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（Rene Descartes）的非凡洞察力将任何方程与一个几何对象相关联，即方程的图形。图的几何性质编码方程的代数性质，反之亦然。在整个十八、十九和二十世纪，许多数学家致力于深化和扩展笛卡尔的思想。代数几何学就是这样发展起来的。这一学科蓬勃发展，取得了许多突出成果。在世纪中期--主要是受到量子理论和其他物理学的启发--出现了一种新的代数（算子代数）。 最近，这个新的代数已经与几何结合起来，形成了非交换几何的新学科。这个项目的数学采用了非对易几何的方法和结果，并将其应用于更古老、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吃惊的发现和结果。在约化p-adic群的表示论中，揭示了一种完全出乎意料的几何结构。这大大简化了表示论，并将鲍姆-康纳斯猜想（非对易几何中的一个猜想）与朗兰兹纲领联系起来。对于几何上产生的Fredholm算子的指数，非对易几何的观点导致了一个令人惊讶的结论，即像Atiyah-Singer指数公式这样的公式适用于椭圆算子之外。因此，椭圆性不是获得这种算子的指数的拓扑公式所需的要点。这个项目将探讨这一广泛的想法的许多相互作用。

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.