Symplectic Representation Theory

这一建议将应用几何学中强大的工具和技术来解决表示论中的某些问题，表示论是代数的一个主要分支，与几何和数学物理有很强的相互作用。纯数学的目的是抽象和提炼熟悉概念的本质：例如，在对称性的情况下，这导致了组的定义，即给定对象的对称性的集合。然而，数学上的定义要公理得多，以至于小组所描述的底层对象几乎消失了。在这些情况下，重要的是尝试恢复这个对象，或者更具体地说，找到其对称性导致问题群的所有对象。这就是表象理论背后的激励思想。尽管这个看似抽象的问题，但表示理论在许多科学领域都是至关重要的，例如物理(如弦理论/镜像对称性)、化学(研究分子振动)和计算机科学，以及数学的核心。代数和几何从现代数学的概念本身就是志同道合的，想法和激励问题一直在来回传递。例如，连续群是李理论和现代表示理论的基石，由于索菲斯·李将代数应用于几何研究和分类的计划，连续群变得突出起来。自从李氏理论的概念提出以来，几何学一次又一次地在推动这一学科向前发展方面发挥了关键作用。相反，交换和同调代数在代数几何的现代发展中起着关键作用，使格罗森迪克等巨人能够在坚实的数学基础上重建这门学科。在这个跨学科的建议中，我们的目标是再次利用代数研究中的强大几何结果，这一次是通过发展圆锥辛流形上的混合Hodge结构理论的基础，从而将混合Hodge结构理论应用于表示理论中的一系列(似乎难以解决的)问题。我们还期望这一理论的发展也将在理解圆锥辛流形的几何方面有无数的应用。发展变形量子化(DQ)模上的混合Hodge结构理论的第一个关键步骤是将D模的邻近和消失函子的构造推广到这一背景下。其次，我们将使用这些函子将感兴趣的范畴重构为粘合在更简单的子商上的范畴。在这种情况下，我们还提出了一种与Soerel的V-函子类似的几何形式，使我们能够将Rouquier的拟遗传覆盖理论应用于DQ-模。

This proposal will apply powerful tools and techniques in geometry to solve certain problems in representation theory, a major branch of algebra interacting strongly with geometry and mathematical physics. Pure mathematics aims to abstract and distil the essence of familiar concepts: for instance, in the case of symmetries this leads to the definition of a group, the collection of symmetries of a given object. However, the mathematical definition is far more axiomatic, to the point that the underlying object that the group is describing all but disappears. In these cases it is important to try to recover this object, or more specifically to find all objects whose symmetries give rise to the group in question. This is the motivating idea behind representation theory. Despite this seemingly abstract problem, representation theory is crucially important in many areas of science such as physics (e.g. string theory / mirror symmetry), chemistry (study of molecular vibrations) and computer science, as well as being central for mathematics.Algebra and geometry have been kindred spirits from the very conception of modern mathematics, with ideas and motivating problems passing to and fro all the time. For instance, continuous groups, the bedrock of Lie theory and modern representation theory, came to prominence thanks to Sophus Lie's program applying algebra to the study and classification of geometries. Since the conception of Lie theory, geometry has played a key role, time and again, in moving the subject forward. Conversely, commutative and homological algebra has been pivotal in the modern development of algebraic geometry, enabling the giants, such as Grothendieck, to rebuild the subject on firm mathematical foundations. In this intradisciplinary proposal we aim once again to exploit powerful geometric results in the study of algebra, this time by developing the foundations of a theory of mixed Hodge structures on conic symplectic manifolds, thereby bringing the theory of mixed Hodge structures to bear on a host of (seemingly intractable) problems in representation theory. We also expect that the development of such a theory would also have myriad applications to the understanding of the geometry of conic symplectic manifolds. The first key step to develop this theory of mixed Hodge structures on Deformatio-Quantization (DQ)-modules, is to generalise the construction of nearby and vanishing functors for D-modules to this setting. Secondly, we will use these functors to reconstruct the categories of interest as categories glued out of simpler subquotients. We also propose to develop a geometric analogue of Soergel's V-functor in this setting, allowing us to apply Rouquier's theory of quasi-hereditary covers to DQ-modules.