Applications of Non-Commutative Geometry

现代数学的中心主题之一是几何与代数的融合。这始于17世纪，勒内·笛卡尔（Rene Descartes）的非凡见解，他将任何方程与几何物体联系起来；也就是方程的图形。图形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。代数几何这门学科就这样发展起来了。这门学科蓬勃发展，取得了许多突出的成果。在20世纪中期——很大程度上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交换几何的方法和结果，并将它们应用于更古老、更传统的数学分支中的问题。非交换几何的观点导致了一些惊人的猜想和结果。在可约p进群的表示理论中，揭示了一个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（非交换几何中的一个猜想）与Langlands程序联系起来。对于几何生成的Fredholm算子的指数，非交换几何的观点得出了令人惊讶的结论，即像Atiyah-Singer指数公式这样的公式远远适用于椭圆算子。因此，椭圆性并不是得到这类算子索引的拓扑公式所必需的要点。这个项目将探索这一系列广泛的思想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.