Mathematical Sciences: Research in Number Theory and Automophic Forms

在这项研究中，Harris将继续研究全纯和非全纯自同构形的算术以及与L函数的特殊值的关系。它将集中讨论Deligne关于代数簇的上同调上同调的L函数在已知这些L函数是可处理的唯一情况下的特殊值的猜想：即它们何时可以实现为自同构的L函数。自同构函数是复杂的解析函数，它编码了数论信息，并允许人们利用强大的分析机器来处理数论问题。在这种情况下，应用领域是算术代数几何，即与多项式方程组的解相关的数论领域。然后，这就成为了一个广泛的研究领域，它的工具涵盖了大多数数学领域。哈里斯是这方面的大师，这项研究将产生许多令人兴奋的数学结果。

In this research Harris will continue to study the arithmetic of holomorphic and non-holomorphic automorphic forms and relations with special values of L-functions. It will concentrate on the conjectures of Deligne on the special values of L-functions attached to the cohomology of algebraic varieties in the only cases in which these L-functions are known to be tractable: namely, when they can be realized as automorphic L- functions. Automorphic functions are complex analytic functions that encode number theoretic information and allow one to bring the powerful machinery of analysis to bear on the number theory problems. In this case the area of application is in arithmetic algebraic geometry, namely in the area of number theory connected with the solutions of systems of polynomial equations. This then becomes a sweeping area of study encompassing most areas of mathematics in its tools. Harris is a master of this and much exciting mathematics will result from this research.