Hyperkaehler Geometry with Applications

Hyperkaehler 几何及其应用

• 批准号: EP/G027110/1
• 负责人: Balazs Szendroi
• 金额: $ 50.99万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2009
• 资助国家: 英国
• 起止时间: 2009 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG027110%2F1
• 关键词: Hyperkaehler; Geometry; Applications

What is common in (1) the existence of a magnet with a single pole (2) the reliability of computer networks and (3) code theory and code breaking? The proposed research provides an answer: these scientific problems can all be attacked using quaternionic geometry. Quaternions are four dimensional analogues of complex numbers. For problem (1) one can study magnetic monopoles using quaternionic equations. The possible existence of these and similar elementary particles could lead to new energy sources. For (2) the proposed research shows that the number of holes on a certain quaternionic surface attached to a graph agrees with the reliability polynomial of a computer network based on the graph. Qualitative properties of this reliability polynomial, obtained from the study of the geometry of quaternionic surfaces, help explain how to make computer networks, like the internet, more reliable. In (3) arithmetic study of certain quaternionic surfaces sheds light on the representation theory of finite groups of Lie type, which are used in various schemes in code theory. Information emerging from the geometry of these quaternionic surfaces, could help devise better codes. In short, the proposed research is two-folded, first it studies fundamental problems in quaternionic geometry, and second it breaths life into these investigations by applying the results to other fields in mathematics and physics. This yields a colourful palette of various fields in mathematics and physics all related in one way or another to quaternionic geometry.This proposal therefore aims to understand the global analysis, geometry, topology and arithmetic of complete hyperkaehler manifolds of non-compact type and find exciting applications in other fields of mathematics and physics, where these manifolds naturally appear. The proposed research has two main aspects: studying fundamental questions for non-compact hyperkaehler manifolds, such as Hodge theory and the Atiyah-Singer index theorem, and applying these methods in other fields. The hyperkaehler spaces appearing in this proposal include: moduli spaces of Yang-Mills instantons on asymptotically locally Euclidean gravitational instantons; more generally Nakajima's quiver varieties; toric hyperkaehler varieties; moduli spaces of magnetic monopoles on R^3; moduli spaces of Higgs bundles on a Riemann surface; and more generally hyperkaehler spaces appearing in the non-Abelian Hodge theory of a curve (like moduli of flat GL(n,C)-connections and character varieties) and in the Geometric Langlands Program. The fields of applications include: combinatorics, representation theory, finite group theory, low dimensional topology, number theory, mathematical physics and string theory.

在（1）单极磁铁的存在（2）计算机网络的可靠性和（3）代码理论和代码破译中有什么共同点？这项研究提供了一个答案：这些科学问题都可以用四元数几何来解决。四元数是复数的四维类似物。对于问题（1），人们可以使用四元数方程来研究磁单极子。这些和类似的基本粒子的可能存在可能导致新的能源。对于（2），本文的研究表明，某个四元数曲面上的孔洞数与基于该图的计算机网络的可靠性多项式一致。这个可靠性多项式的定性性质，从四元数曲面几何的研究中获得，有助于解释如何使计算机网络，如互联网，更可靠。在（3）中，对某些四元数曲面的算术研究揭示了李型有限群的表示理论，这些表示理论用于码理论的各种方案中。从这些四元曲面的几何形状中产生的信息可以帮助设计更好的代码。简而言之，拟议的研究是双重的，首先它研究四元数几何中的基本问题，其次它通过将结果应用于数学和物理的其他领域来为这些研究注入活力。这产生了一个丰富多彩的调色板的各种领域的数学和物理都以这样或那样的方式与四元数几何。因此，这个建议的目的是了解整体分析，几何，拓扑和算术的完整的超凯勒流形的非紧型，并找到令人兴奋的应用在其他领域的数学和物理，这些流形自然出现。本文的研究主要有两个方面：一是研究非紧超kaehler流形的基本问题，如Hodge理论和Atiyah-Singer指标定理，二是将这些方法应用于其他领域。出现在这个建议中的超kaehler空间包括：渐近局部欧氏引力瞬子上的杨-米尔斯瞬子的模空间;更一般的Nakajima的超模簇;复曲面超kaehler簇; R^3上磁单极子的模空间;黎曼曲面上希格斯丛的模空间;更一般地，超凯勒空间出现在曲线的非阿贝尔霍奇理论（如平坦GL（n，C）-连接和特征变种的模）和几何朗兰兹计划中。应用领域包括：组合数学、表示论、有限群论、低维拓扑、数论、数学物理和弦理论。

项目成果

Motivic Donaldson--Thomas invariants of some quantized threefolds

Motivic Donaldson——一些量子化三重的托马斯不变量

• 期刊: Journal of Noncommutative Geometry
• 影响因子: 0.9
• 作者: Cazzaniga,A

Hilbert Schemes as Moduli of Higgs Bundles and Local Systems

作为希格斯丛和局部系统模的希尔伯特方案

• DOI: 10.1093/imrn/rnt167
• 发表时间: 2014
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Groechenig M

Exchange between perverse and weight filtration for the Hilbert schemes of points of two surfaces

两个曲面点的希尔伯特格式的反常过滤和权重过滤之间的交换

• DOI: 10.5427/jsing.2013.7c
• 发表时间: 2013
• 期刊: Journal of Singularities
• 影响因子: 0.4
• 作者: De Cataldo M

Moduli of flat connections in positive characteristic

• DOI: 10.4310/mrl.2016.v23.n4.a3
• 发表时间: 2012-01
• 期刊: arXiv: Algebraic Geometry
• 作者: M. Groechenig

The classical limit of the Heisenberg and time-dependent Hartree-Fock equations: the Wick symbol of the solution

海森堡和瞬态 Hartree-Fock 方程的经典极限：解的 Wick 符号

• DOI: 10.4310/mrl.2013.v20.n1.a11
• 发表时间: 2013
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: Amour L