Characters, Motives, and First-order Logic

研究者和他的同事们应用一种新的整合类型，称为动机整合，来研究p-adic群的表征及其特征。 动机整合是M. Kontsevich和随后由J. Denef，F. Loeser和其他人。 该积分的算术形式取虚周动机环中的值。 在p-adic群的表示论中自然出现的许多对象，包括表示的特征、轨道积分、Shalika芽和轨道的傅里叶变换，都有关于有限域上簇上的点的拓扑描述，或者更一般地说，作为Frobenius算子在动机上的迹。 这一建议的研究将利用动机积分来确认这些对象中的许多对象都具有约束型的几何描述。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一个统一的描述。 直到最近，这个梦想还不可能实现，甚至还不能给这个词赋予精确的含义。 然而，通过结合数学的三个不同分支-代数，几何和逻辑-现在似乎有可能实现这个梦想。 这个研究领域依赖于数理逻辑的方法，对以前通过纯分析考虑的问题给出几何解释。 具体地说，数理逻辑给出了积分和测度的几何解释（称为动机整合）。 这项资助支持的研究将使用这种新工具来统一描述现代代数中的一些基本对象，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.