Reductions & resolutions in representation theory and algebraic geometry

量子力学是20世纪科学的重要组成部分，它使人们认识到物理量是由非交换代数控制的。更准确地说，维尔纳·海森堡用矩阵力学取代了经典力学，在经典力学中，可观测量是成对交换的，在矩阵力学中，关键的可观测量，如位置和动量，不再相互交换。因此，为了研究量子力学，自然也要尝试将点、线、面等经典几何扩展到非交换世界。这就产生了非交换几何的数学领域。后来，数学家赫尔曼·魏尔（Hermann Weyl）意识到，与位置和动量相对应的运算符满足了另一个数学领域的关系，即研究抽象数学对象的“对称性”的表征理论。在这个项目中，我们通过观察它们的对称性来分析（非交换）几何中出现的几个空间，并使用表征理论来阐述一些关于它们的新东西。其基本思想，可以追溯到Alexander Grothendieck，将一个可能的非交换空间与一个代数不变量联系起来，这个代数不变量足够丰富，可以捕捉到空间的很多几何特征，同时又足够灵活，将焦点从几何转移到更代数的观点。为了给出至少一个我们在这个项目中考虑的问题的具体例子，考虑马尔科夫方程，一个由x^2 +y^2 +z^2 = 3xyz给出的丢芬图方程，它是由马尔科夫在1880年研究积分二次型的最小值时引入的。马尔科夫证明了这个方程的所有解都可以从一个简单的归纳过程中得到。1913年，Frobenius提出了一个明显的唯一性问题：给定一个三重体（a, b, c），其中c是最大值，满足方程，c是否唯一地决定了这个三重体？就像数论中经常出现的情况一样，初等问题可以引发数学各个领域的深刻理论，这些理论乍一看与问题无关。本课题的目标之一是研究射影平面非交换对称群的表示理论与马尔科夫方程解之间的联系。

Quantum mechanics is a staple of 20th century science, and has led to the realisation that physical quantities are governed by noncommutative algebra. More precisely, Werner Heisenberg replaced classical mechanics, in which observable quantities commute pairwise, with matrix mechanics, where crucial observables like position and momentum no longer commute with each other. To study quantum mechanics, it is therefore natural to also try and extend the classical geometry of points, lines, planes etc. to the noncommutative world. This gives rise to the mathematical field of noncommutative geometry.Later on, the mathematician Hermann Weyl realised that the operators corresponding to position and momentum satisfied relations that occurred in another area of mathematics called representation theory, which studies the "symmetries" of abstract mathematical objects. In this project we analyse several spaces appearing in (noncommutative) geometry by looking at their symmetries, and use representation theory to say something new about them. The fundamental idea, which goes back to Alexander Grothendieck, is to associate to a possibly noncommutative space an algebraic invariant which is rich enough to capture a lot of the geometry of the space while at the same time being sufficiently flexible, moving the focus from geometry to a more algebraic point of view.To give at least one concrete example of a problem we consider in this project, consider the Markoff equation, a diophantine equation given by x^2 +y^2 +z^2 = 3xyz,which was introduced by Markoff back in 1880 while investigating minimal values taken up by integral quadratic forms. Markoff showed that all solutions to this equation could be obtained from a simple inductive process. An obvious unicity question was formulated by Frobenius in 1913: given a triple (a, b, c), with c as largest value, satisfying the equation, does c uniquely determine this triple? As is often the case in number theory, elementary questions can give rise to deep theories in diverse areas of mathematics, at first glance unrelated to the problem. One of the objectives in the current project is to investigate a connection between the representation theory of the noncommutative symmetry group of the projective plane and the solutions of Markoff's equation.