Reductions & resolutions in representation theory and algebraic geometry

量子力学是20世纪科学的主要内容，它使人们认识到，物理量是由非对易代数支配的。更准确地说，Werner Heisenberg用矩阵力学取代了经典力学，在经典力学中，可观测量成对交换，而在矩阵力学中，位置和动量等关键可观测变量不再相互交换。因此，要研究量子力学，自然也会尝试将点、线、面等的经典几何推广到非对易世界。这就产生了非对易几何的数学领域。后来，数学家赫尔曼·韦尔认识到，对应于位置和动量的运算符满足另一个数学领域--表象理论--中出现的关系，该领域研究抽象数学对象的“对称性”。在这个项目中，我们通过观察它们的对称性来分析出现在(非交换)几何中的几个空间，并使用表示论来描述一些关于它们的新东西。其基本思想可追溯到Alexander Grothendieck，其基本思想是将一个代数不变量与一个可能的非对易空间联系起来，该代数不变量丰富到足以捕捉空间的许多几何信息，同时又足够灵活，将焦点从几何转移到更代数的观点。为了给出我们在这个项目中考虑的一个问题的至少一个具体例子，考虑Markoff方程，一个由x^2+y^2+z^2=3xyz给出的丢番图方程，它是由Markoff在1880年引入的，当时他正在研究通过积分二次形式得到的极小值。马尔科夫证明了这个方程的所有解都可以通过一个简单的归纳过程得到。一个明显的唯一性问题是由Frobenius在1913年提出的：给定一个三元组(a，b，c)，其中c为最大值，满足方程，c唯一地确定这个三元组吗？就像数论中经常出现的情况一样，初等问题可以在数学的不同领域产生深刻的理论，乍一看与问题无关。本课题的目的之一是研究射影平面的非对易对称群的表示理论与Markoff方程的解之间的联系。

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.