Minimal Representations and Functoriality

摘要：SavinGordan Savin继续他在最小表示方面的研究，对一些其他方法无法得到的Langlands泛函的情况进行了应用和显式构造。自同构形式是分析、代数和数论的交叉学科。这是一门植根于不可或缺的数学理论的学科，如傅立叶分析，它在许多方面影响着我们的日常生活。如果没有傅里叶分析，就不可能有电信、数据传输和现代放射学仪器。本研究的目的是利用分析（微积分）的工具来回答数论中的一些问题，例如更好地理解多项式的零。几乎每个人都知道如何解二次方程。虽然不太为人所知，但也有可能找到任何3次或4次多项式的零。一般来说，高次多项式是不能解的，但可以用一些特殊的复变量函数（称为模形式）来研究。这个思想圈，令人惊讶地把乍一看似乎没有关联的数学理论联系起来，大约有30年了，被称为朗兰兹纲领，以普林斯顿高等科学研究所的罗伯特·p·朗兰兹的名字命名。

Abstract: SavinGordan Savin is continuing his work on minimal representations, with applications to and explicit constructions of some cases of Langlands functoriality which can not be obtained by any other method. The subject of automorphic forms is a crossroad of analysis, algebra, and number theory. It is a subject with roots in indispensable mathematical theories such as Fourier Analysis which impact our everyday lives in many ways. Telecommunications, data transmission, and modern instruments of radiology would not be possible without Fourier Analysis. A purpose of this research is to use tools of analysis (calculus) to answer some questions in number theory, such as to get a better understanding of zeroes of polynomials. Almost everybody knows how to solve a quadratic equation. It is also possible, although it is less known, to find zeroes of any polynomial of degree 3 or 4. A higher degree polynomial, in general, cannot be solved, but can be studied using some special functions of complex variable (called modular forms). This circle of ideas, which surprisingly relates mathematical theories which at first glance do not appear related, is about 30 years old, and called the Langlands Program, after Robert P. Langlands of the Institute of Advanced Sciences in Princeton.