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

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.