Symplectic Representation Theory

这一建议将应用强大的工具和几何技术来解决某些问题的表示论，一个主要的分支代数互动强烈的几何和数学物理。纯粹数学的目的是抽象和抽象熟悉概念的本质：例如，在对称性的情况下，这导致了一个群的定义，一个给定对象的对称性的集合。然而，数学定义更加不言自明，以至于群体所描述的基本对象几乎消失了。在这些情况下，重要的是试图恢复这个对象，或者更具体地说，找到所有的对象，其对称性产生了所讨论的群。这就是表征理论背后的动机思想。尽管这个问题看起来很抽象，但表示论在许多科学领域都是至关重要的，比如物理学（例如弦论/镜像对称）、化学（分子振动的研究）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开始就有着相似的精神，思想和激励问题一直在来回传递。例如，连续群，李群理论和现代表示论的基石，由于Sophus Lie的程序将代数应用于几何的研究和分类而变得突出。自从李学的概念提出以来，几何学一次又一次地在推动这门学科向前发展方面发挥了关键作用。相反，交换代数和同调代数在代数几何的现代发展中起着关键作用，使格罗滕迪克等巨人能够在坚实的数学基础上重建这一学科。在这个跨学科的建议，我们的目标再次利用强大的几何结果在代数的研究，这一次通过发展的基础理论的混合霍奇结构的锥辛流形，从而使理论的混合霍奇结构承担的主机（看似棘手的）问题表示论。我们还期望，这样一个理论的发展也将有无数的应用程序的几何形状的圆锥辛流形的理解。发展变形量子化（DQ）模上的混合霍奇结构理论的第一个关键步骤是将D-模的邻近函子和消失函子的构造推广到这种设置。其次，我们将使用这些函子来重建感兴趣的类别，作为从更简单的子项中胶合出来的类别。我们还建议开发一个几何模拟Soergel的V-函子在这种情况下，使我们能够应用Rouquier的理论，拟遗传覆盖DQ-模块。

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.