Characters, Motives, and First-order Logic

这位研究人员和他的同事们将一种新的整合类型，称为Motivic整合，应用于研究p-ady群及其特征的表示。Motivic整合于1995年由M.Kontsevich提出，随后由J.Denef、F.Loeser等人发展起来。这个积分的算术版本取值于一系列虚拟的周动机。在p-Addi群表示理论中自然出现的许多对象，包括轨道的表示、轨道积分、Shalika芽和傅立叶变换的特征，都有关于有限域上簇上的点的猜想描述，或者更广泛地说，作为关于动机的Frobenius算子的迹。多年来，数学家们一直梦想着代数中的一些基本研究对象应该有统一的描述。直到最近，人们还不可能实现这个梦想，甚至不可能给这些话赋予确切的含义。然而，通过将三个不同的数学分支--代数、几何和逻辑--结合起来，现在似乎有可能实现这个梦想。这一研究领域依赖于数理逻辑的方法，对以前通过纯分析所考虑的问题给出几何解释。具体地说，数理逻辑给出了积分学和测量的几何解释(称为动机积分)。由这笔拨款支持的研究将使用这一新工具对现代代数中的一些基本对象进行统一描述，包括通过群表示及其特征的数学来描述对称性。

The investigator and his colleagues apply a new type of integration, called motivic integration, to the study of representations of p-adic groups and their characters. Motivic integration was introduced in 1995 by M. Kontsevich and developed subsequently by J. Denef, F. Loeser, and others. The arithmetic version of this integral takes values in a ring of virtual Chow motives. Many objects that occur naturally in the representation theory of p-adic groups, including characters of representations, orbital integrals, Shalika germs, and Fourier transforms of orbits have conjectural descriptions in terms of points on varieties over finite fields, or more generally as the trace of Frobenius operators on motives. The research of this proposal will make use motivic integration to affirm that many of these objects have geometric descriptions of the conjectured type.For many years, mathematicians have dreamed that some ofthe fundamental objects of study in algebra should have a uniform description. Until recently, it was not possible to carry out this dream, or even to give precise meaning tothe words. However, by combining three different branches of mathematics -- algebra, geometry, and logic -- it now seems possible to bring this dream to fruition. This field of research relies on methods of mathematical logic to give a geometric interpretation of what was previously considered through pure analysis. Specifically, mathematical logic gives a geometric interpretation (called motivic integration) of integral calculus and measure. The research supported by this grant will use this new tool to give a uniform description of some of the fundamental objects in modern algebra, including symmetry through the mathematics of group representations and their characters.