Special Functions and Transcedence

研究者提出了算术几何和超越数论中的几个问题，探讨了解析函数的特殊值的算术性质。在一个项目中，研究者计划研究由Anderson-Drinfeld动机引起的正特征数的超越性和代数独立性。这些超越问题与Frobenius差分方程的伽罗瓦群密切相关，特别是研究者提出追求超越理论与伽罗瓦理论之间的相互作用，以研究函数场的多重zeta值，周期和对数的Drinfeld模块，以及高维版本。研究了模形式的傅里叶系数及其与有限域超几何函数和Calabi-Yau流形的关系。他还计划发现椭圆曲线扩展群的算术性质以及它们与伽罗瓦表示的联系。作为现代数学的基本分支之一，数论是许多应用的基础，包括密码学和编码理论。提出的研究考虑了涉及数论的算术和分析方面之间相互作用的问题，这些问题始于欧拉和高斯的经典工作，并且今天继续对数论和许多其他数学分支中问题的基本理解产生重要影响。这个项目的几个部分自然会给研究生和本科生的研究带来问题。

The investigator proposes several problems in arithmeticgeometry and transcendental number theory that explore the arithmeticnature of special values of analytic functions. In one project, theinvestigator plans to study transcendence and algebraic independence ofnumbers in positive characteristic arising from Anderson-Drinfeld motives. These transcendence questions are closely connected to Galois groups of Frobenius difference equations, and in particular the investigatorproposes to pursue the interaction between transcendence theory and Galoistheory to study function field multiple zeta values, periods and logarithms of Drinfeld modules, and higher dimensional versions. The investigator proposes to study Fourier coefficients of modular forms and their relationships with finite field hypergeometric functions and Calabi-Yau manifolds. He plans also to discover arithmetic properties of the extension groups of elliptic curves and their connections with Galois representations.One of the fundamental branches of modern mathematics, number theoryserves as the basis for many applications, including cryptography andcoding theory. The proposed research considers questions involving theinterplay between arithmetic and analytic aspects of number theory, which have their beginnings in classical work of Euler and Gauss and which continue to have important implications today to the essential understanding of problems both in number theory and in many other branches of mathematics. Several parts of the project lead naturally to problems for graduate and undergraduate research.