Characters, Motives, and First-order Logic

研究者和他的同事们将一种称为动机整合的新型整合应用于p-adic群体及其特征表征的研究中。动机整合在1995年由M. Kontsevich提出，随后由J. Denef、F. Loeser等人发展。这个积分的算术形式取虚周动机环中的值。在p进群的表示理论中自然出现的许多对象，包括表示的特征、轨道积分、Shalika细菌和轨道的傅立叶变换，都可以用有限域上的变异点来推测描述，或者更一般地作为动机上的Frobenius算子的迹。本提案的研究将利用动机整合来确认许多这些物体具有推测类型的几何描述。多年来，数学家们一直梦想着对代数中的一些基本研究对象有一个统一的描述。直到最近，还不可能实现这个梦，甚至不可能给出它的确切含义。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。这个领域的研究依赖于数学逻辑的方法，对以前通过纯分析来考虑的问题给出几何解释。具体地说，数学逻辑给出了积分和测度的几何解释（称为动机积分）。该基金支持的研究将使用这个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对称。

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.