Topics in Number Theory and Geometry: Zeta Functions and Circle Packings

Lagarias博士将研究数论和几何中的三个问题领域。第一个主题是研究lach zeta函数的推广及其在表示理论方面的解释。第二个主题是研究与自同构l -函数相关的整个函数和相关算子的Hilbert空间。第三个主题是探讨圆填料、共形几何和丢番图近似之间的类比。数论和几何的主题在几个领域相互作用，包括包装问题和数的几何。近年来，这些联系导致了数学、物理和材料科学之间的相互作用。数论也在通信、编码理论和密码学中提供了许多有用的应用。ζ函数编码数论不变量和几何不变量。该提案将研究一类这样的函数来探索这类连接。在数学物理中，ζ函数和算符之间也有联系，lagarias博士将从算符的角度研究这种联系。第三，圆填料最近得到了深入的研究，因为它们似乎在共形几何和数论之间提供了一个离散的桥梁。所考虑的问题可能会刺激研究人员之间的互动领域：共形几何，数学物理和数论。拉加里亚斯博士正在培养数论和几何两方面的研究生。

Dr. Lagarias will investigate three problem areas in number theory and geometry. The first topic concerns study of generalizations of the Lerch zeta function and their interpretation in representation theory terms. The second topic concerns the study of Hilbert spaces of entire functions and associated operators related to automorphic L-functions. The third topic concerns exploration of analogies between circle packings, conformal geometry and Diophantine approximation.The subjects of number theory and geometry interact in several areas, including packing problems and the geometry of numbers. In recent years these connections have led to interactions across mathematics, physics and materials sciences. Number theory has also provided many useful applications in communications, coding theory and cryptography. Zeta functions encode both number theoretic and geometric invariants. The proposal will study one class of such functions to explore connections of this kind. There have also been connections noted between zeta functions and operators in mathematical physics, and Dr.Lagarias will investigate connections of this kind from the operator side. Thirdly, circle packings have recently been intensively studied, as they seem to provide a discrete bridge between conformal geometry and number theory. The problems considered may stimulate interactions between researchers in the fields: conformal geometry,mathematical physics and number theory. Dr. Lagarias is training graduate students in both number theoretic and geometric topics.