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

整体远远大于其各部分的总和：一组对象展示了许多更深层次的结构，而不是简单地研究其组成部分就能理解的。我们的视觉直觉赋予了我们感知整体的非常强大的工具。这是几何学。尽管如此，我们最深刻的理解将这与秩序、精确度和计算结合在一起。这就是代数。当这两种观点融合在一起时，试图同时解释小规模和大规模行为，作为一个整体。通常只有将这两个视角结合起来，才能实现对其中任何一个的最有洞察力的理解。在PI和Cois等人的开创下，过去20年见证了一系列壮观的进展，来自我们提案的三个主要主题：稳定性(在表示理论和代数几何中)，非对易变形和增强(在非对易代数和代数几何中)，以及复数的模(Bridgeland稳定性，受弦理论启发)。这其中的每一项都单独导致了过去20年来一些杰出的数学成就。但所有人都达到了单独完成的极限。要采取下一步行动，并解决紧迫的研究问题，需要将这些方法结合起来。这就是这项计划赠款将实现的目标。PI和COI，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，组成了一个具有独特专业知识和广度的鼓舞人心的团队，横跨大部分代数和几何。我们之所以热情，是因为我们现在可以看到相同的结构在数学的不同部分以不同的原因独立出现，这表明存在着深层次的隐藏联系。科学史上充斥着这样的例子，例如在数学和量子物理中对称理论(群论)的发现。我们自己的工作列举了几个例子：在表示理论和代数几何中独立出现的跨越墙；在几何表示理论和最小模型程序中同时发现的非对易代数的使用。我们团队中的每个人都有独特的经验，能够以创造性和原创性的方式应用他们的技能来解决超出我们专业范围的问题。因此，我们的动机不仅是我们期望在已知的未回答的问题上取得的进展，而且也是我们还无法预测的应用。我们相信，通过推进最先进的数学，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，并将通过这一点塑造和影响未来的广泛问题。

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.