Hyperkaehler Geometry with Applications

(1)单极磁铁的存在(2)计算机网络的可靠性和(3)密码理论和密码破译的共同之处是什么？这项拟议的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)，人们可以用四元数方程来研究磁单极子。这些基本粒子和类似基本粒子的可能存在可能会带来新的能源。对于(2)，所提出的研究表明，附于图的某个四元数曲面上的孔数符合基于该图的计算机网络的可靠性多项式。这个可靠性多项式的定性性质来自于对四元数曲面几何的研究，它有助于解释如何使计算机网络像互联网一样更可靠。在(3)中，对某些四元数曲面的算术研究揭示了有限李型群的表示理论，这些有限群被用于码论中的各种方案。从这些四元数表面的几何形状中涌现的信息，可以帮助设计出更好的代码。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到数学和物理的其他领域来为这些研究注入活力。这就产生了一个五颜六色的数学和物理领域的调色板，所有这些领域都以这样或那样的方式与四元数几何有关。因此，这个提议旨在理解非紧致类型的完备的Hyperkaehler流形的整体分析、几何、拓扑和算术，并在数学和物理的其他领域找到令人兴奋的应用，这些流形自然地出现在这些流形上。本文的研究主要包括两个方面：研究非紧超Kaehler流形的基本问题，如Hodge理论和Atiyah-Singer指数定理，并将这些方法应用于其他领域。在这个建议中出现的超Kaehler空间包括：渐近局部欧氏引力瞬子上的Yang-Mills瞬子的模空间；更一般的Nakajima箭簇；Toric Hyperkaehler簇；R^3上磁单极子的模空间；Riemann曲面上的Higgs丛的模空间；以及更一般的出现在曲线的非阿贝尔Hodge理论(如平坦的GL(n，C)-连接和特征簇的模)和几何朗兰兹程序中的Hyperkaehler空间。其应用领域包括：组合学、表示论、有限群论、低维拓扑、数论、数学物理和弦论。

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.