号:
1749447 - 财政年份:2018
- 资助金额:
$ 88.39万 - 项目类别:
Continuing Grant
Analytic Methods in Complex Algebraic Geometry
复杂代数几何中的解析方法
- 批准号:
1707661 - 财政年份:2017
- 资助金额:
$ 88.39万 - 项目类别:
Standard Grant
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