Enhancing Representation Theory, Noncommutative Algebra And Geometry Through Moduli, Stability And Deformations

整体远远大于其部分的总和：一个物体的集合展示了许多更深层次的结构，而不仅仅是通过研究其组成部分来理解。我们的视觉直觉赋予我们一个非常强大的工具来感知整体。这是几何学。尽管如此，我们最深刻的理解将其与秩序、精确度和计算结合起来。这是代数。当这两种观点融合在一起时，它们试图将小尺度和大尺度行为解释为一体。通常只有将这两种观点结合起来，才能对其中任何一种观点有最深刻的理解。在PI和COI以及其他人的引领下，过去二十年来，我们的提案的三个主题带来了一系列引人注目的进步：稳定性（在表示论和代数几何），非交换变形和增强（在非交换代数和代数几何中）和复体的模（Bridgeland稳定性，受到弦理论的启发）。每一个都在过去的二十年里取得了一些杰出的数学成就。但所有人都在达到他们独自所能达到的极限。为了采取下一步行动，解决紧迫的研究问题，需要将这些方法结合起来。这就是这个计划的目的。PI和coIs，再加上我们三个机构的数学专业知识以及我们招募的许多国内外数学家的专业合作，形成了一个鼓舞人心的团队，具有独特的专业知识和广度，跨越了代数和几何。我们之所以充满热情，是因为我们现在可以看到同样的结构在数学的不同部分中独立地、出于不同的原因出现，这表明存在着深层的隐藏联系。科学史上充满了这样的例子，例如数学和量子物理中对称性理论（群论）的发现。我们自己的工作带来了几个例子：墙交叉出现独立的表示理论和代数几何;使用非交换代数发现在同一时间在几何表示理论和最小模型计划。我们团队中的每个人都有将他们的技能以创造性和原创性的方式应用于超出我们自己专长的问题的特殊经验。因此，我们的动力不仅来自于我们期望在已知的未解问题上取得的进展，而且来自于我们尚无法预测的应用。我们相信，通过推动数学的最新发展，并通过接触其他学科，我们的建议将最大限度地发挥其潜力，并通过这一点，它将塑造和影响广泛的未来问题。

The whole is far greater than the sum of its parts: a collection of objects exhibits many deeper structures than can be understood by simply investigating its constituent pieces. Our visual intuition endows us with a remarkably powerful tool to perceive the whole. This is geometry. Nonetheless, our deepest understanding couples this with order, with precision, and with calculation. This is algebra. The two viewpoints, when fused together, seek to explain both small-scale and large-scale behaviour, together, as one. It is often only by combining both perspectives that the most insightful understanding of either can be achieved.Pioneered by the PI and coIs and others, the last two decades have seen a series of spectacular advances coming from our proposal's three main themes: stability (in representation theory and algebraic geometry), noncommutative deformations and enhancements (in noncommutative algebra and algebraic geometry), and moduli of complexes (Bridgeland stability, inspired by string theory). Each of these has individually resulted in some of the stand-out mathematical achievements of the last two decades. But all are reaching the limit of what they can achieve alone. To take the next step, and to solve the pressing research questions, requires bringing together these approaches. This is what this Programme Grant will achieve. The PI and coIs, together with the mathematical expertise at our three institutions and the specialist collaboration of many mathematicians nationally and internationally whom we have enlisted, form an inspiring team with a unique expertise and breadth that straddles much of algebra and geometry. We are enthusiastic because we can now see the same structures arising independently and for separate reasons across different parts of mathematics, which suggests the existence of deep hidden connections. The history of science is filled with such examples, such as the discovery of the theory of symmetries (group theory) in mathematics and in quantum physics. Our own work brings several examples: wall-crossing arising independently in representation theory and in algebraic geometry; the use of noncommutative algebra found at the same time in geometric representation theory and the minimal model programme. Everyone in our team has particular experience of applying their skills in creative and original ways to problems beyond our own specialism. We are therefore motivated not only by the progress that we expect to make on known unanswered questions, but also by the applications that we cannot yet predict. We believe that by pushing forward the mathematical state-of-the-art, and by reaching out to other disciplines, our proposal will maximise its potential, and through this it will shape and influence a broad range of future problems.