Applications of Non-Commutative Geometry

现代数学的中心主题之一是几何和代数的融合。这始于17世纪勒内·笛卡尔的非凡洞察力，他认为任何方程式都有一个几何对象，即方程式的图形。图的几何属性编码方程的代数属性，反之亦然。在整个十八世纪、十九世纪和二十世纪，许多数学家致力于深化和扩展笛卡尔的思想。于是，代数几何学科应运而生。这一学科蓬勃发展，取得了许多突出成果。在二十世纪中叶--主要受到量子理论和其他物理学的启发--一种新的代数(算子代数)出现了。最近，这种新的代数与几何相结合，形成了非对易几何的新学科。这个项目的数学采用非对易几何的方法和结果，并将它们应用于较老的、更传统的数学分支中的问题。非对易几何的观点导致了一些令人震惊的猜想和结果。在约化群的表示理论中，揭示了一种完全意想不到的几何结构。这极大地简化了表示理论，并将Baum-Connes猜想(这是非对易几何中的一个猜想)与朗兰兹程序联系起来。对于几何生成的Fredholm型算子的指数，非对易几何观点导致了一个令人惊讶的结论：像Atiyah-Singer指数公式这样的公式远远超出了椭圆算子的适用范围。因此，椭圆性不是获得这类算子的指数的拓扑公式所需要的本质点。这个项目将探索这一广泛的想法之间的许多相互作用。

One of the central themes of modern mathematics has been the fusion of geometry and algebra. This began in the seventeenth century with Rene Descartes's remarkable insight that associated to any equation a geometric object; namely, the graph of the equation. Geometric properties of the graph encode algebraic properties of the equation, and vice versa. Throughout the eighteenth, nineteenth, and twentieth centuries, many mathematicians worked to deepen and extend Descartes's ideas. Thus the subject of algebraic geometry was developed. This subject has flourished and has achieved many outstanding results. In the mid-twentieth century--largely inspired by quantum theory and other physics--a new kind of algebra (operator algebras) emerged. More recently, this new algebra has been combined with geometry to form the new subject of noncommutative geometry. The mathematics of this project takes methods and results from noncommutative geometry and applies them to problems in the older, more traditional branches of mathematics.The noncommutative geometry point of view has led to some startling conjectures and results. In the representation theory of reductive p-adic groups a totally unexpected geometric structure has been revealed. This greatly simplifies the representation theory and links the Baum-Connes conjecture (which is a conjecture within noncommutative geometry) to the Langlands program. For the index of geometrically-arising Fredholm operators, the noncommutative geometry point of view leads to the surprising conclusion that formulas like the Atiyah-Singer index formula apply well beyond elliptic operators. Hence ellipticity is not the essential point needed to obtain a topological formula for the index of such operators. This project will explore the many interactions of this wide-ranging set of ideas.