Mathematical Sciences: Weights of Semisimple Lie Algebras Arising from the Cohomology of Algebraic Varieties

该基金支持Y教授的工作。Zarhin工作在算术几何特别是问题的l-进上同调。 他将研究如何重量的一个L-进表示可以限制李代数表示所产生的伽罗瓦行动的阿贝尔品种。他还将研究特征值的弗罗贝纽斯行动阿贝尔品种在有限领域。 他还打算研究复杂的阿贝尔品种的排名霍奇集团是接近的层面。 这个项目福尔斯属于算术几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的主题。事实证明，这种结合非常富有成效--最近解决了几代人都无法解决的问题。 在它的许多后果是新的纠错码。 此类代码对于现代计算机（硬盘）和光盘都是必不可少的。

This grant supports the work of Professor Y. Zarhin to work in arithmetic geometry specifically problems in l-adic cohomology. He will study how the weights of an l-adic representation can restrict the Lie algebra representations arising from Galois actions on Abelian varieties. He will also study the eigenvalues of the Frobenius action on Abelian varieties over finite fields. He also intends to study complex Abelian varieties where the rank of the Hodge group is close to the dimension. This project falls into the general area of arithmetic geometry -a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.