Zeta Values and Infinite Dimensional Representations

Bloch教授将研究Zeta函数与无限维李代数和函数群的表示之间的关系。他将专注于与Zeta和L级数的值的关系，拟模性和相关的模性质，推广了已知的一元特征标的模性质，以及与镜像对称的关系。研究人员还将尝试利用代数Chern Simons特征标类发现具有平坦连通的向量丛的Riemann Roch定理。这是表示理论和几何中的一个项目。在数学中，调查会产生一个无限的数字序列，代表被研究对象的基本信息，这在数学上是很常见的。这样的序列可以看起来相当随机地形成，即使它实际上完全由它被识别的方式决定。众所周知，某些数学函数，如zeta和L级数，包含有关此类序列中数的分布的重要信息。Zeta和L级数的重要例子往往与抽象的数学结构、代数和群的具体表示有关。不幸的是，数学家既不能完全理解数列和这些数列之间的联系，也不能完全理解数列和表示法之间的联系。如果这两种联系都正规化，数学就会有一种强大的新发现方法。这个项目将调查第二组这些联系。

Professor Bloch will study relations between zeta functions and representations of infinite dimensional Lie algebras and function groups. He will focus on relations with values of zeta and L-series, quasi-modularity and related modular properties extending known modular properties for characters in one variable and relations with mirror symmetries. The investigator will also attempt to discover a Riemann Roch theorem for vector bundles with flat connections, using algebraic Chern Simons character classes.This is a project in both representation theory and geometry. It is a common occurrence in mathematics that a investigation will produce an infinite sequence of numbers representing basic information about the objects under study. Such a sequence can appear to form quite randomly, even though it is actually completely determined by the way it was identified. Certain mathematical functions, zeta and L-series, have long been known to contain important information about the distribution of numbers in such sequences. Important examples of zeta and L-series are often connected to the concrete representations of the abstract mathematical constructions, algebras and groups. Unfortunately, mathematicians do not fully understand either the connection between number sequences and these series or the connection between the series and the representations. If both of these connections were regularized, mathematics would have a powerful new method of discovery. This project will investigate the second set of these connections.